COUNTING PLANE CURVES OF ANY GENUS

Lucia Caporaso* and Joe Harris

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1. Definitions and notation

2. The formula for the degrees of generalized Severi varieties

3. The main results

4. The formula for irreducible curves

2. Geometry of Severi varieties at a general point . . . . . . . . . . . . . . . . . 10 1. Statement of results

2. Deformations of maps

3. Dimension counts and consequences

4. Normal sheaves and normal bundles

3. Hyperplane sections of Severi varieties: set-theoretic description 23 1. The basic setup

2. Some simplifying assumptions and some corollaries

3. Proof of simplified Theorem 1.2

4. The local picture of the degeneration

5. Verifying the assumptions

4. Hyperplane sections of Severi varieties: local geometry . . . . . . . . . 33 1. Deformation spaces of tacnodes

2. Products of deformation spaces of tacnodes

3. The local geