Lectures on Cremona transformations, Torino, September 19-24, 2005

Alessandro Verra September 18, 2005

ii

0.1 Lecture 1

0.1.1 Generalities

A Cremona transformation is a birational automorphism f : P^{r} → P^{r} of the complex projective space and the Cremona group is the group Bir(P^{r}) of birational automorphisms of P^{r}. The name of Cremona transformations goes back to Luigi Cremona which was a pioneer in the systematic study of Bir(P^{r}) and obtained fundamental results. A Cremona transformation f defines an integral subscheme

Γ_{f} ⊂ P^{r} × P^{r}  (1)

which is the graph of the birational map f. Let π_{1}, π_{2} : Γ_{f} → P^{r} be the natural projection maps onto P^{r}, as a birational map f is defined by the linear system of hypersurfaces

Φ := π_{1∗}π_{2}^{∗} | O_{P^{r}}(1) |,  (2)

while f^{−1} is defined by the linear system

Φ′ := π_{2∗}π_{1}^{∗} | O_{P^{r}}(1) | .  (3)

Following the tradition we will say sometimes that an element of Φ is a Φ-hypersurface and that an element of Φ′ is a Φ′-hypersurface.

Definition 0.1.1. A (m, n) Cremona transformation is a Cremona transformation f : P^{r} → P^{r} such that Φ ⊂| O_{P^{r}}(m) | and Φ′ ⊂| O_{P^{r}}(n) |.

Definition 0.1.2. A homaloidal linear system is an r-dimensional linear system

Φ ⊂| O_{P^{r}}(m) |

which is free of fixed components and defines a birational map f : P^{r} → P^{r}.

The next proposition is an elementary remark of special interest.

Proposition 0.1.1. Let Φ be a homaloidal linear system and let D ∈ Φ be an integral hypersurface then D is rational.

Proof. Let f : P^{r} → P^{r} the Cremona transformation defined by Φ, we have D = f^{∗}H for some hyperplane H ⊂ P^{r}. Let y ∈ H be a general point then the scheme f^{∗}(y) is 0-dimensional of lenghth one: otherwise H would be in the indeterminacy locus of f^{−1}, which is impossible. This implies that f/D : D → H is a birational map. Hence D is rational.

On the other hand the base locus of Φ is a subscheme

B_{f} ⊂ P^{3}.  (4)

We have Φ = PV, where V ⊂ H^{0}(O_{P^{r}}(m)) is a subspace of dimension r+1. Let F_{0}, …, F_{r} be a basis of V, then B_{f} is defined as a scheme by the homogeneous ideal generated by F_{0} … F_{r}. We can start from Γ_{f} to introduce some general properties of Cremona transformations. Often, when no confusion arises, we will use the notations Γ and B respectively for Γ_{f} and B_{f}. Let

I

be the ideal sheaf of B, recall that the blowing up of P^{r} along the scheme B is by definition

P roj (S)

where S is the sheaf of O_{P^{r}}-algebras ⊕_{n≥0}I^{n} and I^{0} := O_{P^{r}}. Just another way of saying this is the following: the blowing up of I is the Proj scheme associated to the Spec scheme Spec S of the sheaf of algebras S. By construction P roj(S) is a scheme over P^{r} and hence it is endowed with an epimorphism

σ : P roj (S) → P^{r}

which is the structural map of the blowing map. It turns out that

σ^{∗}I ≅ O_{P roj(S)}(E)

where E is a Cartier divisor, known as the exceptional divisor of σ. Moreover it holds h^{0}(O_{P roj(S)}(E)) = 1 and E is the pull-back of the scheme B by σ.

Theorem 0.1.2. The blowing up of P^{r} along the ideal sheaf I of B is naturally embedded in P^{r} × P^{r} as the graph Γ of f. Moreover π_{1}/Γ is the structural map of such a blowing up.

Proof. (See Fulton’s book p.82). Let V ⊂ H^{0}(I(d)) be the subspace whose projectivization is Φ, then the natural evaluation map

e : V ⊗ O_{P^{r}}(−d) → I

is an epimorphism. Moreover e induces an epimorphism of O_{P^{r}}-Algebras

Sym(e) : Sym(V ⊗ O_{P^{r}}(−d)) → ⊕_{n≥0} I^{n}.

This induces an embedding

ϕ : P roj (⊕_{n≥0} I^{n}) → P roj (Sym(V ⊗ I(−d)).

Observe that

P roj (Sym(V ⊗ I(−d)) = P^{r} × P(V^{∗})

hence ϕ is a natural embedding of the blowing of B in P^{r} × P(V^{∗}). ϕ is a morphism of schemes over P^{r}: they are respectively endowed with the structure map σ : P(⊕_{n≥0}I^{n}) → P^{r} and with the projection onto π_{1}P^{r} ×PV^{∗} → P^{r}. This implies that π_{1} · ϕ = σ. Hence the first projection of

Γ := ϕ(P roj(⊕_{n≥0}(I^{n})) ⊂ P^{r} × PV^{∗}

is σ. On the other hand the rational map

π_{2} · ϕ : P^{r} → P(V^{∗})

is the map induced by the evaluation O_{P^{r}} ⊗ V → _{P^{r}}(m). Hence it is the rational map defined by Φ that is f. Then the image of the embedding ϕ is the graph Γ and the statement follows.

It is well known that the blowing up of a smooth variety along a smooth subscheme is smooth. The next corollary is therefore obvious.

Corollary 0.1.3. Let B be the indeterminacy scheme of a Cremona transformation f. If B is smooth then the graph Γ of f is smooth.

If B is smooth then the standard sequence

0 → T_{Γ} → T_{P^{r}×P^{r}} ⊗ O_{Γ} → N_{Γ} → 0

is exact and we have the equality of Chern polynomials

c(T_{P^{r}×P^{r}} ⊗ O_{Γ}) = c(T_{Γ})c(N_{Γ}).

As usual this implies some stronger restrictions to the existence of Γ and hence of f. In particular it is interesting the following

Problem 0.1.1. For which Cremona transformations the indeterminacy scheme B is smooth and irreducible?

We will discuss this problem later, for the moment we want to see some easy examples where the blowing up Γ of B is not smooth.

Example 0.1.1. Let (x_{0} : x_{1} : x_{2}) × (y_{0} : y_{1} : y_{2}) be projective coordinates on P^{2} × P^{2} and let f : P^{2} → P^{2} be the rational map defined by the equations

y_{1} = x_{1}^{2} , y_{2} = x_{2}x_{0} , y_{0} = x_{1}x_{0}.

The base locus B is supported on the points o = (1 : 0 : 0) and v = (0 : 1 : 0). The ideal I, defining B as a scheme, is generated by x_{1}^{2}, x_{2}x_{0}, x_{1}x_{0}. In particular I is the intersection of the ideals (x_{1}, x_{2}) and (x_{0}, x_{1}^{2}), so that that B has multiplicity 1 at o and 2 at v. It is easy to check that the graph Γ is defined by the equations

y_{1}x_{0} − y_{0}x_{1} = y_{2}x_{1} − y_{0}x_{2} = 0.

On the affine open set U_{o} = {x_{0}y_{0} ≠ 0} Γ is defined as

Y_{1} − X_{1} = Y_{2}X_{1} − X_{2} = 0,  (5)

where X_{i} = x_{i}/x_{0} and Y_{i} = y_{i}/y_{0}. Hence, on the open set U_{o}, π_{1} : Γ → P^{2} is the blowing up of o. On U_{v} = {x_{2}y_{2} ≠ 0} the graph Γ is defined as

Y_{1}X_{0} − Y_{0}X_{1} = X_{1} − Y_{0} = 0.

In other words U_{v} ∩Γ is the surface Y_{1}X_{0} −X_{1}^{2} in affine coordinates (X_{0}, X_{1}, Y_{1}). Hence, on the open set U_{v}, π_{1} : Γ → P^{2} is the blowing up of the scheme defined by (x_{0}, x_{1}^{2}), wich is supported with multiplicity two on v. Note that Γ is singular at the point (0 : 1 : 0)×(0 : 1 : 0) as follows from equation (5). To obtain Γ one has to blow up o and then a point on the exceptional line E_{o} over o. The strict transform of E_{o} is a −2-curve and its contraction is a copy of Γ.

Example 0.1.2. It is easy to see that the converse of corollary 0.1.3 is not true. We may have very well that Γ is smooth and that B is not smooth. The simplest example is probably realized by the homaloidal net of plane cubics passing through 5 general points o, p_{1}, p_{2}, p_{3}.p_{4}: with multiplicity ≥ 2 at o and ≥ 1 at p_{i}. Prove that Γ is a smooth surface of degree 8 contained in a hyperplane section of the Segre embedding of P^{2} × P^{2}.

Remark 0.1.1. More in general it is appropriate to remark the following fact. Let

σ : X → Y

be the blowing of the scheme B ⊂ Y defined by the ideal sheaf I and let B_{s} be the scheme defined by J := I^{s}, s ≥ 2. Then there exists a natural biregular map

ϕ : P roj (⊕_{n≥0}I^{n}) → P roj (⊕_{n≥0}J^{n}).

So, even if it is smooth, X is also the blowing up of a non reduced scheme B_{s}. Notice also that X is obtained from every ideal sheaf J generated by a vector space W ⊂ H^{0}(O_{P^{r}}(m)) such that W ⊗ O_{P^{r}} → I(m) is an epimorphism.

Example 0.1.3. It may also happen very well that B_{red} is smooth but that Γ is not the blowing up of B_{red}. To see this specialize the net Φ of plane cubics considered in the previous example as follows: Φ is the net of cubics having multiplicity ≥ 1 at p_{1}, p_{2}, p_{3}, multiplicity ≥ 2 at o and a fixed tangent direction at o (i.e. the tangent cone of each D ∈ Φ has a fixed line). Prove that Γ is a smooth surface of degree 8 in the Segre embedding of P^{2} × P^{2} and that the fibre of π_{1}/Γ over o is a conic of rank two. This implies that π_{1}/Γ is not the blowing up of B_{red} at o.

Example 0.1.4. Key formula (see Fulton p.114). At least in the case where B is smooth we can profit of the fact that π_{1} : Γ → P^{r} is a the structure map of a blowing up as follows: assume codim B = s and let E = π_{1}^{∗}B be the exceptional divisor of π_{1}. Then

E = PN

where N is the normal bundle of B: we will see more directly this in the next section. Moreover there is a Cartesian square

\begin{matrix}E & ⟶_{j} & Γ \\ ↓^{g} & & ↓^{π_{1}} \\ B & ⟶_{i} & P^{r} \end{matrix}

Theorem 0.1.4. Let z be the class of a cycle of codimension k in the Chow ring of B, then

g^{∗}i_{∗}(z)j_{∗}(c_{s−1}(Q) ∩ g^{∗}z.

where

Q := g^{∗}N/O_{E}(−E).

Always from Fulton’s book we have a split exact sequence

0 → CH^{k}(B) → CH^{k}(E) ⊕ CH^{k}(P^{r}) → CH + (Γ) → 0

Moreover we know that the Chow ring

CH^{∗}(E)

is generated by the class e = c_{1}(Q) with the basic relation

e^{s} + a_{1}e^{s−1} + ⋯ + a_{s} = 0

where a_{i} = g^{∗}(c_{i}(N)). The key fomula and the latter remarks make possible to compute the numerical condition so that a linear system Φ with smotth base locus B is homaloidal or not: if Φ ⊂| O_{P^{r}}(m) | we have the basic condition

(mh − e)^{r} = 1

where h = c_{1}(O_{Γ}(0, 1)). This equality corresponds to classical set of conditons, on a rational map f : P^{r} → P^{r} known as equivalence conditions. We will see more in details some examples. If r = 3

B = ∪C_{j} ∪ p_{1} ∪ ⋯ ∪ p_{t}

where C_{j} is a smooth integral curve of genus g_{j} and p_{i} is a point, it follows

1 = m^{3} − t + (4 − 3m) ∑ deg(C_{j}) + ∑(2g_{j} − 2).

We will discuss with too many details some solutions in the next lectures. For instance:

- one line and three points, r = 3.

- one conic and one point, r = 3.

- a unique sextic of genus 3.

The previous examples already indicate that the pair (Γ, B) may have many kinds of behaviours. We will discuss this later with some further examples.

Let H ⊂ P^{r} be a hyperplane: recall that the Chow ring CH^{∗}(P^{r} × P^{r}) is generated by the elements

h_{1}^{i} h_{2}^{j}  (6)

0 ≤ i, j ≤ r, where h_{1} and h_{2} respectively denote the rational equivalence classes of H × P^{r} and P^{r} × H. In particular this implies that

h_{1}^{i} h_{2}^{j} = [L_{1} × L_{2}]

where L_{1} and L_{2} are linear subspaces of P^{r} respectively of codimension i and j. Since Γ_{f} has codimension r we have

[Γ_{f}] = ∑_{0≤i≤r} d_{i} h_{1}^{i} h_{2}^{r−i}  (7)

and hence

[Γ_{f}] h_{1}^{r−i}h_{2}^{i} = d_{i}.

By transversality of the general translate there exist products L_{1} × L_{2} as above which are transversal to Γ_{f} and have class h_{1}^{r−i} h_{2}^{i}. This implies that d_{i} ≥ 0.

Definition 0.1.3. The sequence \underline{d} := (d_{0}, …, d_{r}) is the multidegree of f. The number d_{i} is the i-th degree of f.

We will also refer to d_{1}, …, d_{r−1} as to the characteristic numbers of f. Since f is birational it is clear that

d_{0} = d_{r} = 1.  (8)

Remark 0.1.2. For every subscheme Γ ⊂ P^{r} × P^{r} of pure dimension r the multidegree of Γ can be defined exactly as above. Of course, for such a Γ, we only have [Γ]h_{1}^{r} ≥ 0 and [Γ]h_{2}^{r} ≥ 0 instead of the equalities [Γ]h_{1}^{r} = [Γ]h_{2}^{r} = 1.

Let Γ be integral and let [Γ]h_{1}^{r} = [Γ]h_{2}^{r} = 1, then Γ is the graph of the Cremona transformation

f : P^{r} → P^{r}

defined in the following way: f(x) = x × P^{r} ∩ Γ, for a general x ∈ P^{r}.

In particular it follows that the classification of all Cremona transformations is equivalent to the classification of all integral subschemes Γ ⊂ P^{r} × P^{r} satisfying the conditions [Γ]h_{1}^{r} = [Γ]h_{2}^{r} = 1. Let

γ = h_{1}^{r} + d_{1}h_{1}^{r−1}h_{2} + ⋯ + d_{r−1}h_{1}h_{2}^{r−1} + h_{2}^{r}

be a rational equivalence class satisfying γh_{1}^{r} = γh_{2}^{r} = 1, concerning the classification problem for Cremona transformations the following questions are natural:

(1) Find all the sequences (d_{1}, …, d_{r−1}) for which there exist integral subschemes of rational equivalence class γ.

(2) Find all the Hilbert schemes containing an element which is integral and of rational equivalence class γ.

(3) For each of the previous Hilbert schemes find all the irreducible components whose generic member is integral.

We will see that most of the classes γ as above do not contain cycles defined by an integral subscheme. This is due to some inequalities that the characteristic numbers d_{i}’s have to to satisfy if γ is the class of the graph of a Cremona transformation. These inequalities are due to Cremona in the case r = 3, for the full general result see [?] and the next theorem. Not very much is known on questions (2) and (3): we will discuss in detail examples and some classical results.

A geometric meaning to the i-th degree d_{i} of a Cremona transformation f can be given as follows: fix a general flag

Λ_{0} ⊃ Λ_{1} ⊃ ⋯ ⊃ Λ_{r}  (9)

of linear subspaces Λ_{k} of codimension k = 0, … r, then d_{i} is the length of the zero dimensional scheme

Γ_{f} · Λ_{r−i} × Λ_{i}.

This means that, counting multiplicities, d_{i} is the number of points x ∈ Λ_{i} such that f is regular at x and f(x) ∈ Λ_{r−i}.

Proposition 0.1.5. Let (1, d_{1}, …, d_{r−1}, 1) be the multidegree of the graph of a Cremona transformation f. Then:

(1) d_{i} is the degree of the strict transform by f of a general subspace L ⊂ P^{r} of codimension i.

(2) d_{i} is also the degree of the image by f of a general subspace M ⊂ P^{r} of codimension r − i.

Proof. Let L ⊂ P^{r} be a general subspace of codimension i, then the intersection scheme

S = Γ · P^{r} × L

is proper and integral. This follows from Bertini’s theorem since S is the complete intersection of i general elements of | O_{Γ}(0, 1) |. We can also assume that π_{1} is generically unramified on S and generically injective. This implies that

F = π_{1}(S) = π_{1∗}(S),

in particular F is the strict transform of L by f. It is also clear that

[S] = [Γ]h_{2}^{i}

in CH^{∗}(P^{r} × P^{r}). On the other hand h_{1} = π^{1}[H] where H ⊂ P^{r} is a hyperplane. Therefore we have by projection formula

d_{i} = deg π_{1∗}([Γ]h_{2}^{i}h_{1}^{r−i}) = deg π_{1∗}([S]h_{1}^{r−i}) = deg [π_{∗}S][H]^{i} = deg F.

This proves (1). To prove (2) just observe that the image of M by f is the strict transform of M by f^{−1}: then (2) follows applying (1) to f^{−1}.

The next poposition is an obvious, useful remark.

Proposition 0.1.6. Let i be the involution interchanging the factors of P^{r} → P^{r} and let Γ′ = i^{∗}(Γ). Then Γ′ is the graph of a Cremona transformation f′ such that

d′_{k} = d_{r−k}

where d′_{k} is the k-th degree of Γ′.

Proof. Just observe that i^{∗} : CH^{∗}(P^{r} × P^{r}) → CH^{∗}(P^{r} × P^{r}) intechanges h_{1} and h_{2}. Hence

d′_{k} = [i^{∗}Γ]h_{1}^{r−k}h_{2}^{k} = [Γ]h_{1}^{k}h_{2}^{r−k} = d_{r−k}.

Let Σ be a linear system of dimension ≥ 1 over an integral variety X and let B its base locus. For i ≤ dim X consider the Grassmannian G_{k} of (i − 1)-dimensional subspaces of Σ. Then consider the Zariski closure

R ⊂ X × G_{k}

of the subset of pairs (x, Λ) such that x ∈ X − B and x is in the base locus of Λ. One can give the following

Definition 0.1.4. Let D_{1}, …, D_{i} ∈ Σ be independent elements, the variable intersection of D_{1} … D_{i} is the fibre of the natural projection

p : R → G_{k}

at the space Λ generated by D_{1}, …, D_{i}. The degree of the variable intersection of i elements of Σ is the degree of a general fibre of p.

So we can rephrase the previous proposition as follows:

Proposition 0.1.7. Let (1, d_{1}, …, d_{r−1}, 1) be the multidegree of a Cremona transformation f. Then d_{i} is the degree of the variable intersection of i Φ-hypersurfaces as well as the degree of the variable intersection of r − i Φ′-hypersurfaces.

Theorem 0.1.8. (Cremona inequalities) The characteristic numbers d_{1}, …, d_{r−1} of a Cremona transformation f : P^{r} → P^{r} satisfy the following conditions

1 ≤ d_{a} ≤ d_{b}d_{c} , 1 ≤ d_{r−a} ≤ d_{r−b}d_{r−c}, a = b + c.

Proof. To prove the first set of inequalities consider a general subspace L_{a} ⊂ P^{r} of codimension a and its strict transform F_{a} by f. Then F_{a} is integral of codimension a and, by the proposition, its degree is d_{a}. Assume a = b + c, then L_{a} = L_{b} ∩ L_{c} where L_{b} and L_{c} are again general subspaces of codimension b and c. Denoting by F_{b} and F_{c} their strict transforms we have

F_{a} ⊂ F_{b} · F_{c}.

Finally observe that F_{b} and F_{c} are integral of codimension b and c and that F_{a} is a component of proper codimension of the intersection scheme F_{b} · F_{c}. Applying Bézout theorem as stated in Fulton’s book it follows

deg F_{a} ≤ (deg F_{b})(deg F_{c})

i.e. d_{a} ≤ d_{b}d_{c}. The second set of inequalities then follows from proposition 0.1.5 and the first set of inequalities.

Proposition 0.1.9. Let f : P^{r} → P^{r} be a Cremona transformation such that the base locus of Φ is 0-dimensional, then f has multidegree

(1, m, m^{2}, …, m^{r−1}, 1).

Proof. Let F_{k} be the strict transform by f of a general subspace L_{k} of codimension k ∈ {1 … r−1}. Since Φ has a 0-dimensional base locus it follows that F_{k} is the complete intersection of k Φ-hypersurfaces. Hence d_{k} = deg(F_{k}) = m^{k}.

Notice also that, for a Cremona transformation defined by a homaloidal linear system of degree m, we have

(1, m, d_{2}, …, d_{r−1}, 1) ≤ (1, m, m^{2}, …, m^{r}, 1)

for the multidegree (1, m, d_{2}, …, d_{r−1}, 1) of f. It is not difficult to prove that:

Proposition 0.1.10. Let r ≥ 2, then for each m ≥ 1 there exist Cremona transformations of maximal multidegree (1, m, m^{2}, …, m^{r}, 1).

Proof. By induction on r. For r = 2 see the next example of De Jonquieres transformations. Let r ≥ 3 and let g : P → P be a Cremona transformation of multidegree (1, m, m^{2}, …, m^{r−1}) for the hyperplane P ⊂ P^{r}. Fix o ∈ P^{r} − P and consider the linear system Φ_{P} of all the cones of vertex o over the curves of the homaloidal linear system defining g. Then consider the linear system Φ generated by Φ_{P} and by a further hypersurface having multiplicity m − 1 in o and intersecting properly, i.e. along a zero dimensional scheme, the cone of vertex o over the base locus B_{P} of g. One can easily check that Φ is a homaloidal linear system. It is obvious that its base locus zero dimensional. Hence the Cremona transformation defined by Φ has maximal multidegree.

The next question has not a general answer

Problem 0.1.2. Let (1, d_{1}, …, d_{r−1}, 1) be a sequence of integers satisfying Cremona inequalities: do they exist Cremona tansformations of this multidegree ?

Example 0.1.5. Space Cremona transformations Let f : P^{3} → P^{3} be a birational map, then f is a (m, n)-Cremona transformation and its multidegree is (1, m, n, 1). The Cremona inequalities yeld in this case

n ≤ m^{2} , m ≤ n^{2}.

In other words the degree of the hypersurfaces defining f^{−1} cannot exceed the square of the degree of the hypersurfaces defining f and conversely.

- Assume m = 2 i.e. that Φ is a web of quadric surfaces. Then the possible values of n are 2, 3 or 4. The classification of all these space Cremona transformations is known: we will see this in lecture 3.

- Assume f is a space Cremona transformation with 0-dimensional indeterminacy locus, then we know from the above proposition that f is a (m, m^{2})-Cremona transformation. Space Cremona transformations with 0-dimensional indeterminacy locus were classified by Margherita Beloch in [?], in particular it turns out that m ≤ 4.

- For space Cremona transformations the answer to the previous problem is yes. Ivan Pan has produced an example of (m, n) space Cremona transformation for each admissible pair (m, n).

Example 0.1.6. Plane Cremona transformations A plane Cremona transformation f : P^{2} → P^{2} has multidegree (1, m, 1), In particular we have d_{1} = d_{r−1}. Therefore it follows

Proposition 0.1.11. For every plane Cremona transformation the Φ-curves and the Φ′-curves have the same degree.

To have a less indirect proof of this property just observe that a general Φ′-curve of a plane Cremona transformation f is f(L), where L ⊂ P^{2} is a general line. Moreover f(L) is the birational image of L under the morphism defined by the linear series which is cut on L by Φ. Such a linear series has degree m hence m is the degree of f(L).

Example 0.1.7. (De Jonquieres transformations in the plane ). Consider m − 1 distinct lines of the same pencil of center o = {x_{1} = x_{2} = 0} that is the curve g = l_{1} ⋯ l_{m−1} = 0, where l_{1} … l_{m−1} are non zero linear forms in x_{1}, x_{2}. Then fix an integral curve of degree m having multiplicity m−1 at o. Let f_{0} be the equation of such a curve and let f_{1} = x_{1}g, f_{2} = x_{2}g. Then the net generated by f_{0}, f_{1}, f_{2} is a homaloidal linear system. It is the net of all curves of degree m passing through 2m − 2 points with multiplicity one and through o with multiplicity m − 1.

0.1.2 Linear automorphisms of P^{r}.

Let f : P^{r} → P^{r} be a Cremona transformation, the multidegree of f is a vector

(1, d_{1}, …, d_{r−1}, 1) ∈ Z^{r+1}.

On Z^{r+1} we have a natural partial ordering which will be denoted by < . Since d_{i} ≥ 1 the minimum of the multidegrees of all possible Cremona transformations is realized when each d_{i} = 1. This implies d_{1} = 1 and hence that

Φ =| O_{P^{r}}(1) | .

Therefore f is a linear automorphism of P^{r}. Notice also that Γ_{f} is rationally equivalent to the diagonal Δ ⊂ P^{r} × P^{r} so that

[Γ_{f}] = [Δ] = Σ h_{1}^{i}h_{2}^{r−i}.

On the other hand let Γ be an integral subscheme rationally equivalent to Δ. Then we have [Γ]h_{1}^{r} = [Γ]h_{2}^{r} = 1 and hence Γ is the graph of a Cremona transformation. Moreover, by the previous remarks, Γ is the graph of a linear automorphism of P^{r}. We have shown the following

Proposition 0.1.12. The family of all integral schemes of rational equivalence class Σh_{1}^{i}h_{2}^{r−i} is irreducible. Furthermore the following conditions are equivalent:

(1) Γ is in such a family,

(2) Γ is the graph of a linear automorphism of P^{r}.

We will classify in section [?] Cremona transformations of multidegree (1, 2, …, 2, 1). What about Cremona transformation of multidegree (1, d_{1}, …, d_{r−1}, 1) such that

(1, 1, …, 1, 1) < (1, d_{1}, …, d_{r−1}, 1) < (1, 2, …, 2, 1)?

The next proposition excludes this possibility

Proposition 0.1.13. Let Γ ⊂ P^{r} × P^{r} be an integral scheme such that

[Γ] = h_{1}^{r} + d_{1}h_{1}^{r−1}h_{2} + ⋯ + d_{r−1}h_{1}h_{2}^{r−1} + h_{2}^{r}

and (1, 1, …, 1, 1) ≤ (1, d_{1}, …, d_{r−1}, 1) < (1, 2, …, 2, 1). Then Γ is the graph of a linear automorphism of P^{r}.

Proof. Assume that d_{i} = 1 for some i ∈ {1 … r − 1}. Since d_{0} = d_{r} = 1, Γ is the graph of a Cremona transformation. If i = 1 there is nothing else to show. Hence we can assume d_{1} = 2 and d_{i} = 1 for some i ≥ 2. This means that the intersection scheme of i general quadrics Q_{1} … Q_{i} ∈ Φ is

N + B

where B is the base locus of Φ and N is a general subspace of codimension i. We claim that this implies that the only possible subscheme of B of codimension ≤ i is a linear space of codimensdion ≥ 2. But then, taking a general plane P, the surface f(P) has degree 3 or 4 and hence this is the value of d_{r−2}: a contradiction. To complete the proof we pove our claim: Since N is general we can assume that it is transversal to B, moreover we can assume the same for a general space M ⊃ N of codimension i−1. We can also assume that M is not in Q_{k}, k = 1 … i, so that Q_{k} · M = N ∪ N_{k} is the union of two hyperplanes of M. It follows from these assumptions that

M · B = N_{1} ∩ ⋯ ∩ N_{i}

is a linear space of codimension ≤ i. Then the only possible subscheme of B of codimension ≤ i is a linear space Z (of codimension ≥ 2).

The case of Cremona transformations of multidegree (1, 2, …, 2, 1) is also completely understood: see the next section.

0.1.3 Elementary quadratic transformations

Several generalizations of plane quadratic transformations to higher dimension are possible. The one given here keeps fixed to degree 2 the Φ-hypersurfaces and the Φ′-hypersurfaces. We start with a triple

(Q, o_{1}, o_{2})  (10)

where Q ⊂ P^{r+1} is an integral quadric and o_{1}, o_{2} ∈ Q−Sing Q are distinct points. Let σ : X → Q be the blowing up of o_{1}, o_{2} and let E_{1} = σ^{−1}(o_{1}), E_{2} = σ^{−1}(o_{2}). We consider the r-dimensional linear system | H − E_{t} | (t = 1, 2), where H is the pull-back of a hyperplane by σ. In addition we fix an identification

| H − E_{1} |^{∗}=| H − E_{1} |^{∗}:= P^{r}

via a projective isomorphism. Then | H − E_{t} | defines a birational morphism

p_{t} : X → P^{r},  (11)

hence the product p_{1} × p_{2} defines a generically injective morphism

p : X → P^{r} × P^{r}.  (12)

In addition, under the homomorphism p^{∗} : CH^{∗}(P^{r} × P^{r}) → CH^{∗}(X), we have

p^{∗}h_{t} = [H − E_{t}]

Let Γ ⊂ P^{r} × P^{r} be the image of p, it follows from projection formula that

[Γ] h_{1}^{k} h_{2}^{r−i} = (H − E_{1})^{i} (H − E_{2})^{r−i}.  (13)

It is clear that (H − E_{1})^{r} = (H − E_{2})^{r} = 1 because p_{1}, p_{2} are birational morphisms. Moreover it is easy to check that

(H − E_{1})^{i} (H − E_{2})^{r−i} = 2, i = 1 … r − 1,  (14)

just because Q has degree two. To see this more geometrically just observe that [H − E_{t}]^{i} is the class in CH^{∗}(X) of the strict transform of a general subspace L_{t,i} having codimension i and passing through o_{t}. Then the line

l = L_{1,i} ∩ L_{2,r−i}

does not contain o_{1}, o_{2} for i = 1 … r − 1 and hence [H − E_{1}]^{i}(H − E_{2})^{r−i} = deg(σ^{∗}l) = 2. So we have shown that the class of Γ is

[Γ] = 2 ∑_{i=0...r} h_{1}^{i} h_{2}^{r−i} − h_{1}^{r} − h_{2}^{r}.  (15)

In particular Γ is the graph of a 2 − 2 Cremona transformation f : P^{r} → P^{r}, it is also clear from the construction that f is the birational map p_{2} · p_{1}^{−1}.

Definition 0.1.5. An elementary quadratic transformation f : P^{r} → P^{r} is a Cremona transformation constructed as above from a triple (Q, o_{1}, o_{2}).

It has been shown few lines above that an elementary quadratic transformation has multidegree

(1, 2, 2, …, 2, 2, 1).

As in the case of linear automorphisms, that is Cremona transformations of multidegree (1, 1, …, 1, 1), also elementary quadratic transformations are characterized by their multidegree. This is an immediate consequence of the next result.

Theorem 0.1.14. Let γ = Σ d_{i} h_{1}^{i} h_{2}^{r−i} be a rational equivalence class such that d_{0} = d_{r−1} = 1 and let S_{γ} be the set of all integral subschemes Γ ⊂ P^{r} × P^{r} such that [Γ] = γ. Then the following conditions are equivalent:

(1) S_{γ} is non empty and d_{i} ≤ 2, i = 1 … r − 1.

(2) S_{γ} is non empty and each Γ ∈ S_{γ} is either the graph of a linear automorphism or of an elementary Cremona transformation.

(3) S_{γ} is non empty and either d_{1} = ⋯ = d_{r−1} = 1 or d_{1} = ⋯ = d_{r−1} = 2.

Proof. (1) ⇒ (2): Let Γ ∈ S_{γ}, since it is integral and d_{0} = d_{r} = 1 it follows that Γ is the graph of a Cremona transformation f. If d_{1} = 1 or d_{r−1} = 1, f is a linear automorphism and there is nothing else to show. So we can assume d_{1} = d_{r−1} = 2. Take two general Φ-hypersurfaces D_{1} and D_{2}, then

[D_{1} · D_{2}] = [F] + [M]

where F ∪ M is the intersection scheme D_{1} · D_{2}, F is contained in the base locus B and M is the residual component not contained in B. In other words M is the variable intersection of two elements of Φ and hence its degree is d_{r−2}. We know that d_{r−2} ≥ 1 for each Cremona transformation and we also know that d_{r−2} = 1 implies that f is a linear automorphism. Hence d_{r−2} = 2 and, since D_{1}, D_{2} are quadrics, F has also degree two. Then F is a quadric hypersurface of dimension r − 2 in a hyperplane of P^{r} or F = F_{1} ∪ F_{2} where F_{1}, F_{2} are codimension 2 subspaces spanning P^{r}. The two cases coincide for r ≤ 2. Let us see that, for ≥ 3, the latter case cannot appear: indeed in this case we have dim | I_{F} (2)) |= 3 for the ideal sheaf of F. Since Φ ⊂| I_{F} (2) | this case is impossible if r ≥ 4. For r = 3 F_{1}, F_{2} are two disjoint lines and Φ =| I_{F} (2) |. But it is easy to see that the map defined by | I_{F} (2) | is not birational, hence the case r = 3 is also excluded. Choosing appropriate coordinates (x_{0} : ⋯: x_{r}) on P^{r} we can therefore assume that F = {x_{0} = q = 0}, where q is a quadratic form in x_{1} … x_{r}. This implies that H^{0}(I_{F} (2)) has dimension r +2 and that a basis for it is {q, x_{0}x_{1}, …, x_{0}x_{r}, x_{0}^{2}}. Let β : P^{r} → P^{r+1} be the map defined | I_{F} (2) | then β is defined by the equations

u_{0} = q, u_{1}x_{0}x_{1}, …, u_{r} = x_{0}x_{r}, u_{r+1} = x_{0}^{2},

The image of β is the quadric Q = {q(u_{1}, …, u_{r}) − u_{0}u_{r+1} = 0} and β^{−1} is the projection of Q from its point o_{1} = (1 : 0 : ⋯: 0). Finally we observe that Φ is a hyperplane in | I_{F} (2) | so that f = α · β where α : Q → P^{r} is the restriction to Q of the projection from some point o_{2} ∈ P^{r}. Since f is birational o_{2} must be in Q − Sing Q, moreover o_{2} is not o_{1} because f is not linear. Hence f is the elementary Cremona transformation defined by the triple (Q, o_{1}, o_{2}).

(2) ⇒ (3): It has been proved previously that an elementary quadratic transformation has multidegree (1, 2, …, 2, 1) and that a linear automorphism has multidegree (1, 1, …, 1, 1). Finally (3) ⇒ (1) is obvious.

Remark 0.1.3. Geometric construction. Let (Q, o_{1}, o_{2}) be a triple as above and let P ⊂ P^{r+1} be a hyperplane not containing o_{1}, o_{2}. Consider the birational map i : P → P which is so defined: for a general x ∈ P let z =< x, o_{1} > ∩Q and let y =< z, o_{2} > then y = i(x). i is an elementary Cremona transformation, defined by the triple (Q, o_{1}, o_{2}) and by the natural identification

| H − E_{1} |^{∗}= P =| H − E_{2} |^{∗}

induced by the projection of center o_{t}. Observe that i is a birational involution and that for each elementary quadratic transformation f one has f = h · i where h is a linear automorphism of P.

Remark 0.1.4. Standard form of an elementary quadratic transformation It follows from the previous arguments that the classification of elementary quadratic transformations up to conjugation by linear automophism is equivalent to the classification of the orbits of the opens set of Hilbert scheme Hilb_{2}(Q − SingQ) parametrizing smooth 0-dimensional subschemes of length two under the action of Aut(Q). Fixing the quadric Q we have only three orbits:

(1) Z = o_{1} + o_{2} with o_{1} ≠ o_{2}, < o_{1}, o_{2} >⊂ Q, Sing Q∩ < o_{1}, o_{2} >≠ ∅.

(2) Z = o_{1} + o_{2} with o_{1} ≠ o_{2}, < o_{1}, o_{2} >⊂ Q, Sing Q∩ < o_{1}, o_{2} >= ∅.

(3) Z = o_{1} + o_{2} with o_{1} ≠ o_{2} and < o_{1}, o_{2} > is not contained in Q.

In case (3) p is an embedding. In case (2) the image Γ of the map p : X → P^{r}×P^{r} is singular: p is an embedding on Q− < o_{1}, o_{2} > and contract this line to a double point o. The same in case (3), the difference is that in this case p is not a resolution of the singular point o. Up to changing coordinates (x_{0} : ⋯: x_{r}) the equations of f in the cases (1), (2) and (3) can be reduced to thhe following standard form :

(1) f(x) = (x_{1}^{2} + ⋯ + x_{s}^{2} : x_{0}x_{1} : ⋯: x_{0}x_{r})

(2) f(x) =

(3) f(x) =

where s is the rank of Q.

Example 0.1.8. De Jonquieres transformations. De Jonquieres transformations can be considered as a generalization of the previous geometric construction: let Q ⊂ P^{r+1} be an integral hypersurface of degree d with two points o_{1}, o_{2} of multiplicity d − 1. Fix a hyperplane P ⊂ P^{r+1} not containing o_{1}, o_{2} and define a Cremona transformations

i_{d} : P → P

exactly as in the case considered above where Q is a quadric. i_{d} is a birational involution and a Cremona transformation of type (d, d). The proof is again exactly as in the case of quadrics. The multidegree of i_{d} is

(1, d, …, d, 1).

Definition 0.1.6. A De Jonquieres transformation is a Cremona transformation f = h · i_{d} where i_{d} is a birational involution as above and h ∈ P GL(r + 1).

Maybe the following question admits some easy answer: is it true that any Cremona transformation of multidegree (1, d, …, d, 1) is De Jonquieres?

0.2 Lecture 2

0.2.1 Base locus and exceptional divisor

Let V ⊂ H^{0}(O_{P^{r}}(m)) be any subspace of dimension r + 1 and let q_{0}, …, q_{r} be a basis of V. Fixing coordinates (x_{0} : ⋯: x_{r}) × (y_{0} : ⋯: y_{r}) on P^{r} × P^{r} we can assume that the map defined by V is the rational map

f : P^{r} → P^{r}

defined by the equations

y_{0} = q_{0}, …, y_{r} = q_{r}.

The ideal generated by q_{0} … q_{r} defines the base locus B of the linear system Φ := PV. Often B has a bad behaviour and we will see some concrete examples. Before we want to make some general remarks on B and on the exceptional divisor

E = π_{1}^{∗}B.

Though not necessary in what follows, we will also assume that f is a Cremona transformation. Then E is the exceptional divisor of the blowing up π_{1} : Γ → P^{r}, where Γ is the graph of f. At first we remark that the equations

y_{i}q_{j} − y_{j}q_{i} = 0

define a subscheme

S = Γ ∪ B × P^{r}.

This essentially follows from the explicit well known construction of the blowing up of the scheme defined by q_{0}, …, q_{r}. The projection π_{1} establishes a biregular map

π_{1}/U; U → P^{r} − B,

where

U = Γ − (Γ ∩ π_{1}^{−1}(B)).

If B is smooth then it turns out that

E = Sing(S) ∩ B × P^{r},

we want now to use this remark to describe E. For any, non necessarily smooth, B let x ∈ B and let Γ_{x} be the fibre of π at x. Then it is clear that

Γ_{x} ⊂ Sing(S)

and that

Sing(S) ⊂ B × P^{r}.

Let (x, y) ∈ B × P^{r} then

T_{B,x} ⊕ T_{P^{r},y} ⊆ T_{S,(x,y)} ⊂ T_{P^{r},x} ⊕ T_{P^{r},y}.

Since Sing(S) is contained in the component B × P^{r} of S it is clear that:

Lemma 0.2.1. (x, y) ∈ Sing(S) iff x ∈ Sing(B) or (x, y) ∈ Γ ∩ B × P^{r}.

Moreover we have

Proposition 0.2.2. (x, y) ∈ Sing(S) if and only if

dimT_{S,(x,y)} ∩ T_{P^{r},x} ⊕ 0 > dim O_{B,x}.

Proof. Observe that

T_{B,x} ⊕ 0 ⊆ T_{S,(x,y)} ∩ T_{P^{r},x} ⊕ 0

and that

0 ⊕ T_{P^{r},y} = T_{S,(x,y)} ∩ 0 ⊕ T_{P^{r},y}.

By the lemma (x, y) ∈ Sing(S) iff x ∈ Sing(B) i.e. dim T_{B,x} > dim O_{B,x} or (x, y) ∈ Γ ∩ B × P^{r} i.e. the previous inclusion is proper. These two conditions together are equivalent to the condition in the statement.

Let (x, y) ∈ B × P^{r}, the space

T_{S,(x,y)} ∩ T_{P^{r},x} ⊕ 0

admits a standard projective completion

P_{x} ⊂ P^{r} × y.

One can show that the linear equations

y_{i}dq_{j}(x) − y_{j}dq_{i}(x) = 0

actually define P_{x} in P^{r} × y. Since T_{B,x} ⊂ T_{S,(x,y)} ∩ T_{P^{r},x} ⊕ 0 we have also the inclusion

τ_{x} ⊂ P_{x}

τ_{x} being the projective completion of T_{B,x}. Let c be the codimension of τ_{x}, then

Corollary 0.2.3.

(x, y) ∈ Sing S ⟺ codim P_{x} ≤ c − 1 or x ∈ Sing(B).

Proof. Immediate from the previous lemma

Theorem 0.2.4. Assume for simplicity that B is smooth, connected and of codimension c. Then

codim P_{x} = c − 1

for each x ∈ B.

Proof. Since B is smooth at x, the linear forms dq_{0}(x) … dq_{r}(x) generate a vector space of dimension c. Up to reindexing we can assume that dq_{1}(x) … dq_{c}(x) is a basis of it and that y_{c} ≠ 0. Then P_{x} is contained in the linear space {y_{c}dq_{1}(x) − y_{1}dq_{c}(x) = ⋯ = y_{c}dq_{c−1}(x) − y_{c−1}dq_{c}(x) = 0} which is of codimension c − 1.

Geometric description of Γ ∩ B × P^{r}. Let G be the Grassmannian of the subspaces of P^{r} having codimension c − 1, we have proved that

E = {(x, P) ∈ B × G / τ_{x} ⊂ P }.

E is the projectivization of a vector bundle: it turns out that the fibre of E at x is

E_{x} = P(T_{P^{r},x}/T_{B,x})^{∗}

and that

E = P roj(N_{B}) = P(I/I^{2})

where N_{B} is the normal bundle defined by the exact sequence.

0 → T_{B} → T_{P^{r}} ⊗ O_{B} → N_{B} → 0.

We will develop with some more detain in an example the following general fact: consider the Euler sequence

0 → T_{B} → V ⊗ O_{B}(1) → T_{P^{r}} ⊗ O_{B} → 0

and the standard exact sequence

0 → T_{B} → T_{P^{r}} ⊗ O_{B} → N_{B} → 0.

Using the two exact sequences we obtain an exact sequence

0 → P → V ⊗ O_{B}(1) → N_{B} → 0

where P is known as the sheaf of the principal parts (Fulton pag. 144). The right hand epimorphism induces an embedding

P roj(N_{B}) ⊂ B × P^{r} ⊂ P^{r} × P^{r}.

One can show that, under this embedding,

P roj(N_{B}) = E = B × P^{r} · Γ.

The latter equality holds more in general:

Proposition 0.2.5. Let f be a Cremona transformation then

Γ · B × P^{r} = P roj(I/I^{2}).

Let f : P^{r} → P^{r} be a Cremona transformation, the smoothness of B_{red} is however not sufficient to guarantee too many things. Plane Cremona transformations already provide examples where we may have that:

(1) Γ is smooth, B_{red} is smooth but Γ is not the blowing up of B_{red}.

(2) Γ is singular and B_{red} is smooth. The next examples describe further examples of ’bad’ behaviour for space or higher dimensional Cremona transformations. It could appear much more difficult, but however it is not impossible, to find Cremona transformations such that:

(i) B_{red} is smooth and also integral, (ii) B ≠ B_{red}, (iii) Γ is the blowing up of B_{red}. An example of such a situation will be produced in a moment.

Example 0.2.1. Consider for instance the family of web of quadrics of P^{3}:

Φ_{t} :=< x_{3}^{2} − x_{1}x_{2}, x_{0}(x_{1} − x_{2}), x_{0}(x_{2} − x_{3}), x_{0}(tx_{3} − x_{0}) >, t ∈ C.

For each t we can study the base locus B_{t} of Φ_{t} that is the scheme defined by the previous four equations. We can distinguish two cases:

(1) t ≠ 0. Then B_{t} is smooth and disconnected: it is the union of the conic C = {x_{0} = x_{3}^{2} − x_{1}x_{2} =} and of the point o_{t} = (t : 1 : 1 : 1).

(2) t = 0. Then B_{0} is not reduced: its support is the smooth conic C. But B_{0} contains the embedded point (0 : 1 : 1 : 1).

In both cases we study the rational map f_{t} : P^{3} → P^{3} defined by the equations

y_{0} = x_{3}^{2} − x_{1}x_{2}, y_{1} = x_{0}(x_{1} − x_{2}), y_{2} = x_{0}(x_{2} − x_{3}), y_{3} = x_{0}(tx_{3} − x_{0}).

CASE (1). Let σ_{t} = π_{1}/Γ_{t}, we know that σ_{t} is the blowing up of the smooth variety C ∪ o_{t}. Let

E := σ_{t}^{−1}(C) , F := σ_{t}^{−1}(o_{t})

be the exceptional divisors of σ_{t}: we compute as an exercise the degree of the map defined by the linear system Φ_{t}, (t ≠ 0). The strict transform of Φ_{t} on Γ_{t} is

| 2H − E − F | .

We have

E = PN_{C}^{∗} = P roj O_{P^{1}}(2) ⊕ O_{P^{1}}(4)

so that the self intersection of c_{1}(O_{E}(E)) is −degN_{C} = −6. In other words E^{3} = −6. On the other hand recall that F^{3} = 1. So we have

(2H − E − F)^{3} = 8H^{3} − E^{3} − F^{3} + 6HE^{2} = 8 + degN − 1 − 12 = 1.

Hence Φ_{t} is homaloidal and f is a Cremona transformation.

A very simple way of proving this fact goes as follows: for two general elements Q_{1}, Q_{2} ∈ Φ_{t} we have Q_{1} · Q_{2} = C + D, where D is a smooth conic passing through o_{t}. Taking a third general Q_{3} ∈ Φ_{t} we have Q_{1} · Q_{2} · Q_{3} = C + p + o_{t}. Hence the variable intersection of three elements of Φ_{t} is one point.

We conclude with a brief geometric description: we have f_{t} = α·β where β = σ^{−1} and α = π_{2}/Γ_{t}. This implies that

P ic Γ_{t} = Z[H] ⊕ Z[E] ⊕ [F].

Since P ic P^{3} ≅ Z and α_{∗} is surjective, it follows that α contracts two divisors which generate a primitive subgroup of rank two of P ic Γ_{t}. More concretely the two divisors are the following ones:

- let F′ be the strict transform by β of the plane containing C. Then

β(F′) = a point o′_{t}.

- Let E′ be the strict transform by β of the cone of vertex o_{t} and containing C. Then

β(E′) = a conic C′.

CASE (2) Now the problem is to describe the specialization of this map. Actually the simplest way for understanding this goes probably as follows: Note that a general element of Φ_{0} is still smooth and that these quadrics have a fixed tangent plane at the base point o = (0 : 1 : 1 : 1) namely the plane 2x_{3} − x_{1} − x_{2} = 0. This implies that the strict transform Φ_{1} of Φ_{0} by the blowing up

σ_{1} : Q_{1} → P^{3}

of C has a unique base point o_{1}. Such a point is in the exceptional divisor E_{1} = σ_{1}^{−1}(C): it is the point corresponding to plane 2x_{3} − x_{1} − x_{2} = 0 in the fibre

f = σ_{1}/E_{!}^{−1}(o)

of E_{1}. Let σ_{2} : Q_{2} → Q_{1} be the blowing up of o_{1} and let E be the strict transform of E_{1}, F = σ_{2}^{−1}(o_{1}). Let σ = σ_{1} · σ_{2}, the strict transform of Φ_{1} on Q_{2} is the linear system

| 2H − E − F |

where H is the pull-back of a hyperplane by σ. Let

π : Q_{2} → P^{3} × P^{3}

be the map defined by | H | × | 2H −E −F |. It turns out that π is an embedding on Q_{2} − f and that f is contracted to a singular point. Moreover the image of π is Γ.

To see all this more geometrically observe that the rational map ψ : P^{3} → P^{4}

y_{0} = x_{3}^{2} − x_{1}x_{2}, y_{1} = x_{0}x_{1}, y_{2} = x_{0}x_{2}, y_{3} = x_{0}x_{3}, y_{4} = x_{0}^{2}

has image Q′ of equation

y_{3}^{2} − y_{1}y_{2} − y_{0}y_{4} = 0.

Q′ is just σ_{1} followed by the contraction to the point p = (1 : 0 : 0 : 0 : 0) of the strict transform of the plane {x_{0} = 0}. Notice also that E_{1} maps onto the tangent hyperplane section to Q′ at p. This is a cone E′. The tangent line l

x_{0} = 2x_{3} − x_{1} − x_{2} = 0

to C blows down to p. The tangency point l ∩ C blows up to a line

f′ ⊂ E′.

Such a line parametrizes the pencil of planes through L. Let o′ be the point corresponding to the plane 2x_{3} − x_{1} − x_{2} = 0, then the strict transform of Φ is the linear system of hyperplane sections through o′. Conclusion: let π′ : Q′ → P^{3} ×P^{3} be the product of the stereographic projections of centers p and o′, then π′(Q′) = Γ. The line f′ contracts to the unique singular point of Γ.

Example 0.2.2. The Cremona space transformation of rational normal cubic. Let

C ⊂ P^{3}

be a rational normal cubic, the normal bundle of C is

N = O_{P^{1}}(5) ⊕ O_{P^{1}}(5).

We apply to C a more general construction which works for every B ⊂ P^{r} which is locally complete intersection. Let V = H^{0}(O_{P^{3}}(1))^{∗}, consider the Euler exact sequence

0 → O_{P^{3}} → V ⊗ O_{P^{3}}(1) → T_{P^{3}} → 0.

Such a sequence defines an embedding

P roj(T_{P^{3}}) ⊂ P^{3} × P^{3∗}

of P roj(T_{P^{3}}) as a hyperplane section of the Segre embedding. More precisely

P roj(T_{P^{3}}) = {(x, H) ∈ P^{3} × P^{3∗} / x ∈ H}

is the universal plane over P^{3∗}. Restricting the Euler sequence to C we can construct in particular the exact diagram

0

↓

V ⊗ I_{C}(1)  0
↓  ↓
V ⊗ O_{P^{3}}(1)  T_{C}
↓  ↓
0 ⟶ O_{C} ⟶ V ⊗ O_{C}(1) ⟶ T_{P^{3}} ⊗ O_{C} ⟶ 0
↓  ↓
0  N

↓ The diagram defines the embeddings

P roj(N) ⊂ P roj(T_{P^{3}}) ∩ P^{3} × C ⊂ P^{3} × P^{3}.

The diagram of the associated long exact sequences defines a natural map

μ : V ⊗ V^{∗} → H^{0}(N)

which is just the map induced by the Euler sequence. This second diagram immediately implies the next lemma.

Lemma 0.2.6. μ is surjective.

Since h^{0}(N) = 12 it follows that

E ⊂ P^{3} × P^{3}

as a smooth surface of degree 10, which is the degree of the self intersection of the tautological bundle of P roj(N). Notice also that E_{x} is the pencil of planes through the tangent line to C at x. So we have a canonical section

s : C → E

sending x ∈ C to P_{x} ∈ E_{x}, where P_{x} is the plane such that P_{x} · C = 3x. Let

Λ ⊂ P^{15} = P(V ⊗ V^{∗})

be the orthogonal space to Ker μ. Then Λ is a linear space of codimension 4.

Lemma 0.2.7. The scheme Λ · P^{3} × P^{3} is proper.

Proof. This follows from the diagram in Fulton’s book p. 144, we have the exact sequence

0 → P → V ⊗ O_{C}(1) → N_{C} → 0.

Dualizing it we have E = P roj(N_{C}) ⊂ Λ · PV^{∗} × P^{r}. Assume that there exists (x, y) ∈ Λ · PV^{∗} × P^{r} which is not in E, then the map V^{∗} ⊗ O_{P^{r}}(−1) → P_{x}^{+} is not surjective

Since deg = 10 and the degree of the Segre embedding of P^{3} × P^{3} is 20 we have

Λ · P^{3} × P^{3} = E ∪ E′

where E′ is a second surface.

Proposition 0.2.8. E′ is a copy of E, embedded in the same way. E ∩ E′ = C′ where C′ is the image of the section s : C → E considered above.

Let Γ be a general codimension three linear section containing E′ ∪ E, then Γ is integral and smooth away from E ∩ E′: this essentially follows from Bertini’s theorem. We can say more:

Lemma 0.2.9. Γ is smooth.

Notice also that, by Lefschetz theorem,

P ic Γ = Z[H_{1}] ⊕ Z[H_{2}].

Γ is the graph of a Cremona transformation which we will consider more in general later: for each r a general linear section of codimension r of the Segre embedding of P^{r} × P^{r} defines a Cremona transformation. This type of Cremona transformations are known as bilinear transformations. Thery are defined by the order r minors of a matrix r × (r + 1) of linear forms. In our situation Γ is the graph of an f : P^{3} → P^{3} defined by a web of cubic surfaces through C. Since H_{1} + H_{2} ∼ E + E′, it follows

E′ ∼ 4H_{2} − E , E ∼ 4H_{1} − E′.

Therefore S = π_{1}(E′) is a quartic surface: what is this quartic surface?

Proposition 0.2.10. S is the union of the tangent lines to C.

Proof. Following Fulton p. 144 we have the exact diagram

It is an interesting point that f is defined by a homaloidal polynomial:

Theorem 0.2.11. Let F be an equation of the quartic surface S. Then the polar linear system of F is a homaloidal web of cubics and define f.

Remark 0.2.1. Interestingly we have ∞^{3} codimension three linear sections L of the Segre embedding of P^{3} × P^{3} containing E. A general L defines a Cremona transformation f : P^{3} → P^{3}, any other Cremona transformation L′ defined by another linear space containing E is just l · f where l ∈ Aut(C) ⊂ Aut(P^{3}).

Remark 0.2.2. We will discuss more general constructions of this type later. For the moment we only point out two related possible constructions to do:

(1) The only rational normal curve C ⊂ P^{r} defining still a similar construction seems to be the rational normal quartic. It seems interesting to study the corresponding bilinear Cremona transformations, though it does not seem defined by a homaloidal polynomial.

(2) The construction of the rational normal cubic probably generalizes to (P^{1})^{n} (suitably embedded), but only if n ≤ 3 for some trivial reasons.

(3) More in general one should consider P^{a} × P^{b} such that

(a + 1)^{2} + (b + 1)^{2} ≥ (a + 1)(b + 1)

and reply the same construction done fo C. The next example is very much related to the present one and its geometry will be described in full generality.

Example 0.2.3. Classification of space Cremona transformations such that B_{red} is smooth, connected and moreover π_{1}/Γ is its blowing up. We first recall from section [?] the intersection numbers of some standard ways of blowing up P^{3}:

(1) Let σ : P → P^{3} be the blowing up of a simple point o and let E = σ^{−1}(o). Then O_{E}(E) ≅ O_{P^{2}}(−1) and hence

(dH − qE)^{3} = d^{3} − q^{3}.

Proof. One can remark that 0 = (H − E)^{3} = H^{3} − E^{3}. hence E^{3} = 1 and the result follows.

(2) Let σ : P → P^{3} be the blowing up of an integral curve C wich is locally complete intersection. Then

−E^{3} = deg(N) = 4deg(C) + 2p_{a}(C) − 2 = deg(N)

where E = σ^{−1}(C) = PN^{∗} and N is the normal bundle of C.

Proof. Recall that [E^{2}] = −[C′] where C′ is an element of | O_{PN}(1) |. This implies that we have the equality of Chern polynomials restricted to C′

c(T_{X}) = c(T_{C′})(1 − Et)(1 + Et).

From this we deduce (4H − E)C′ = 2 − 2p_{a}(C). Since [C′] = −[E^{2}] it follows 4deg(C)+E^{3} = 2−2p_{a}(C). The second equality follows from the standard exact sequence

0 → T_{C} → T_{P^{r}} ⊗ O_{C} → N → 0

(3) In particular (2) implies

(mH − sE)^{3} = m^{3} + 3ms^{2}HE^{2} − s^{3}E^{3} = m^{3} + 3ms^{2}d(degC) + s^{3}(degN)

that is

(mH − sE)^{3} = m^{3} − 3ms^{2}d + s^{3}(2g − 2 + 4d)

with d = deg C, g = genus of C.

Assume now that f : P^{3} → P^{3} is a Cremona transformation such that:

(i) B_{red} is a smooth, integral curve. (ii) Γ is smooth. (iii) The divisor π_{1}/Γ^{−1}(B_{red}) is irreducible. We know that then Γ is the blowing up of B_{red}. In particular P ic(Γ) is generated by H and E, where H is the pull-back of a hypeplane by π_{1}/Γ is the exceptional divisor over B_{red}. The morphism π_{2}/Γ : Γ → P^{3} is therefore defined by a homaloidal linear system

Φ ⊆| mH − sE | .

For checking the possible cases all what we have to do is to put

m^{3} − 3ms^{2}d + s^{3}(2g − 2 + 4d) = 1

and look for the solutions. We can add a second very natural numerical condition: let L ⊂ Γ be the strict transform of a general line. Then L is the complete intersection of two surfaces of the linear system

| mH − sE | .

Since L has genus zero adjunction formula yelds

(mH − sE)^{2}(2(mH − sE) − 4H + E) = −2

that is

−1 = m^{2}(m − 2) − [s^{2}(m − 2) + sm(2s − 1)]d + s^{2}(2s − 1)[2d + g − 1] So we have to solve the Diophantine system with m, s, d > 0 and g ≥ 0. Eliminating g between the two equations we get

(m^{2} − s^{2}d)(4s − m) = (4s − 1).

Note that (m^{2} − s^{2}d) = H(mH − sE)^{2}, so this number is the degree of σ(L) and it is strictly positive. hence

4s − m > 0.

On the other han we have from the first equality that s divides m^{3} − 1. But s divides also 1 + m^{2}(m − 2). Hence s divides

m(m^{3} − 2m^{2} + 1) − (m − 2)(m^{3} − 1) = 2m − 2.

So 2m − 2 = ks and m < 4s imply k ≤ 8. With some more work one obtains that the only possible solution is for

m = 3 , s = 1 , d = 6 , g = 3.

We will discuss the geometry of this unique case, the so called cubo-cubic transformation, in a moment.In particular note the rational normal cubic considered in the previous example appears as a curve of arithmetic genus 3 with a double structure. This double structure is defined by a suitable web of cubic surfaces with the following property: any two of these cubics touch along C.

0.2.2 Geometry of cubo-cubic space transformations

We have to deal with a smooth integral sextic

C ⊂ P^{3}

of genus 3 and our purpose is to reconstruct a Cremona transformation of P^{3} from C. Actually everything is relatively easy and well known, we can construct such a transformation as follows:

C is a projectively normal curve and its ideal I_{C} is generated by cubic forms. The projective normality of C makes possible to compute

dim | I_{C}(3) |= 3

and the fact that C is generated by cubics implies that the general D ∈| I_{C}(3) | is smooth. Now use the geometry of the smooth cubic surface D to prove that

| O_{D}(3H − C) |

is a linear system of rational normal cubic curves of self intersection 1. More precisely | 3H − C | defines the contraction

σ : D → P^{2}

of six lines of D to P^{2}. Notice by the way that these six lines are trisecant lines to C and that σ(C) is an octic curve with six triple points. What is interesting us is that the linear system of cubic surfaces containing C is homaloidal and defines a Cremona transformation

f : P^{3} → P^{3}

with B = C. So this shows the existence of the Cremona transfomation predicted by the numerical condtions we have considered.

However we want to see some more of the big amount of geometry related to this construction. Due to the connection with the previous example of the rational normal cubic curve, we start with the exact diagram considered there. In particular the long exact sequence of its horizontal row yelds

0 → H^{0}(O_{C}) → V ⊗ V^{∗} → H^{0}(T_{P^{3}} ⊗ O_{C})) → H^{1}(O_{C}) → 0.

Differently from the previous example we have here H^{1}(O_{C}) ≅ C^{3}, hence

μ : V ⊗ V^{∗} → H^{0}(N)

cannot be surjective and its codimension is three. It quite clear what happens: consider the three bilinear forms

a_{i0}y_{0} + a_{i1}y_{1} + a_{i2}y_{2} + a_{i3}y_{3} ∈ Ker(μ) ⊂ H^{0}(O_{P^{3}×P^{3}}(1, 1)),

i = 1, 2, 3. Then we have a 3 × 4 matrix of linear forms on P^{3} = PV which is

A = (a_{ij}).

The four 3 × 3 minors of A are lineary independent and the cubic surfaces defined by them contain C, hence they generate the ideal of C. To see this just observe that all the previous bilinear forms are zero on E := PN^{∗} and that

π_{1}(E) = C.

But then A has rank ≤ 2 at each point x ∈ C. Hence C is contained in each of the previous cubic surfaces. This Cremona transformation is very much related to linear algebra: fix the Segre embedding

P^{3} × P^{3} ⊂ P^{15} = P(V ⊗ V^{∗}).

Then the graph Γ of the Cremona transformation

f : P(V) → P(V^{∗})

defined by C is just a codimension 3 linear section. The inverse transformation is again defined by a linear system of cubic surfaces vanishing on a second projective model of C.

Remark 0.2.3. The embedding

Γ ⊂ P^{3} × P^{3} ⊂ P^{15}

is the image of the rational map

ϕ : P^{3} → P^{15}

defined by the linear system | I_{C}(4) |. Indeed Mumford-Castelnuovo’s theorem implies that the multiplication map

V ⊗ H^{0}(I_{C}(3)) → H^{0}(I_{C}(4))

is surjective and this implies our claim.

Fundamental loci. To understand the fundamental loci of f and of f^{−1} we study the surface

Sec_{3}(C)

of trisecant lines to C.

Lemma 0.2.12. The singular locus of Sec_{3}(C) is C and C is a curve of triple points for Sec_{3}(C).

Proof. Two distinct trisecant lines L_{1}, L_{2} to C do not meet in P^{3} − C. Indeed let l_{i} = L_{i} · C, if L_{1} ∩ L_{2} is a point p ∈ P^{3} − C then we have P · C = t_{1} + t_{2} for the plane spanned by L_{1}, L_{2}. But then it is easy to see that dim | t_{i} |= 1 and that C is contained in a quadric, which is impossible under our assumption O_{C}(1) ≅ ω_{C}(t), t non effective. This implies that C = Sing(Sec_{3}(C)). Projecting C from one of its points, we obtain a plane quintic with three nodes. Hence there are exactly three trisecant lines passing through a general point of C. This implies that Sec_{3}(C) has multiplicity three along C.

The lemma implies that a general plane section of Sec_{3}(C) is plane model of degree d of C with six triple points. From genus formula for plane curves we then obtain

\frac{1}{2}(d − 1)(d − 2) − 18 = 3

and hence d = 8. Let D be a general cubic though C then P = f(D) is a general plane and intersects the fundamental locus

C′

of f^{−1}. We know from the construction that Φ′ is a homaloidal linear system of cubics. In particular it turns out that C′ is an integral curve: we omit the details of the poof. On the other hand

f/D : D → P

is the usual contraction of six lines of D. Which are these lines? We have

D · Sec_{3}(C) = 3C + L_{1} + ⋯ + L_{6}

where the L_{i}’s are these lines. Now there is a natural isomorphism

ϕ : C” → C

where C” parametrizes the trisecant lines to C. Let L be such a trisecant and let l = CL̇. Then h^{0}(O_{C}(1 − l)) = 2 that is

h^{0}(ω_{C}(l − 1)) = 1.

But ω_{C}(l − 1) has degree one, hence it defines a unique point

ϕ(L) := x_{l} ∈ C.

This map is clearly invertible, hence it is biregular. Notice also that

O_{C}(x_{l}) ≅ O_{C}(l − t).

It is clear that

f(Sec_{3}(C)) = C′.

In addition we have now all the elements to understand what is the line bundle O_{C′}(1): it is clear that

O_{C′}(1) ≅ O_{C}(Σx_{l_{i}}) ≅ O_{C}(Σl_{i} − 6t).

What is such a line bundle? For a general C P ic^{0}(C) has no automorphism, with the exception of translations and of −1 multiplication. This implies that

O_{C}(1) ⊗ O_{C′}(1)

does not depend on the choice of O_{C}(1) in P ic^{6}(C). Since there is no rationally connected line bundle on the universal curve over M_{∋} with the exception of the relative cotangent bundle or of its pwers and duals, it follows that

O_{C}(1) ⊗ O_{C′}(1) ≅ ω_{C}^{⊗3}.

A further remark is about the normalization of Sec_{3}(C) and its construction: take a Poincaré bundle

U → C × P⟩⌋^{∋}(C)

and then the restriction

E

to C = W_{3}^{1}(C) of its push down by the second projection. One can compute that degE = 4 and we can choose E so that its determinant is ω_{C}. Tensoing by O_{C}(t) we obtain E(t) of degree 8 and a tautological morphism

s : P rojE(t) → PH^{0}(E(t))^{∗} = P^{3}.

The image of s is Sec_{3}(C). Let t′ + t ∼ K_{C} then the tautological morphism

s′ : H^{0}(E(t′)) → PH^{0}(E(t′))^{∗} = P^{3}

has image equal to Sec_{3}(C′).

Moduli of vector bundles of rank two over C.

Remark 0.2.4. We want now to discuss some relations of the space cubo-cubic transformation to the moduli space

SU(2, O_{C}(1))

of semistable rank two vector bundles of fixed determinant. We use a slightly different description with respect to Bertram’s description. The projective bundle

P = PT_{P^{3}}(−1)

is endowed with the relative tangent bundle

T

of the projection map π : P → P^{3}. Let O_{P} (1) := π^{∗}O_{P^{3}}(1), then T (−1) restricts on each fibre of π to a vector bundle with determinant O_{P^{2}}(1). Notice also that the fibre of π at x is naturally identified to the projective plane

P_{x} = { all the lines through x }.

Let U = P^{3} − C, we have a morphism

λ : U × C → P

sending (x, y) to the line < x, y >. The pull-back

λ^{∗}T (−1)

is therefore a family of rank two vector bundles on C with fixed determinant O_{C}(1). Let x ∈ U and let

F := λ_{x}^{∗}T_{P^{2}}

be the corresponding vector bundle, where λ_{x} : P^{3} → P^{2} is the projection of center x. The pull-beck of the Euler sequence by λ_{x} yelds the exact sequence

0 → O_{C}(−1) → W ⊗ O_{C} → F → 0

where W = λ_{x} ∗ H^{0}(T_{P^{2}}(−1)). We know that dim W = 3, moreover λ_{x}^{∗} is injective because λ_{x}(C) is not contained in a line. Then, passing to the long exact sequence, it follows

h^{0}(F) = 3.

We have also the determinant map

det : ∧^{2}W → H^{0}(det F),

its image equal is the linear system defining the projection λ_{x} : C → P_{x}. In particular P_{x} = P ∧^{2} W^{∗}. Taking

s_{1}, s_{2} ∈ λ_{x}^{∗}W = H^{0}(F)

s_{1} ∧ s_{2} is the pull-back by λ_{x} of the divisor cut on a line on λ_{x}(C). In particular:

Assume s is zero on the effective divisor d then λ_{x∗}(d) is the base point of a pencil of lines of P_{x}.

Note that

h^{0}(F) = 3 ⟺ h^{1}(F) = h^{0}(ω ⊗ F^{∗}) = 1.

Hence we have a uniquely defined non tivial extension

0 → O_{C} → ω_{C} ⊗ F^{∗} → O_{C}(t′) → 0

where, as previously

t′ + t ∼ K_{C}.

So we started from a general

x ∈ P^{3} = PH^{0}(ω_{C}(t))^{∗} = PExt^{1}(O_{C}(t′), O_{C}))

and we have constructed explicitely the extension defined by x. In general we can forget the pevious construction and simply consider, for any embedded curve C ⊂ P^{r}, the space

PExt^{1}(O_{C}(t′), O_{C}))

where O_{C}(t′) ≅ ω_{C}^{∗}(1). We describe how to derive the cubo-cubic transformation of C via this construction: We claim, without the simple proof, that F is stable for a geneal x. let

f : P^{3} → SU(2, ω_{C}(t)) ⊂| 2Θ |= P^{7}

be the map in the moduli space, embedded by its generalized theta divisor. This map sends x to the divisor

Θ_{F} = {u ∈ P ic^{1}(C) / h^{0}(F (−u)) ≥ 1} ∈| 2Θ | .

Here Θ is the theta divisor on P ic^{1}(C) which is so define

Θ = {u ∈ P ic^{1}(C) / h^{0}(M(−u)) ≥ 1, M^{⊗2} ≅ ω_{C}(t′)}

The linear system | 2Θ | does not depend on the choice of M. Let o ∈ P ic^{1}(C), then

| 2Θ − o |

is a hyperplane of P^{7}. Since h^{0}(F) = 3 we have h^{0}(F (−x)) ≥ 1 for each x ∈ C ⊂ P ic^{1}(C). Hence

f(x) ∈ ∩ | 2Θ − x |, x ∈ C.

Now C embeds in P^{7}: it is not difficult to show that

O_{C}(Θ) ≅ M

so that M^{⊗2} ≅ ω_{C}(t′). So we have constructed a rational map

f : PH^{0}(ω_{C}(t))^{∗} → PH^{0}(ω_{C}(t′))^{∗}

The map f is generically injective: indeed a general point of f(P^{3}) is a stable point, hence defines a unique stable bundle F. We know that h^{0}(F) = 3 and moreove the image of

w : ∧^{2}H^{0}(F) → H^{0}(O_{C}(1))

is injective. By the previous construction the orthogonal of Im(w) is a unique point x ∈ P^{3} and

f(x) = y.

Exercise: note that this implies that only the strictly semistable locus SSL is contracted by f. Prove that

SSL = Sec_{3}(C).

Remark 0.2.5. The genus two case. The previous construction applies to any smooth, integral curve in P^{3}. Applying it to a smooth, integral genus two quintic we obtain

f : P^{3} → SU(2, O_{C}(1)) = Q_{1} ∩ Q_{2} ⊂ P^{5}

where Q_{1} and Q_{2} are two transversal quadrics. (See Tjurin’s description or Griffiths-Harris book). In this case f is defined by the linear system of all cubics th4rough C. Let

L ⊂ P^{3}

be a bisecant line to C. Then C ∪ L is in the same Hilbet scheme of an integ4al smooth genus three sextic.

0.2.3 elementary cubo-cubic transformations.

0.2.4 Bilinear Cremona transformations

Definition 0.2.1. A bilinear Cremona transformation is a Cremona transformation f : P^{r} → P^{r} whose graph is an r-dimensional linear section of the Segre embedding of P^{r} × P^{r}.

Let (x_{0} : ⋯: x_{r}) × (y_{0} : ⋯: y_{r}) be projective coordinates on P^{r} × P^{r}, the graph of a bilinear Cremona transformation f is then defined by r bilinear forms

b_{1}(x, y) = … = b_{r}(x, y) = 0.

Each p ∈ P^{r} defines the linear forms b_{1}(p, y), …, b_{r}(p, y) in (y_{0} : ⋯: y_{r}). Then f(p) is the intersection of the hyperplanes {b_{1}(p, y) = 0}, …, {b_{r}(p, y) = 0}. The multidegree of f is easily computed, we have [Γ_{f}] = (h_{1} + h_{2})^{r} hence

d_{i} = (h_{1} + h_{2})^{r} h_{1}^{r−i}h_{2}^{i} = \binom{r}{i}, i = 0, …, r

A very special example of bilinear Cremona transformation is obtained when

b_{i} = x_{i}y_{i}, i = 0 … r.

Example 0.2.4. r = 4 The base locus of the linear system of determinantal quartic hypersurfaces is a surface

B ⊂ P^{4}

of degree 10.

The degree of the hypersurfaces of the linear system is

d = \binom{r + 2}{2} − r − 1 = d.

Indeed fixing a general P^{2} ⊂ P^{r} we have that d is the number of points of P^{2} so that the linear system of plane curves through d points is n-dimensional. This yelds the above formula.

0.2.5 Numerical conditions

Let

B ⊂ P^{r}

be the base locus a linear system Φ of hypersurfaces of degree d. If the map f defined by Φ is a Cremona transformation then some numerical characters related to B must satisfy three fundamental equalities. Roughly speaking these three equalities reflect the fact that, if f is a Cremona transformation, then:

(I) The contribution of B to the intersection number of r hypersurfaces of Φ must be d^{r−1} − 1, so that f has degree one. (II) The dimension of the vector space of the forms of degree d vanishing on B must be r + 1. (III) The variable intersection of r − 1 Φ-hypersurfaces must be an integral curve of genus zero

The numerical conditions deriving from (I), (II), (III) were respectively called equivalence, postulation and genus equalities for Cremona transformations, [?]. In the following we will discuss them with some more details and precision. For this we need to recall the notion of Segre class

s(B, P^{r})

of the scheme B. Recall that the normal cone of B in P^{r} is defined as

C_{B/P^{r}} = Spec(⊕_{n≥0} I^{n}/I^{n+1}),

where I is the ideal sheaf of B. Taking the projective completion

P := P(C_{B/P^{r}} ⊕ O_{B})

we can consider the natural projection p : P → B and the tautological line bundle O_{P} (1). Then the (total) Segre class is defined as

s(B, P^{r}) = p_{∗}(∑_{i≥0} c_{1}(O_{P} (1)^{i}[P])

in the Chow group CH_{∗}(B).

I Equivalence equality The fact that f must have degree one can be expressed in terms of the Segre class s(B, P^{r}). Any r-dimensional linear system Φ ⊂| O_{P^{r}}(m) | defines a scheme B ⊂ P^{r} which is its base locus and a rational map f : P^{r} → P^{r}. The relation between the Segre class of B and the degree of f is given by the following theorem: see Fulton’s book.

Theorem 0.2.13.

d^{r} − deg (i^{∗}(1 + c_{1}O_{P^{r}}(m))^{r} ∩ s(B, P^{r})) = deg f,

where i : B → P^{r} is the inclusion and ∩ denotes the cap product.

So the equality condition for a Cremona transfomration f is expressed by the formula

m^{r} − deg (i^{∗}(1 + c_{1}O_{P^{r}}(m))^{r} ∩ s(B, P^{r})) = 1.

II Postulation equality Assume that f is defined by the homaloidal linear system Φ then Φ ⊂ I(m) |, where I is the ideal sheaf of B and m = d_{r−1}. Since f has degree one and B is the base locus for both Φ and I_{B}(m) |, it follows that Φ =| I_{B}(m) |: otherwise | I_{B}(m) | would define a non doominant rational embedding of P^{r} in P^{s}, for some s > r, with image of degree one. This is clearly impossible. hence the postulation equality can be expressed by the following simple formula:

h^{0}(I_{B}(m)) = r + 1.

II Genus equality Another very simple observation, already used in example ..., is that the variable intersection of r − 1 general Φ-hypersurfaces D_{1} … D_{r−1} is an integral curve

C′ = D_{1} ∩ ⋯ ∩ D_{r−1}

of genus zero. By Bertini’s theorem the strict transform C of C′ by π_{1}/Γ is a smooth copy of C′. So the genus eqality simply means that we want

p_{a}(C) = 0.

At least when Γ is smooth the genus formula is therefore the usual adjunction formula provided by the standard exact sequence

0 → T_{C} → T_{Γ} ⊗ O_{C} → O_{C}(D1) ⊕ … O_{C}(D_{r−1}) → 0

that is

c(T_{Γ}) = (1 + 2h)(1 + (d_{1}d_{r−1}(r − 1))h)

where d_{1} = deg C, d_{r−1} = m and h is the class of one point on C.

Example 0.2.5. If r = 2 then B is a 0-dimensional scheme and CH_{∗}(B) = Z^{c} where c is the number of connected components of B. In this case s(B, P^{2}) is the length of B, hence the equivalence equality is just d^{2} − 1 = length (B).

Example 0.2.6. Equivalence equality for generic space transformations We first discuss some typical blowing up’s of P^{3}:

(1) Let σ : P → P^{3} be the blowing up of a simple point o and let E = σ^{−1}(o). Then O_{E}(E) ≅ O_{P^{2}}(−1) and hence

(dH − qE)^{3} = d^{3} − q^{3}.

Proof. One can remark that 0 = (H − E)^{3} = H^{3} − E^{3}. hence E^{3} = 1 and the result follows.

(2) Let σ : P → P^{3} be the blowing up of an integral curve C wich is locally complete intersection. Then

−E^{3} = deg(N) = 4deg(C) + 2p_{a}(C) − 2 = deg(N)

where E = σ^{−1}(C) = PN^{∗} and N is the normal bundle of C.

Proof. Recall that [E^{2}] = −[C′] where C′ is an element of | O_{PN}(1) |. This implies that we have the equality of Chern polynomials restricted to C′

c(T_{X}) = c(T_{C′})(1 − Et)(1 + Et).

From this we deduce (4H − E)C′ = 2 − 2p_{a}(C). Since [C′] = −[E^{2}] it follows 4deg(C)+E^{3} = 2−2p_{a}(C). The second equality follows from the standard exact sequence

0 → T_{C} → T_{P^{r}} ⊗ O_{C} → N → 0

(3) In particular (2) implies

(mH − sE)^{3} = m^{3} + 3ms^{2}HE^{2} − s^{3}E^{3} = m^{3} + 3ms^{2}d(degC) + s^{3}(degN)

that is

(mH − sE)^{3} = m^{3} − 3ms^{2}d + s^{3}(2g − 2 + 4d)

with d = deg C, g = genus of C.

(4) Let σ : P → mathbfP^{3} be σ_{o} · σ_{t} where σ_{o} : P_{o} → P^{3} is the blowing up of a point o as in (1) and σ_{t} : P → P_{o} is the blowing up of an element

t ∈| O_{E_{o}}(−E_{o}) |

that is of a line in the plane E_{o} = σ_{o}^{−1}(o). Let F = σ_{t}^{−1}(t), then we have the following formula

(mH − pE − qF)^{3} =

(5) More in general let σ : P → P^{3} be such that

σ = σ_{o} · σ_{t_{1}} · · · · · σ_{t_{k}},

where σ_{o} is as above and σ_{i}, i = 1 … k blows up an element t_{i} of the tautological linear system of the exceptional divisor of σ_{i−1}. Then we have the formula Let Φ be a homaloidal linear system of P^{3} such that

Supp(B) = C_{1} ∪ ⋯ ∪ C_{a} ∪ o_{1} ∪ ⋯ ∪ o_{b} ∪ p_{1} ∪ … p_{c}

where C_{1}, …, C_{a} are smooth integral curves, o_{1} … o_{b} are isolated points defined by an ideal (x, y, z)^{q_{i}}, i = 1 … b and finally p_{1} … p_{c} are isolated contact points for the surfaces of Φ, respectively of order . We assume that Γ is smooth and that

π_{1}/Γ : Γ → P^{3}

has irreducible divisors over C_{1} … C_{a} and ecah point o_{j}. Moreover we assume that over each p_{j} the map π_{1}/Γ is as in (5). then the following formulae hold:

TO BE COMPLETED

Example 0.2.7. Other numerical characters. Definition 0.2.2. The curvilinear genera of a birational map f : P^{r} → P^{r} are the geometric genera

g(f) and g′(f)

of integral plane sections of the hypersurfaces respectively of Φ and Φ′. Let ψ : X → Γ_{f} be a desingularization of Γ_{f} and let

| D | and | D′ |

be the strict transforms by ψ of the linear systems | O_{Γ_{f}} (1, 0) | and | O_{Γ_{f}} (0, 1) |. The g(f) is the arithmetic genus of the zero locus of a general global section of

O_{X}(D_{1})^{r−2} ⊕ O_{X}(D_{2})

So, by adjunction formula, we have

2g(f) − 2 = K_{X}D_{1}^{r−2}D_{2}.

as well as

2g′(f) − 2 = K_{X}D_{1}D_{2}^{r−2}.

If r = 3 it is obvious that g(f) = g′(f). In such a case we will say that g(f) is the genus of the space Cremona transformation f. More in general the same remark implies the following Proposition 0.2.14. Let s ≤ [\frac{r}{2}], then the integral linear sections of codimension s of the hypersurfaces of Φ and of Φ′ have the same birational invariants.

0.3 Lecture 3

0.3.1 Quadro-quadric transformations and syzygies

The name of quadro-quadric transformations is reserved to those Cremona transformations f such that both f and f^{−1} are defined by homaloidal systems of quadrics. They are of special interest. We will see in particular that the classification is affordable in dimension ≤ 5. Moreover quadro-quadrics transformations with smooth and connected base locus appear as beautifully related to the geometry of the four Severi varieties, [?]. Let

γ = h_{1}^{r} + 2h_{1}^{r−1}h_{2} + d_{2}h_{1}^{r−2}h_{2}^{2} + ⋯ + d_{r−2}h_{1}^{2}h_{2}^{r−2} + 2h_{1}h_{2}^{r−1} + h_{2}^{r}

be the rational equivalence class of a quadro-quadric transformation f, we have

d_{i} ≥ 2, i = 1 … r − 1.

by proposition ??. Moreover we know from theorem ?? that

d_{1} = ⋯ = d_{r−1} = 2

if and only if f is an elementary quadratic transformation. Elementary quadratic transformations have therefore minimal multidegree in the set of the multidegrees of all quadro-quadric transformations. We first recollect some basic properties on rational maps defined by a linear system of quadrics For any scheme

B ⊂ P^{r}

we can consider the Hilbert scheme Hilb_{2}(B) of its zero dimensional subschemes Z of length two. For each Z the evaluation map e_{Z} : H^{0}(O_{B}(1)) → H^{0}(O_{Z}(1)) is obviously surjective, therefore the Kernel of r_{Z} defines a line l_{Z} ⊂ P^{r}. We will say that such a line is bisecant to B at Z. This defines a projective bundle

I_{B} = {(x, Z) ∈ P^{r} × B / x ∈ l_{Z}}

over Hilb_{2}(B) and hence a subscheme

Sec(B) ⊂ P^{r}

which is just the projection of I in P^{r}. Sec B is the secant scheme of B or the scheme of secant lines to B. A more subtle object to be considered is the union of the lines l such that f/l is not an embedding. It is easy to define this for every linear system of quadrics Φ = PV, with V ⊂ H^{0}O_{P^{r}}(2)). Consider the morphism of vector bundles

e : H^{0}(I_{B}(2)) ⊗ O_{G} → Sym^{2}U

where U is the universal bundle and e is the natural evaluation map. The degeneration scheme

Δ_{2} ⊂ G

of the map of vector bundles e parametrizes lines l such that e_{l} has rank ≤ 2. We have the inclusions

Δ_{1} ⊆ C(B) ⊆ Δ_{2}

where Δ_{1} parametrizes lines l such that e_{l} has rank ≤ 1 and C(B) is the Chow scheme of B. Note that Δ_{1} is the image of Hilb_{2}(B) under the morphism l → l_{Z}, so Δ_{1} defines Sec(B). Let Δ′ be the Zariski closure of Δ_{2} − C(B), we will put by definition

S(B) = β(α^{∗}(Δ′))

where α : PU → G is the stucture map and β : PU → P^{∇} is the natural morphism of the universal line PU onto P^{r}. Note that B is the scheme defined by V, hence the definition makes sense even if Supp V is empty. The next lemma is an immediate consequence of the definitions

Lemma 0.3.1. Let Φ be any linear system of quadrics of P^{r} and let B be its base scheme. Then the map f defined by Φ is an embedding on P^{r} −(Sec(B)∪S(B)).

Proof. If a subscheme of length two Z ⊂ P^{r} − (B ∪ S(B)) is not separated by f, then f/l_{Z} is a morphism but not an embedding on l_{Z}. Hence l_{Z} ⊂ S(B).

So the main question for us is

Problem 0.3.1. Let Φ be a r-dimensional linear system of quadrics and let B be its base locus: when S(B) ≠ P^{r}?

Of course it is possible that S(B) itself is empty: we will see that this happens in many good situations. Before we want to see one example.

Example 0.3.1. Take for instance r = 3 and the homaloidal linear system of quadrics passing simply through 3 distinct points p_{1}, p_{2}, p_{3} and having a common tangent plane P at o. The birational map f : P^{3} → P^{3} defined by Φ maps a general plane P to the a Steiner quartic surface

S = f(P) ⊂ P^{3},

S contains three double lines: they are images of three lines of P. These lines are in S(Φ) and not in Sec(Φ) which is a surface union of planes in G. One can show that

S(B) =< p_{1}, p_{2}, o > ∪ < p_{1}, p_{3}, o > ∪ < p_{2}, p_{3}, o > .

On the other hand Sec(B) is the union of P and of the six edges of the tetrahedron defined by o, p_{1}, p_{2}, p_{3}. In the Grassmannian G of lines of P^{3} Δ_{1} is a reducible surface of class (7, 3): Sp_{1} +Sp_{2} +Sp_{3} +So union the dual planes of the previous three ones. It is a degeneration of an Enriques surface. Note that in this case S(B) does not contain Sec(B) because it does not contain P.

Let B ⊂ P^{r} be any projective subscheme. The Green’s condition N_{2} for B can be defined as follows: (i) B is projectively normal, (ii) its homogeneous ideal is generated by the quadrics q_{0} … q_{s}, (iii) all the relations between the q_{i}’s are generated by the linear syzygies. This condition guarantes that S(B) is empty:

Theorem 0.3.2. Assume that a linear system of quadrics Φ has a base locus B which satisfies the N_{2} condition. Then S(B) is empty.

Proof. The following condition K_{2} is weaker than N_{2}: (i) and (ii) hold and: (iii)’ all the Koszul relations between the q_{i}’s are generated by the linear syzygies. Condition (iii)’ is obviously preserved by restriction to any linear subspace of P^{r}, in particular to each line. Assume that S(B) is not empty, then U := Δ′ −C(B)∩ Δ′ is a dense open subset of Δ′. Let L ∈ U then f/L : L → f(L) is a morphism of degree two defined by q_{0}/L = x^{2}, q_{1}/L = y^{2}. The Koszul complex of x^{2}, y^{2} is not defined by linear terms, differently from the case x^{2}, y^{2}, xy. So L cannot exists and U = S(B) = ∅

Example 0.3.2. (Applications ).

(1) The property N_{2} holds for the embedding of a curve C in P^{r} of genus g and degree ≥ 2g + 3. So one can check when for such a curve C the linear system Φ of quadrics containing C has the same dimension of the ambient space: this condition, as we have already observed, is necessary so that Φ is homaloidal. Under the assumptions N_{2} and r ≥ 4 the condition is also sufficient because then Sec(C) ≠ P^{r}. One computes that the only possible case is when C is a quintic elliptic curve in P^{4}.

(2) The property N_{2} holds also for a 2-Veronese embedding B of P^{s} in P^{r}, r = \binom{s+2}{2} − 1. The only possible case where h^{0}(I_{B}(2)) = r + 1 is for s = 2 and r = 5. In this case Sec(B) ≠ P^{5} and we have again an example of Cremona transformation defined by a linear system of quadrics.

0.3.2 Further elementary properties

To afford the classification of all quadro-quadric transformations one can try to make effective a more general inductive classical method, which appears to be quite useful when dealing with quadrics. The general basic idea is the following:

let f : P^{r} → P^{r} be defined by the homaloidal linear system Φ and let Q be a general element of Φ. Then f(Q) is a hyperplane P ⊂ P^{r}. Moreover Q is integral and rational hence there exists a birational map g : P^{r−1} → Q. Therefore the composite map

f · g : P → P

is a Cremona transformation of the r − 1 dimensional projective space P. In particular the base locus of Φ can be described, at least in principle, if there exist sufficient informations from the classification of Cremona transformations of P^{r−1} and if the structure of the birational map g is known.

We assume now that f is a (2, n) Cremona transformation. We want to discuss some elementary properties with the purpose of classifying quadro-quadric Cremona transformations in P^{4} and P^{5}.

Proposition 0.3.3. Let b be a base point of Φ, then b is smooth for the general quadric Q ∈ Φ.

Proof. Assume that b ∈ Sing Q for each Q ∈ Φ and fix a general point x ∈ P^{r} − B. Since f is birational the base locus of the linear system Φ_{x} := {Q ∈ Φ / x ∈ Q} is B∪x. On the other hand each Q ∈ Φ is a cone of vertex b, therefore the line < bx > is contained in each Q ∈ Φ_{x} and hence in the base locus of Φ_{x}. Since < bx > is not in B this is a contradiction.

Let o ∈ P^{r} consider the polar linear map

p_{o} : Φ →| O_{P^{r}}(1) |

which associates to Q the polar hyperplane of Q with respect to o. Such a map is not defined at Q if and only if o ∈ Sing Q. Notice also that o ∈ Sing Q for some Q exactly in the following two cases: (i) o is a base point of Φ, (ii) f is ramified at o. This implies the next two statements.

Lemma 0.3.4. o is a point where f is biregular iff p_{o} is an isomorphism

Proposition 0.3.5. Let q_{0}, …, q_{r} be a basis of V ⊂ H^{0}(O_{P^{r}}(2)) and let f be the rational map defined by PV. Assume that f is a Cremona transformation then the fundamental locus of f is defined by the matrix of linear forms

(\frac{∂q_{i}}{∂x_{j}}).

Let

o ∈ P^{r}

be a point where f is biregular and moreover such that f(o) ≠ o. In particular o is not in B and hence, by Bertini theorems, a general hyperplane

T ⊂ P^{r}

passing through o intersects properly each schematic component of B. Since the polar map p_{o} is an isomorphism there exists a unique quadric Q ∈ Φ such that T is the tangent hyperplane to Q at o. Let

P := f(Q)

then P is a hyperplane through f(o). Note that the map Q → P establishes a projective isomorphism between the linear system of quadrics Q ∈ Φ passing through o and the linear system of hyperplanes through f(o). This follows because f is biregular at o. Since o ≠ f(o) we can choose Q so that o is not in P. Let

σ : Q → P

be the stereographic projection of center o. We know very well this map and its inverse: σ^{−1} blows up the quadric

Q_{P} := Q ∩ T ∩ P

and then blows down the strict transform by this blowing up of the hyperplane

T_{P} := T ∩ P.

Let U = Q − (Q ∩ T) and V = P − Q_{P}, then the map

σ : U → V

is biregular and we can conclude that

Lemma 0.3.6. σ : B ∩ U → σ(B) ∩ V is biregular.

Thus, in the case of a Cremona transformation defined by quadrics, most of the essential informations on B are preserved in σ(B) ∩ V, in particular it is worth to study the Cremona transformation

f · σ^{−1} : P → P.

Fix projective coordinates (t_{0} : ⋯: t_{r−1}) on P and (x_{0} : ⋯: x_{r}) on P^{r}. Then σ^{−1} is defined by the equations

x_{0} = p_{0}(t_{0} …, t_{r−1}), …, x_{r} = p_{r}(t_{0}, …, t_{r−1})

where p_{0}, …, p_{r} are quadratic forms vanishing on Q_{P}. On the other hand f is defined by the quadratic forms q_{0}, …, q_{r} in x_{0}, …, x_{r} and we can assume that q_{r} is the equation of Q. Therefore f · σ^{−1} is defined by the quartic forms

s_{0} = q_{0}(p_{0}, …, p_{r}), …, s_{r} = q_{r}(p_{0}, …, p_{r}).

s_{0}, …, s_{r} span a r − 1-dimensional linear system

Ψ ⊂| I_{Q_{P}} (4) |

which defines f · σ^{−1} and possibly has some fixed components. This clearly implies that the classification of (2, n)-Cremona transformations of P^{r} can be in principle rebuilt from the classification of all homaloidal linear systems of degree ≤ 4 of P^{r−1}. We will apply these remarks in the next series of examples. It is also useful to remark the following fact:

Proposition 0.3.7. Let (1, 2, …, d_{r−1}, 1) be the multidegree of f, then the multidegree of f · σ^{−1} : P → P is (1, d_{2}, d_{3}, …, d_{r−1}, 1).

Proof. d_{k} is the degree of the variable intersection of k quadrics of the linear system Φ. Let Φ_{Q} be the restriction of Φ to Q and let M be its moving part, then M is the strict transform by σ of the homaloidal linear system Φ_{P} defining f · σ^{−1}. Obviously the variable intersection Z of k elements D_{1}, …, D_{k} of M is the variable intersection of k + 1 elements of Φ. Hence its degree is d_{k+1}. Note that, for each D ∈ Φ_{Q}, σ_{∗}D contains Q_{P} = T ∩ P ∩ Q. Therefore Q_{P} is in the base locus of Φ_{P}. The variable intersection of σ_{∗}D_{1}, …, σ_{∗}D_{k} is σ_{∗}Z. Hence d′_{k} = d_{k+1}.

0.3.3 Space Cremona transformations of type (2, n)

This case can be easily afforded as follows: observe that the strict transform of a line by f is an integral rational curve R which is contained in the complete intersection of two integral quadrics. Notice also that the degree of R is n so that n ≤ 4. The case n = 1 is impossible because f would be a linear automorphism, see proposition ??. Hence n ≥ 2 and only the following cases are possible:

- R is a smooth conic.

- R is a smooth twisted cubic.

- R is a quartic with one node.

For each of the previous cases the Hilbert scheme of the corresponding graphs is irreducible, in particular it defines a class

γ = h_{1}^{3} + 2h_{1}^{2}h_{2} + dh_{1}^{2}h_{2} + h_{2}^{3}

with d = degR. Denoting such a Hilbert scheme as H_{d} it is possible to describe it geometrically, proving many of its properties as well as the construction of the corresponding Cremona transformations. We are going to sketch this. It is also interesting to address the following:

Problem 0.3.2. What elements of H_{d} correspond to Cremona transformations defined by homaloidal polynomials?

TYPE (2, 2) Elementary quadratic transformation. Description: B is the union of a conic and a point. The family depends on 11 parameters. f^{−1} is in the same family. Construction: see the previous examples and the section on elementary quadratic transformations. Homaloidal polynomial: B = C ∪ o where C is a conic and o a point. If o is not in < C > then f is defined by the reducible homaloidal polynomial F = F_{1}F_{2} where F_{1} is a linear equation of < C > and F_{2} a quadratic equation of the cone of vertex o and directrix C.

TYPE (2,3) Quadro-cubic transformation. Description: For a general f B is the union of a line and three simple points. The family depends on 13 parameters. Φ′ is the linear system of the cubics passing with multiplicity ≥ 2 through a line. A general element of Φ′ is the general projection of rational cubic scroll of P^{4}. Construction: The complete intersection of two integral quadrics containing a smooth rational normal cubic R is B_{1} ∪ R where B_{1} is a line. Let I be the ideal sheaf of B_{1}, then | I(2) | and defines a rational map

β : P^{3} →| I(2) |^{∗}:= P^{6}

whose image is rational normal scroll Q′ of degree 4. This implies that

Q′ = P(O_{P^{1}}(2) ⊕ O_{P^{1}}(1)^{⊕2})

and that O_{Q′}(1) is the tautological bundle. Φ is a subspace of codimension 4 in | I(2) | hence its orthogonal space is a plane

Φ^{⊥} ⊂ P^{6}.

Finally the linear projection

α : Q′ → P^{3}

is birational and α · β is the Cremona transformation f. To study these type of space transformation is therefore equivalent to study the birational linear projections of Q′ with center a plane. Let

Z = ϕ^{⊥} · Q′

then Z is not empty: otherwise f would have degree 4. We have the following possibile cases:

(1) Z = Φ^{⊥}: impossible because Φ has no fixed component.

(2) Z contains an integral curve C. Note that β = i · σ^{−1}, where i is the inclusion of Q′ in P^{6} and

σ : Q′ → P^{3}

is the blowing up of the line B_{1}. In particular the exceptional divisor of σ is a smooth quadric

E ⊂ Q′ ⊂ P^{6}.

Since f = α · β it follows that

C ⊂ E ∩ Φ^{⊥}.

Assume that σ_{∗}C = B_{1}: this implies that any two quadrics D_{1}, D_{2} ∈ Φ are not transversal at each point of B_{1}. But then D_{1} · D_{2} contains B_{1} with multiplicity 2 and the residual intersection R has degree 2: a contradiction. Hence σ(C) is a point and C is a line in the quadric E. Let

γ : Q′ → P^{4}

be the linear projection of center C, it is easy to see that the image of γ is a 2dimensional cubic scroll. Hence γ is not birational. On the other hand f clealy factors through: a contradiction.

(3) Z is 0-dimensional. Then the length of Z is 3. So we have now to stratify all the elements

Z ∈ Hilb_{3}(Q′)

as above via the action of Aut(Q′), (which is the subgroup of P GL(4) of the linear automoprhisms fixing a line). Fixing projective coordinates (x : y : z) in the plane < Z > the list can be given as follows:

- For the ideal I_{Z} we have I_{Z} = (xy, xz, yz) or (x^{2}, yz, xz) or (x^{2}, y^{2}, xy).

- For the length l of E · Z we have 0 ≤ l ≤ 2. (l = 3 implies < Z >⊂< E > which is impossible). Combining the two conditions one obtains all possible cases. Homaloidal polynomials: This case is interesting. Assume that F is an irreducible cubic polynomial the polar map of which defines f. Then the cubic surface X defined by F has a double line, (since X is irreducible no tiple line is possible). Assume that SuppZ ∩ Q′ − E contains o′. Then sigma(o′) = o is a singular point of X: this is again impossible if X is irreducible. So we must have

Supp(Z) ⊂ E and < Z > ∩ < E >= a line.

The latter equality is clear if SuppZ consists of two points. If it is one point then < Z > ∩ < E > is a line because othewise Q′ would be singular. To conclude it is now convenient to consider the polar linear system Φ of F, then to take its strict transform Φ′ on Q′ and then its restriction Σ to E. Φ′ is a linear system of curves of type (1, 2) on E = P^{1} × P^{1}. For F general it has two simple base points: these corresponds to the only two uniplanar double points of X, (uniplanar simply means that the tangent cone is x^{2}). We leave the details for the moment. Since they are simple there is no hope that a general F is homaloidal. But we can specialize to have only one uniplanar double point along the singular line B_{1} of X. Actually this happens:

(1) the strict transform of X′ by σ intersects E along a conic B′_{1} = E · X′.

(2) If B′_{1} has a node o′ then the two uniplanar double points of X coincide. One can check that the base locus of Σ has length 2. Moeove thee is still one direction more where the elements of Φ′ intersect. Hence the base locus of Φ′ has length 3 and it is of the required type.

TYPE (2,4) Quadro-quartic transformation. B is the union of the 4 point o, o_{1}, o_{2}, o_{3}. The length of B is 7. o_{1}, o_{2}, o_{3} are simple base points. The family depends on 15 parameters. The faces < o, o_{1}, o_{2} >, < o, o_{2}, o_{3} >, < o, o_{1}, o_{3} > of the tetrahedron are contarcted to the union of 3 lines l_{1}, l_{2}, l_{3} intersecting in one point. Φ′ is the linear system of quartic surfaces passing with multiplicity ≥ 2 through l_{1} ∪ l_{2} ∪ l_{3}. A general element of Φ′ is a general projection of the Veronese surface of P^{5} i.e. it is a Steiner quartic surface. Homaloidal polynomials: take the list of integral cubic surfaces with isolated singularities and check that never the polar linear system has degree two. The E_{6} surface achieves the best degree: degree two.

To use the above mentioned ’principle’ one has to consider, with the same notations, the plane Cremona transformation

σ^{−1} · f : P → P.

In our situation f · σ^{−1} is defined by a homaloidal net of curves of degree n ≤ 4. The classification of these linear system is well known: (1) n = 1 net of lines.

(2) n = 2 net of conics with 3 simple base points.

(3) n = 3 net of cubics with 6 simple base points and 1 double base point.

(4) n = 4 net of quartics with 3 simple base points and 3 double base points.

(5) n = 4 net of quartics with 5 simple base points and 1 triple base point.

Discussing each of the previous case one retrieves the previous result. We leave this to the reader, noting that this approach will be not completely useless in te forthcoming example.

0.3.4 The quadro-quadric transformations of P^{4}

Let r = 4, we keep our usual notations and consider again the space Cremona transformation

f · σ^{−1} : P → P.

It follows from the arguments of the previous example that f · σ^{−1} is a space Cremona transformation of type (m, 2) with

m ≤ 4.

Therefore f · σ^{−1} is just the inverse h^{−1} of one of the three Cremona transformations

h : P → P

considered in the previous example. Let us discuss these three possibilities:

h OF TYPE (2,2). We have h = σ · f^{−1}: let B′ be the indeterminacy scheme of f^{−1} and let B_{h} be the indeterminacy scheme of h, then

B_{h} − f(o) = B′ · P.

Since P is a general hypeplane the intersection B′ · P is proper at each schematic component of B′. Since B_{h} − f(o) is a conic it follows that B′ is the union of a quadric surface S and a zero-dimensional scheme Z. To understand Z we consider the inverse map h^{−1} : P → P: the base locus is the union of a second conic T_{o} ∩ P ∩ Q and of σ(Z) which is one point. Hence Z is a point and B = S ∪ Z. Hence f^{−1} is an elementary quadratic transformation of P^{4} and the same is true for f.

h OF TYPE (2, 3). B_{h} is the union of a line and of three simple points. Hence B contains a plane and a curve Z of degree two. To understand Z we study its projection σ(Z): this is in the base locus of h^{−1}. The multidegree of h is (1, 2, 3, 1).

Hence the multidegree of h^{−1} is (1, 3, 2, 1) and the variable intersection of two cubic sufaces of the linear system defining h^{−1} is a smooth conic C. These two cubic surfaces F_{1}, F_{2} intersect along the following curve:

L ∪ L′ ∪ C ∪ σ(Z)

where L = Sing F_{1} = Sing F_{2} is a line and L′ is a second line. To see this obseve that F_{i} = σ(R_{i}) where R_{i} ⊂ Q is a rational cubic scroll. Moreover T_{o} · Q = Π ∪ Π′, the union of two planes intersecting a long a line containing o and such that: Π·F_{i} = a conic, Π′ ·F_{i} = a line. Both these curves does not contain o and their projection via σ are respectively L and L′. Hence σ(Z) is the union of two skew lines and Z is also union of two skew lines. We can see the projected scheme σ(B) as the union of two skew lines and a plane. Provided no embedded point exists the base locus of the homaloidal system of cubics defining h is L ∪ L′ ∪ σ(Z), L with multiplicity four. Conclusion: Φ is the linear system of all quadrics passing through Mcupl_{1} ∪ l_{2} where M is a plane and l_{1}, l_{2} are skew lines intersecting it in one point. The multidegree of f is

(1, 2, 3, 2, 1).

Note that the space < l_{1}, l_{2} > is contracted to a plane and that the spaces < M, l_{1} > and < M, l_{2} > are contracted to two skew lines intersecting f(< l_{1}, l_{2} >). The construction is self dual and f^{−1} is of the same type.

Remark 0.3.1. We point out that f factors through β : P^{4} → P^{2} × P^{2} and a kind of ’stereographic’ projection α : P^{4} × P^{4} → P^{4} centered at the strict transform of the base plane by β: see generalization at the end.

h OF type (2, 4). In this case h : P → P is defined by the linear system of quadrics with 3 simple base points o_{1}, o_{2}, o_{3} and tangent to a plane in a fixed point o. f(o) is one of the simple base points, say o_{3}. This implies the base locus B′ of f^{−1} is such that

B′ · P = {o_{1}, o_{2}, o}

where o is defined by the ideal (x, y^{2}, z^{2}) and o_{1}, o_{2} are smooth. We deduce from this that B′ contains a double line of ideal (x, y^{2}, z^{2}) and a smooth curve of degree 2. Hence B′ is not the base locus of one of the two previous quadro-quadric transformations and therefore f^{−1} is in the same family of f. Let us consider now the inverse map h^{−1}: this is defined by the linear system of Steiner surfaces passing doubly through 3 edges l_{1}, l_{2}, l_{3} of the same vertex of a tetrahedron and simply through a smooth conic c. Hence σ(B) is contained in l_{1} ∪ l_{2} ∪ l_{3} ∪ c. On the other hand the conic b_{o} = T_{o} ∩ P ∩ Q is a component of σ(B). We have the following cases:

- b_{o} = c, then Q is smooth. Let F be a general surface of the system defining f · σ^{−1} then σ^{−1}(F) is a complete intersection. Let Ψ be the homaloidal linear system defining f · σ^{−1} and let ψ : Ψ →| O_{F} (2) | be the restriction map. Let us study the linear system Imψ: the normalization of F is P^{2}, the pull-back of | Imψ | is a linear system of quartics with c as a fixed component. Hence its free part is a linear system of cubics of P^{2}. We claim that this has 3 simple base points: indeed let S be a smooth quadric through < c > then

S · F = c + h

where h pulls back to one of these cubics. h is a curve of type (3, 3) on S passing simply through the points u = l_{1} ∩c, v = l_{2} ∩c, t = l_{3} ∩c. u, v, t pull-back via the normalization map to points u′, u”, v′, v”, t′, t” with the obvious meaning of the notations. Then u, v, t belong to the pull-back of h and u′, v′, t′ to the pull-back of c. Since the base locus of Im ψ consists of 3 simple base point the image of F by σ^{−1} is a sextic surface, hence a (2, 3) complete intersection. Since σ^{−1}(F) must have degree 4 this case is excluded.

- b_{o} = l_{i} ∪ l_{j}, i ≠ j. This case is possible: the general element of | Imψ | is a nodal quartic curve of type (2, 2) on S. | Im ψ | is base-point-free on F: the image is the complete intersection of two quadrics Q, Q′ such that Sing Q′ is a line contained in Q.

- b_{o} = 2l_{i} for some i. This is just a specialization of the previous case. The conclusion is the following:

B = L ∪ C

where L is a double line of equations (x, y^{2}, z^{2}) and C is a conic tangent to {x = 0} at one point L ∩ C.

Remark 0.3.2. Note that each Cremona transformation (2, n) of a hyperplane P ⊂ P^{4} ’lifts’ to a Cremona transformation of P^{4} as follows: fix p ∈ P^{4} − P and the cone C(B, p) over the base locus B of the space transformation f : P → P we are considering . The homaloidal linear system Ψ defining f lifts to a web of quadric cones of vertex p. Add to such a linear system one more quadric which is smoot at p. The linear system spanned by this quadric and the web is homaloidal. This trasformation is not quadro-quadric unless n = 2.

Theorem 0.3.8. There are exactly three irreducible families of quadro-quadrics transformations of P^{4}: (I) (1, 2, 2, 2, 1) Elementary quadratic transformation. (II) (1, 2, 3, 2, 1) B is the union of a plane and two skew lines intersecting it in one point. (III) (1, 2, 4, 2, 1) B = L ∪ C where C is a smooth conic and L is a double line contained in a hyperplane H. Moreover C ∩ L is one point and H is tangent to C at L ∩ C.

0.3.5 (2, n) Cremona transformations of P^{4}

After the warming up of the last example we go further in the discussion of Cremona transformations of P^{4} which are defined by quadrics. Recall that for the multidegree we have (1, 2, d_{2}, n, 1) so that, by Cremona inequalities,

d_{2} ≤ 4, n ≤ 8.

We will apply again the previous ’principle’ and moreover the classification of homaloidal linear systems of plane curves of degree ≤ 6. Let f : P^{4} → P^{4} be a Cremona transformation defined by the homaloidal linear system of quadrics Φ and let Q_{1}, Q_{2} ∈ Φ be two general elements. Then

Q_{1} · Q_{2} = S ∪ T

where S is an integral surface of degree d_{2} ≤ 4 and T is the residual intersection contained in the base locus of Φ.

Lemma 0.3.9. For the surface S the following cases are possible:

(1) S is a Del Pezzo surface of degree 4.

(2) S has a double line and it is a projection of a Veronese surface V ⊂ P^{5} from a point lying in the plane spanned by a conic contained in V.

(3) S has a double line and it is a projection of a rational quartic scroll F ⊂ P^{5} from a point lying in the plane spanned by a conic contained in F.

(4) S is a non degenerate cone over a rational quartic with one double point.

(5) S is a non degenerate smooth cubic scroll.

(6) S is a non degenerate cone over a rational cubic curve,

(7) S is a quadric.

Proof. S ∪ T is the base locus of a pencil of quadrics. The lemma then follows from the classification of pencils of quadrics: see Hodge and Pedoe.

For each S in the above list we can fix a (non necessarily minimal) desingularization

τ : S′ → S  (16)

endowed with a birational morphism

σ : S′ → P  (17)

with the projective plane P = f(S). Let

τ^{∗} : Φ →| τ^{∗}O_{S}(2) |

be the pull-back map, then its image is a net of curves

Φ_{S′} ⊂| τ^{∗}O_{S}(2) |  (18)

whose associated map is the birational map f · τ : S′ → P. On the other hand σ defines a linear embedding

σ_{∗} : S′ → P

hence a net Φ_{P} of plane curves of degree d. Moreover σ : S′ → P is the contraction of the exceptional divisors E_{1}, …, E_{k} to the points o_{1} = σ(E_{1}), …, o_{k} = σ(o_{k}). Let m_{i} be the multiplicity of a general curve of Φ_{P} at o_{i}, then we can write

Φ_{S′} = Φ′_{P} + E

where

Φ′_{P} ⊂| dL − m_{1}E_{1} − ⋯ − m_{k}E_{k} |

is the strict transform of Φ_{P} and E is the maximal fixed part of Φ_{S′} which is contracted by σ, (as usual L is the pull-back of a line by σ).

Lemma 0.3.10. For each surface S in the previous list we can choose σ : S′ → P so that

Φ_{P} ⊂| O_{P} (d) |

with d ≤ 6.

Proof. This will be proved case by case in the next part.

0.3. LECTURE 3 59 It is clear that the moving part

N_{P}

of Φ_{P} is the homaloidal net of plane curves of degree d ≤ 6 which defines the plane Cremona transformation

f · τ · σ^{−1} : P → P.

The strict transform of N_{P} is a net N′_{P} such that

N′_{P} + B′_{1} = Φ_{S′}

where B′_{1} is the fixed part of Φ_{S′}. Hence

B_{1} := τ_{∗}B′_{1}

is a Weil divisor and it is the cycle associated to the component of dimension one of B. Fortunately, the cases to be discussed are relatively simple. We rely on the following result: let N be a homaloidal net of plane curves of degree d ≤ 6. Then, as a point of the Grassmannian of planes of PH^{0}(O_{P^{2}}(d)), N is an irreducible closed set whose general point is in the following list (see Hudson’s book and others):

(1) | 6h − 3(e_{1} + e_{2}) − 2(e_{3} + e_{4} + e_{5} + e_{6}) − e_{7} |

(2) | 6h − 3(e_{1} + e_{2} + e_{3}) − 2e_{4} − (e_{5} + e_{6} + e_{7} + e_{8}) |

(3) | 6h − 4e_{1} − 2(e_{2} + e_{3} + e_{4} + e_{5}) − (e_{6} + e_{7} + e_{8}) |

(4) | 6h − 5e_{1} − (e_{2} + e_{3} + e_{4} + e_{5} + e_{6} + e_{7} + e_{8} + e_{9} + e_{10} + e_{11}) |

(5) | 5h − 2(e_{1} + e_{2} + e_{3} + e_{4} + e_{5} + e_{6}) |

(6) | 5h − 3e_{1} − 2(e_{2} + e_{3} + e_{4}) − e_{5} − e_{6} − e_{7}) |

(7) | 5h − 4e_{1} − (e_{2} + e_{3} + e_{4} + e_{5} + e_{6} + e_{7} + e_{8}) |

(8) | 4h − 2(e_{1} + e_{2} + e_{3}) − (e_{4} + e_{5} + e_{6}) |

(9) | 4h − 3e_{1} − (e_{2} + e_{3} + e_{4} + e_{5} + e_{6} + e_{7}) |

(10) | 3h − 2e_{1} − (e_{2} + e_{3} + e_{4} + e_{5}) | (11) | 2h − (e_{1} + e_{2} + e_{3}) |

(12) | h |

Finally, for each type of surface S, we will be able to choose the previous birational morphism σ : S′ → P so that the following property holds: σ is the contraction of the exceptional divisors E_{1}, …, E_{k} and moreover

| τ^{∗}O_{S}(2) |=| dH − n_{1}E_{1} − ⋯ − n_{k}E_{k} |  (19)

where d ≤ 6 and H is the pull-back of a line of P. But then the net of plane curves

Φ_{P} := σ_{∗}Φ_{S′}

defines the birational map

f · τ · σ^{−1} : P → P

and moreover its moving part

N_{P}

is a homaloidal net of curves of degree d ≤ 6. So far our net N_{P} is a, non necessarily general, point in one of the previous irreducible families of homaloidal nets. We discuss each possible case. In particular we show that the choice of N_{P} can be sufficiently general. This is a more difficult point still to be completed. However we can make the list of all the possible examples where N_{P} is general.

CASE (1) S is a quartic Del Pezzo surface. Let τ : S′ → S be a minimal desingularization, then

| τ^{∗}O_{S′}(2) |

is the pull back by a birational morphism

σ : S′ → P

of a linear system whose general member is an integral plane sextic of geometric genus 5. Moreover such a curve is not trigonal nor hyperelliptic, hence all its singularities have multiplicity two. (All this facts are known: see classification of Del Pezzo and [ACGH]).

For the moment we assume that S is smooth. Then we can assume that σ blows five distinct lines down to five points on a smooth conic and that S = S′, τ = id. Let D ∈ Φ_{S} then the curve

σ_{∗}D

is a, possibly reducible, sextic:

this sextic moves in Φ_{P} and has five fixed distinct points o_{1} … o_{5} of multiplicity ≥ 2,

as any curve of σ_{∗} | O_{S}(2) |. If Φ_{P} has fixed components then:

it is not forbidden that some of these singular points of σ_{∗}D are on the fixed component and not in the base locus of N_{P}.

Let us discuss the possible cases:

(1.1) σ_{∗}D is an integral sextic. Cases (2) and (4) are not possible. Cases (1), (3) are possible.

(1.2) Φ_{P} = N_{P} + M, M a line. Hence N_{P} is a homaloidal net of plane quintics. Case (5), (6) are possible. (7) is not possible. In case (6) M is must contain one of the simple base point of the net.

(1.3) Φ_{P} = N_{P} + M, M a conic. Assume M is smooth. Then (8) is possible: M contains two simple base points. (9) is possible, M contains 5 of the simple base points. Assume M is singular: (8) and (9) are again possible: see the next discussion.

(1.4) Φ_{P} = N_{P} + M, M any cubic. The only case (10) is possible. Assume M is smooth. M contains 4 of the simple base points. See the next discussion for the singular cases.

(1.5) Φ = N_{P} + M, M a quartic. The only case (11) is possible. M needs to have two double points distinct from the three base points of N_{P}.

(1.6) Φ = N_{P} + M, M a quintic. M needs to have 5 distinct double points. So this case (12) is still possible.

We ill have various repetitions, as it happens with this method. We briefly sketch the construction of every possible linear system

Φ_{S} ⊂| O_{S}(2) |

from the above list. We will also sketch the corresponding Cremona transformation. Then we will afford the case of S singular. D denotes the general moving element of Φ_{S} i.e. the vaiable intersection of 3 general elements of Φ. B the base locus of Φ. B_{i} the i-dimensional component.

(1.1.1) N_{P} as in (1). {o_{1} … o_{5}} = {e_{1}e_{2}e_{3}e_{4}e_{5}}. p_{a}(D) = 1. deg(D) = 6. B_{1} = E_{1} ∪ E_{2}. Supp(B_{0}) = {σ^{−1}(e_{6}, e_{7}}). M denotes the fixed curve of Φ_{P}. M′ is its strict transform on S. - Φ is the linear system of quadrics containing two skew lines E_{1}, E_{2}, a simple point σ^{−1}(e_{7}), a fat point of length 3 σ^{−1}e_{7}).

- σ^{−1}(e_{7}) has ideal (x^{2}, y^{2}, xy), where (x : y : z) are projective coordinates on the tangent plane to S at σ^{−1}(e_{7}).

(1.1.2) N_{P} as in (1). {o_{1} … o_{5}} = {e_{2}e_{3}e_{4}e_{5}e_{6}}. p_{a}(D) = 3. deg(D) = 7. B_{1} = E_{1}. Supp(B_{0}) = {σ^{−1}(e_{1}, e_{7}}).

- Φ is the linear system of quadrics containing a line E_{2}, a simple point σ^{−1}(e_{7}), a fat point of length 6 σ^{−1}(e_{1}).

- σ^{−1}(e_{7}) has ideal (x^{3}, xy^{2}, x^{2}y, y^{3}), where (x : y : z) are projective coordinates on the tangent plane to S at σ^{−1}(e_{7}).

(1.1.3) N_{P} as in (3). {o_{1} … o_{5}} = {e_{1}e_{2}e_{3}e_{4}e_{5}}. p_{a}(D) = 0. deg(D) = 6. Supp(B_{1}) = E_{1}. Supp(B_{0}) = {σ^{−1}(e_{6}, e_{7}, e_{8}}).

- Φ is the linear system of all quadrics containing three simple base points and a double line L whose ideal is the square of the ideal of Supp(L). -Furter remarks: D ∈| 2(H−E_{1}) | is linearly equivalent to twice a rational normal cubic. Every smooth, non degenerate rational sextic D ⊂ P^{4} is contained in two quadrics. E_{1} is a 4-secant line to D: the general smooth rational sextic in P^{4} has not this property. A smooth, rational sextic endowed with a quadrisecant line defines a three selected points defines a Cremona transformation of this type.

(1.2.1) N_{P} as in (5). {o_{1} … o_{5}} = {e_{1}e_{2}e_{3}e_{4}e_{5}}. p_{a}(D) = 1. deg(D) = 5. B_{1} = σ^{−1}(M). Supp(B_{0}) = σ^{−1}(e_{6}).

- Φ is the linear system of quadrics containing the smooth rational normal cubic σ^{−1}(R) and a fat point of length 3 σ^{−1}(e_{6}).

- σ^{−1}(e_{6}) has ideal (x^{2}, y^{2}, xy), where (x : y : z) are projective coordinates on the tangent plane to S at σ^{−1}(e_{6}).

(1.2.2) N_{P} as in (6). {o_{1} … o_{5}} = {e_{1}e_{2}e_{3}e_{4}e_{5}}. p_{a}(D) = 0. deg(D) = 3. B_{1} = E_{1} ∪ E_{5} ∪ σ^{−1}(M). B_{0} = σ^{−1}({e_{6}, e_{7}}). e_{5} ∈ M: σ^{−1}(M) = M′ ∪ E_{5}, M′ = strict transform of the line M.

- Φ is the linear system of all quadrics containing B = C ∪L∪p∪q. B is smooth. C is a conic. L is a line. p, q two points.

(1.3.1) N_{P} as in (8). {o_{1} … o_{5}} = {e_{1}e_{2}e_{3}e_{4}e_{5}}. p_{a}(D) = 0, deg(D) = 4. B_{1} = M′. B_{0} = σ^{−1}(e_{6}).

- Φ is the linear system of all quadrics containing B = C ∪ p. B is smooth. C is a rational normal quartic. p is a point.

- Further remarks: the linear system of all quadics though C birationally embeds P^{4} in P^{5}. The image is the Grassmannian G of lines of P^{3}, (prove this), i.e. a smooth quadric. f factors through such a birational embedding β and the stereographic projection of G fom β(p).

(1.3.3) N_{P} as in (8). {o_{1} … o_{5}} = {e_{1}e_{2}e_{3}e_{4}n_{5}}. p_{a}(D) = 0, deg(D) = 3. B_{1} = M′. B_{0} = σ^{−1}{e_{5}, e_{6}}. n_{5} = Sing(M). M rank two conic. -Φ is again the linear system of all quadrics passing through B = C ∪ L ∪ p ∪ q, M′ = C ∪ L is the dijoint union of a line and a smooth conic.

(1.3.3) N_{P} as in (9). {o_{1} … o_{5}} = {e_{1}e_{2}e_{3}e_{4}e_{5}}. p_{a}(D) = 0, deg(D) = 4. B_{1} = M′. B_{0} = σ^{−1}{e_{5}, e_{6}}.

- For each M again as in the previous case (1.3.2).

(1.4.1) N_{P} as in (10). {o_{1} … o_{5}} = {e_{1}e_{2}e_{3}e_{4}e_{5}}. p_{a}(D) = 0, deg(D) = 3. B_{1} = M′. -Φ is the linear system of all quadrics through M′. M′ is an elliptic normal quintic curve. See: previous and subsequent comments on this case.

(1.4.2) N_{P} as in (11). {o_{1} … o_{5}} = {e_{1}e_{2}e_{3}n_{4}n_{5}}. p_{a}(D) = 0, deg(D) = 3. B_{1} = M′. SingM = n_{4}, n_{5}. M quartic of genus 1. -As previously B is an elliptic normal quintic.

(1.4.3) N_{P} as in (12). {o_{1} … o_{5}} = {n_{1}n_{2}n_{3}n_{4}n_{5}} = Sing(M). -As in the previous case: M′ is an elliptic normal quintic.

Now we discuss the case when S is a singular Del Pezzo surface.

CASE (2) S is a projected Veronese surface with a double line. We can put τ : S′ → S := normalization map. In particular we have

S′ = P^{2} = P

hence we can also put σ := id_{P}. The plane Cremona transformation to be considered is f · τ. We know that τ^{∗}O_{S}(2) ≅ O_{P} (4), hence the moving part N_{P} of τ^{∗}Φ_{S} is a homaloidal net of cubics or of curves of lower degree. Let M be the singular line of S, then M′ = τ^{−1}(M) is a line in P. The equations of our projected 2-Veronese surface S are available:

Lemma 0.3.11. S is the complete intersection of two integal quadrics touching along a line M which are generically non singular along M.

Proposition 0.3.12. Let f be a (2, n) Cremona transformation of P^{4} such that the variable intersection of two elements of Φ is a Veronese surface S as above. Then M is an irreducible component of the base locus of Φ.

Proof. By the lemma a general quadric Q containing S has rank 4. On the other hand, by Bertini theorem, M is contained in the base locus of Φ restricted to Q i.e. in B. Since B has no component of dimension 2, M is an irreducible component.

(2.4.1) N_{P} as in (11), M_{red} a line. M has equations (x, yz, z^{2}, y^{2}). p_{a}(D) = 0, deg(D) = 4.

- Φ is the linear system of all quadrics through the scheme M and three points.

- f · τ/S′ is a plane quadratic transformation. Note the following facts: the vector space W of quadratic forms through M has the basis xu, xv, x^{2}, y^{2}, z^{2}, xy, xz, yz.

τ^{∗} :| W |→ O_{P} (4) |

has image 2M′+ | O_{P} (2) |. All the quadrics of PW have tangent hyperplane x = 0 or are singular along the line M_{red}. Notice also that the singular Veronese S is not contained in any quadric of rank 3 Q such that M = Sing Q.

(2.4.2) N_{P} as in (12), M_{red} a line. M has equations (x, yz, z^{2}, y^{2}). B = M ∪ N where N is a conic touching M_{red}. p_{a}(D) = 0, deg(D) = 4.

- Φ is the linear system of all quadrics through the scheme M ∪ N.

- f · τ/S′ is a (projective) automorphism of P.

- f is the third type of quadro-quadric transformation: see a previous theorem. The multidegree is (1, 2, 4, 2, 1).

CASE (3) In this case the normalization map τ : S′ → S gives the Hirzebruch surface

S′ = F_{0} orF_{2}.

We leave the details since this case is analogous to the previous one: take the same linear system of quadrics

PW

considered above. then fix three general points o_{1}, o_{2}, o_{3} ∈ P^{4} and consider the linear system Φ of all quadrics of PW containing o_{1}, o_{2}, o_{3}. Then the map f defined by Φ is birational: f ⋯ τ is the projection of S′ embedded as a rational normal quartic scroll from three points. Hence f · τ is birational and the same holds for f.

CASE (4) As in case (3) and (4): using the obvious notations, the normalization τ : S′ → S is biregular to a cone F_{4} over a rational normal quartic curve. S is the projection of such a cone from a point spanned by two lines of F_{4}. Consider again the linear system PW and fix three general points as above: this yelds the only possible Cremona transformation. (To be performed ).

CASE (5)

CASE (6)

TO BE COMPLETED

0.4 Lecture 4

0.4.1 Examples of quadro-quadric transformations of P^{5}

0.4.2 Special Cremona transformations

All the previous examples have been useful to understand that B is sometimes a quite complicated object. Let f : P^{r} → P^{r} be a Cremona transformation, we know that

σ : Γ → P^{r}

is the blowing up of the ideal I of B. In other words we have

Γ = P roj(⊕I^{n}).

On the other hand it is well known that

P roj (⊕I^{n}) = P roj(⊕J^{n})

where J = I^{s} for some s ≥ 1. We have also seen one example of different type, namely the double structure of degree 6 and arithmetic genus 3 on a rational normal cubic C, where the blowing up of B is biregular to te blowing up of B_{red}. In view of all this a natural question to be considered is the following one:

QUESTION Which Cremona transformations f exist such that B_{red} is smooth and connected and moreover π_{1}/Γ : Γ → P^{r} is the blowing up of B_{red} ?

Definition 0.4.1. A Cremona transformation is special if B_{red} is smooth and connected and if Γ is biregular to the blowing up of B_{red}.

In the following of this section f : P^{r} → P^{r} always denotes a special Cremona transformation. We will use the following notations:

- H_{i} = pull-back of a hyperplane by π_{i}/Γ.

- E_{1} = pull-back by π_{1}/Γ of the indeterminacy scheme B_{1} of f.

- E_{2} = pull-back by π_{2}/Γ of the indeterminacy scheme B_{2} of f^{−1}. In particular we will have B = B_{1}. We want to recollect some properties of special Cremona transformations. Before of doing this we prove a theorem, due to Ein and Shepherd-Barron, giving some useful sufficient conditions so that a birational morphism is the blowing up of a smooth fundamental locus B.

Theorem 0.4.1. Let σ : X → Y be a proper birational morphism between two smooth integral quasi projective varieties. Consider the closed sets

Z = {y ∈ Y / dim f^{−1}(y) ≥ 1} and σ^{−1}(Z).

Assume that Z is smooth and σ^{−1}(V) is an irreducible divisor for each irreducible component V of Z. Then σ is the blowing up of Z and σ^{∗}(Z) = σ^{−1}(Z).

Proof. Let E = σ^{∗}Z and let x ∈ E: the main object to be considered is the map

ψ_{x} : T_{X,x} → N_{Z,f(x)}

which is defined by the commutative diagram

0 ⟶ T_{E,x} ⟶ T_{X,x}
↓  ^{dσ_{x}}↓
0 ⟶ T_{Z,σ(x)} ⟶ T_{Y,σ(x)} ⟶ N_{Z/Y,σ(x)} ⟶ 0

(1) Since E is a divisor and X is smooth the rank of ψ_{x} is ≤ 1.

(2) ∀ < v >∈ PN_{Z,σ(x)} there exists x′ ∈ E such that σ(x′) = σ(x) and

Im ψ_{x′} =< v > .

To prove this note that there exists a smooth integral curve C ⊂ Y passing through σ(x) and such that T_{C,σ(x)} s mapped onto < v > via the map T_{Y,σ(x)} → N_{Z,σ(x)}. This follows just because T_{X,σ(x)} → N_{Z,σ(x)} is surjective. Observe also that T_{C,σ(x)} is transversal to T_{Z,σ(x)}. Let C′ be the strict transform of C and let x′ ∈ C′ ∩ E, then dσ_{x′} : T_{C′,x′} → T_{C,x} is an isomorphism. Hence ψ_{x′} has rank one.

(3) Let

U = {x ∈ E / rk ψ_{x} = 1}.

By semicontinuity U is open and it is non empty by (2). Applying separately the argument in (2) to σ^{∗}V for each irreducible component V of Z, it follows that U is also dense in σ^{−1}(Z).

(4) Each point x ∈ U is smooth for E, otherwise we would have T_{E,x} = T_{X,x} and hence ψ_{x} = 0. Since U is dense in σ^{−1}(Z) it follows that E is reduced, in particular σ^{∗}(Z) = σ^{−1}(Z). We consider now the birational map σ/X′ : X′ → Y where

X′ = (X − E) ∪ U.

Let β : X_{1} → Y be the blowing up of Z, then the exceptional divisor of β is

P := PN_{Z}.

Let The universal property of the blowing up implies that there exists a rational map

γ : X′ → X_{1}

which has the following properties:

(i) σ = β · γ, (ii) γ : X − E → Y − P is biregular. (iii) γ is regular at each x ∈ U and moreover γ(x) = Im ψ_{x} ∈ PN_{Z,σ(x)}. Now, because of (2) and of (iii), the map

γ/U : U → P

is surjective. On the other hand we have σ^{∗}Z = γ^{∗}P and we know that σ^{∗}Z is reduced. Hence γ is generically unramified along U. In particular the ramification locus of γ is contained in the closed set E − U of codimension ≥ 2. Since X′ is smooth such a ramification locus must be a divisor or empty. Hence it is empty and γ does not contract any curve. Then, by Zariski main theorem, γ : X → X_{1} is an isomorphism. Finally β is a proper map and γ is an isomorphism. Therefore β · γ is proper and X′ = (β · γ)^{−1}(Y) is proper in X. Since X′ is also dense it follows X′ = X.

Remark 0.4.1. Let V ⊂ Z be an irreducible component, the assumption that σ^{−1}(V) is irreducible is obviously crucial to deduce that σ is the blowing up of Z.

Corollary 0.4.2. Let f be a Cremona transformation, assume that Γ and B_{red} are smooth and that

B_{red} × P^{r} · Γ

is smooth. Then Γ is the blowing up of B_{red}.

Proof. By Zariski main theorem the fibre over each irreducible component of B_{red} is connected. Hence it is irreducible. Moreover it must be a divisor because ı_{1} : Γ → P^{r} maps onto a smooth variety. The theorem then implies the statement.

Proposition 0.4.3. Γ is smooth and P ic(Γ) = Z[H_{1}] ⊕ Z[E_{1}].

Proof. Since B_{1} is smooth these properties are well known.

Proposition 0.4.4. Let Φ_{1} be the strict transform of Φ by π_{1}/Γ, then

Φ_{1} =| m_{1}H_{1} − s_{1}E_{1} |

where s_{1} is the multiplicity along B_{1} of a general Φ-hypersurface and m_{1} is its degree.

Proof. The inclusion Φ_{1} ⊆| m_{1}H_{1} − s_{1}E_{1} | is obvious. to prove that this is an equality note that Φ_{1} defines the second projection

π_{2}/Γ : Γ → P^{r}.

Hence Φ_{1} is base-point-free. On the other hand an element D ∈ Φ_{1} has selfintersection 1. Therefore the image in | m_{1}H_{1} − s_{1}E_{1} |^{∗} of the morphism defined by | m_{1}H_{1} − s_{1}E_{1} | has degree one. Let p be the morphism defined by | m_{1}H_{1} − s_{1}E_{1} |, then p(Γ) has degree one. Since the image of a morphism is never degenerate, it follows p(Γ) =| m_{1}H_{1} − s_{1}E_{1} |^{∗} and dim | m_{1}H_{1} − s_{1}E_{1} | = dim Γ = r. Hence Φ_{1} =| m_{1}H_{1} − s_{1}E_{1} |.

Since f is special, the indeterminacy of f is solved after the blowing up

β := π_{1}/Γ

and f = β · α, where the morphism

α := π_{2}/Γ

is defined by | m_{1}H_{1} − s_{1}E_{1} |. We want now to study the structure of α. Let

E_{2} := α^{∗}(B_{2,red}).

Since Γ is smooth and E_{2} is the fundamental scheme of α, it follows that E_{2} is a divisor. A priori E_{2} could be reducible or not reduced, the fact that P ic(Γ) ≅ Z^{2} guarantees that this does not happen.

Proposition 0.4.5. Let f : X → Y be a proper birational morphism bewteen two smooth, integral varieties. Assume that P ic(X) = Z^{2} and let E = f^{−1}(Z) where

Z = {y ∈ Y / dimf^{−1}(y) ≥ 1}.

Then E is irreducible.

Proof. Let C ⊂ X be the pull back by f of a curve in Y which is complete intersection of very ample divisors. Then we can assume C ∩ Z = ∅. Let

ϕ : P ic(X) → Z

be the (surjective) degree map with respect to C and let O_{X}(D) be a generator of Ker ϕ. Assume (E_{2})_{red} = D_{1} + D_{2} where D_{1}, D_{2} have no common component. Then D_{1}, D_{2} ∈ Kerϕ and hence

D_{1} ∼ aD and D_{2} ∼ bD.

The conclusion is that | abD | contains the independent elements bD_{1} and aD_{2}. Each divisor of the pencil λbD_{1}+μaD_{2} is contracted by f for intersection reasons. This is a contradiction.

Corollary 0.4.6. B_{2,red} is irreducible.

Proof. E_{2} is irreducible. Since B_{2,red} = α(E_{2}) the statement follows.

However E_{2} could be not reduced i.e. we could have E_{2} = bE_{2,red} for some b ≥ 2. This is excluded if B_{1} is reduced:

Proposition 0.4.7. Assume B_{1} = B_{1,red} then E_{2} = E_{2,red}.

Proof. just write bE_{2,red} ∼ m_{2}H_{2} − H_{1} ∼ m_{2}(m_{1}H_{1} − E_{1}) − H_{1} ∼ (m_{1}m_{2} − 1)H_{2}−m_{1}E_{1}. then observe that b divides m_{1}m_{2}−1 and m_{1}m_{2}. Hence b = 1.

The fact that E_{2} is irreducible and integral does not imply yet that B_{2} is reduced: see the example of the rational normal cubic. This can be howevere proved. So, to understand α, we start to study B_{2}. If B_{2,red} is smooth then the theorem proved at the beginning of this section implies that α is the blowing up of B_{2,red}. We do not know this a priori and we will prove something less. This will be however useful for the applications. We fix the open sets

U = P^{r} − Sing(B_{2,red})

and

V = α^{−1}(U).

then α/V : V → U is a proper morphism of quasi projective varieties, moreover let

Z = {y ∈ U / dim(α/V)^{−1}(y) ≥ 1

then

Z = U ∩ B_{2,red}.

Since Z is smooth the previous theorem implies that

α/V : V → U

is the blowing up of Z. Hence Z is smooth and, by the theorem, α/V is the blowing up of

U ∩ B_{2,red}.

Still this does not imply that the base locus B ∩ U of f^{−1}/U is reduced: it is clear that its support is Z, but nevertless f^{−1}, and hence f^{−1}/U, could be defined by a linear system of hypersurfaces having multiplicity s_{2} ≥ 2 along Z. So we show now that s_{2} = 1.

Proposition 0.4.8. Z = B ∩ U that is B is generically smooth.

Proof. f^{−1} is defined by a vector space W ⊂ H^{0}(O_{P^{r}}(m_{2})) of dimension r + 1. Let I_{Z} be the ideal sheaf of Z in U, it suffices to show that the natural morphism

ϕ : W ⊗ O_{U} → I_{Z}(m_{2})

is an epimorphism. This just means that W generate the ideal of Z = B_{red} ∩ U i.e. that B is generically smooth. To prove this consider the pull-back by α/V of the Euler sequence:

0 → σ^{∗}Ω_{P^{r}}(1) ⊗ O_{V} → O_{V}^{r+1} → O_{V} (H_{1}) → 0

Applying (α/V)_{∗} to the above exact sequence we obtain in particular the previous morphism ϕ. Hence ϕ is an epimorphism if

R^{1}(α/V)_{∗}((α/V)^{∗}Ω_{P^{r}}(1)) = 0.

The vanishing of R^{1}σ′_{∗}(σ^{∗}Ω_{P^{r}}(1) follows from the theorem on formal functions: it suffices to show that H^{1}(σ^{∗}Ω_{P^{r}}(1) ⊗ O_{F}) = 0 is zero for each fibre F of α/V : V → U. This is obvious if F is a point. Otherwise F is biregular to a linear space L = α/V (F) ⊂ P^{r}. Then

σ^{∗}Ω_{P^{r}}^{1} = Ω_{L}^{1}(1) ⊕ O_{L}^{dim L+1}

and the required vanishing easily follows Proposition 0.4.9. P ic Γ = Z[H_{1}] ⊕ Z[E_{1}] = Z[H_{2}] ⊕ Z[E_{2}]

Proof. the first equality has been already shown. Note that Γ−V has codimension ≥ 2: indeed it is the pull-back of SingB_{2} which is a proper closed set of B_{2}. Since E_{2} is irreducible, Γ − V is not a divisor. Hence its codimension is ≥ 2 and therefore the natural map

i^{∗} : P ic(Γ) → P ic(V) = Z[H_{2} ∩ V] ⊕ [E_{2} ∩ V]

is an isomorphism. This implies the second equality.

Proposition 0.4.10. K_{X} ∼ (−r − 1)H_{1} + (codimB_{1} − 1)E_{1} ∼ (−r − 1)H_{2} + (codimB_{2} − 1)E_{2}.

Proof. the first part is clear. For the second one use again V instead of X using the fact that X − V has codimension ≥ 2.

Proposition 0.4.11. Assume that B_{1} itself is reduced, then:

(1) E_{1} ∼ (m_{1}m_{2} − 1)H_{2} − m_{2}E_{1}

(2) E_{2} ∼ (m_{1}m_{2} − 1)H_{1} − m_{1}E_{2}.

Proof. (1) Just observe that then E_{1} ∼ m_{1}H_{1} − H_{2} ∼ m_{2}(m_{2}H_{2} − E_{2}) − H_{2} ∼ (m_{1}m_{2} − 1)H_{2} − E_{2}. (2) In the same way E_{2} ∼ m_{2}H_{2} − H_{1} ∼ m_{2}(m_{1}H_{1} − E_{1}) − H_{1} ∼ (m_{1}m_{2} − 1)H_{1} − E_{1}.

The previous proposition yelds some geometric properties of the hypersurfaces

π_{1}(E_{2}) and π_{2}(E_{1}).

At first they are both hypersurfaces of degree m_{1}m_{2} − 1. Moreover they are varieties of multisecant lines respectively to B_{1} and to B_{2}.

Definition 0.4.2. Let B ⊂ P^{r} be an integral subscheme, the variety of s-secant lines is the union

Sec_{s}(B)

of all the lines l such that l · B has length ≥ s.

Theorem 0.4.12. π_{1}(E_{2}) = Sec_{m_{1}}(B_{2}) and π_{2}(E_{1}) = Sec_{m_{2}}(B_{2}).

Proof. Take a general p ∈ B_{2}, then α^{−1}(p) is a projective space of dimension r − dimB_{2} − 1. The restriction of f^{−1} to α^{−1}(p) is a linear embedding: this follows because: (i) B_{2} is integral, (ii) E_{2} is integral, (iii) E_{2} is not contracted by β because the fundamental locus of β is E_{1}. Let

L = β(α^{−1}(p))

and let x ∈ L − (B_{1} ∩ L). Then each line through l is contracted by f and hence it must be a s-secant line. This implies π_{1}(E_{2}) ⊆ Sec_{s}(B_{1}). Conversely let l be a ssecant line to B_{1} not contained in B_{1}, then l is contracted. Hence (not completely correct ) its strict transform is in E_{2}.

Theorem 0.4.13. f is a Cremona transformation of type (m_{1}, m_{2}) such that:

(1) m_{1}m_{2} − 1 = E_{1}H_{2}^{r−1} = E_{2}H_{1}^{r−1}.

(2) 2 + dim B_{1} = m_{2}[(r + 1) − m_{1}(codimB_{1} − 1)]

(3) 2 + dim B_{2} = m_{1}[(r + 1) − m_{2}(codimB_{2} − 1)]

Proof. (1) is an obvious consequence of the latter proposition. To obtain (1) fix a general line l ⊂ P^{r} and its strict transform L ⊂ Γ by β. Then L is the complete intersection of r − 1 elements of

| H_{1} |=| m_{2}H_{2} − E_{2} | .

Therefore, from the standard exact sequence

0 → T_{L} → T_{Γ} ⊗ O_{L} → N_{L} → 0

we obtain that

K_{L} ∼ K_{Γ}_{L} + (r − 1)H_{1}_{L}.

Statement (2) follows from this equivalence and the expression for K_{Γ} and its restriction to L. The proof goes in the same way for statement (3).

We arrive in this way to main conclusion:

Theorem 0.4.14. Let f be a special Cremona transformation of type (m_{1}, m_{2}). Assume that f is quadro-quadric i.e. m_{1} = m_{2} = 2. Then B_{1} is a smooth, integral variety satisfying the following equation:

\frac{3}{2}dimB_{1} + 2 = r.

Moreover the same equality is satisfied by B_{2} and the variety of secant lines to B_{1} is a cubic hypersurface. The same holds for B_{2}.

This is the starting point for a further discussion concerning the classification of special quadro-quadric transformation via the classification of the so called Severi varieties. See a subsequent section.

0.4.3 f quadro-quadric of P^{r} with smooth-connected B

0.4.4 f with B smooth of codimension two

0.5 Lecture 5

0.5.1 Homaloidal polynomials via matrices of linear forms

We have seen that the four Severi varieties define examples of Cremona transformations with smooth and irreducible base locus. One can generalize three of these examples constructing Cremona transformations with integral base locus. They are defined by homaloidal polynomials which are determinants of matrices of linear forms:

(1) V_{t} = Veronese image of P^{t}: in this case there is a determinantal hypersurface D_{t} containing V_{t}: namely D_{t} is the determinant of a symmetric matrix of linear forms. The linear forms are the coordinates of the ambient space. The multiplicity of V_{t} in D_{t} is t. The partials of the deteminantal equation of D_{t} define a Cremona transformations: it contracts

Sec_{2}(V_{t})

This map, in terms of matrices, is A to A^{∗}. It is a birational involution.

(2) Segre embedding of P^{t} × P^{t}: in this case the situation is similar.

(3) Grassmannian of lines of P^{t}.

0.5.2 Cremona transformations via rational normal scrolls

Let

σ : P → P^{r}

be the blowing up of a linear space L of codimension two. Then

P = P roj(O_{P^{1}}^{r−1}(1) ⊕ O_{P^{1}} = P(O_{P^{1}}^{r−1}(−1) ⊕ O_{P^{1}})

and the tautological bundle O_{P} (1) is O_{P} (E) where E = σ^{−1}(L) is the exceptional divisor of σ. Let H be the pull-back of a hyperplane by σ, notice also that each

F ∈| H − E |

is a fibre of the projection map p : P → P^{1}. We consider the linear system

| kH − (k − 1)E |=| H + (k − 1)F |,

for k ≥ 2 such a linear system defines a projective embedding of P as a rational normal scroll

P_{k} ⊂ P^{kr}

of degree (k − 1)(r − 1) + k. Indeed O_{P} ((kH − (k − 1)E) is just the tautological bundle of E := O_{P^{1}}(k − 1)^{r−1} ⊕ O_{P^{1}}(k) so that

h^{0}(O_{P} (kH − (k − 1)E)) = h^{0}(E) = rk + 1

and

(kH − (k − 1)E)^{r} = deg E = (k − 1)(r − 1) + k.

The very ampleness of the latter tautological bundle is well known for k ≥ 2. Let I_{L} be the ideal sheaf of L, it is easy to check that

| kH − (k − 1)E | +(k − 1)E = σ^{∗} | I_{L}^{k} |

so that P_{k} is the image of the rational map

β : P^{3} → P^{kr}

defined by the linear system of the hypersurfaces having multiplicity ≥ k along L. On the other hand we can fix a smooth, 0-dimensional subscheme Z ⊂ P^{r} supported on (k − 1)r general points. Under this assumption the linear projection

α : P_{k} → P^{r}

is birational. Hence f := α · β is a Cremona transformation. We only sketch the conclusion leaving the details to the reader: the indeterminacy scheme of B is supported on L ∪ Z. B has multiplicity k − 1 along B and it is smooth on Z. On the other hand the graph Γ of f is smooth.

0.5.3 Homaloidal polynomials via projected rational scrolls

0.5.4 Cremona transformations and Segre products

This example produces somehow a third generalization of plane quadratic transformations. It relies on the fact that a general plane quadratic transformations is defined by a smooth quadric Q ⊂ P^{3} via two different stereographic projections of Q. Now Q is just one example of Segre product, that is P^{1} × P^{1}, and for a general P^{a} × P^{b} a natural generalization of a stereographic projection is given by the following construction: take a pair of hyperplanes

L_{1} ⊂ P^{a} , L_{2} ⊂ P^{b},

then consider the ideal sheaf I_{L_{1}×L_{2}} of L_{1} × L_{2} and the the linear system

| I_{L_{1}×L_{2}}(1, 1) | .

Such a linear system has dimension a + b: by definition the rational map

p : P^{a} × P^{b} → P^{a+b}

defined by | I_{L_{1}×L_{2}}(1, 1) | is the stereographic projection of center L_{1} × L_{2}.

Lemma 0.5.1. p is birational.

Proof. Fix projective coordinates (u_{o} : ⋯: u_{a}) × (v_{0} : ⋯: v_{b}) on P^{a} × P^{b} so that L_{1} × L_{2} is defined by the equations u_{0} = v_{0} = 0. Then p is defined by the a + b + 1 equations

w_{00} = u_{0}v_{0}, w_{01} = u_{0}v_{1}, …, w_{0s} = u_{o}v_{s}, w_{10} = u_{1}v_{0}, …, w_{s0} = u_{s}v_{0}.

It is easy to check that they are invertible on an open set.

Notice that, as in the case of the stereographic projection of P^{1} × P^{1}, p contracts two integral divisors that is

L_{1} × P^{b} and P^{a} × L_{2}.

p contracts L_{1} × P^{b} and P^{a} × L_{2} to the disjoint linear subspaces

F_{a} and F_{b} ⊂ P^{a+b},

respectively of codimension a + 1 and b + 1. More precisely the restrictions of p to L_{1} × P^{b} and to P^{a} × L_{2} are the projections onto the factors L_{1} and L_{2}. In particular, with the pevious choice of coordinates, we have

F_{a}{w_{00} = w_{01} = ⋯ = w_{0a}} and F_{b} = {w_{00} = w_{10} = ⋯ = w_{1b} = 0}.

Lemma 0.5.2. p^{−1} is defined by the linear system of quadrics

| I_{F_{a}∪F_{b}}(2) |

where I_{F_{a}∪F_{b}} is the ideal sheaf of F_{a} ∪ F_{b}.

Fixing two different ’centers’ L_{1} × L_{2} and M_{1} × M_{2} we have two stereographic projections

p : P^{a} × P^{b} → P^{a+b} and q : P^{a} × P^{b} → P^{a+b}.

Let

f := q · p^{−1} : P^{a+b} → P^{a+b}.

It is clear that f is a quadro-quadric transformation: indeed p^{−1} is defined by a linear system of quadrics Σ and q is a linear projection of the image of p^{−1}. Hence q · p^{−1} is defined by a subspace of Φ ⊂ Σ. For the same reason f^{−1} is also defined by quadrics.

Remark 0.5.1. It is interesting to remark that Aut(P^{a} ×P^{b}) injects in Bir(P^{a+b}) as a continuous group of Cremona transformations: this injective homomorphism sends g to f · g · f^{−1}. This is a classical point of vies for studying continuous subgroups of Bir(P^{r}).

The multidegree is easily computed: TO BE CONTINUED

0.5.5 Cremona via complete intersection of quadrics

0.5.6 Cubic Cremona transformations of P^{5}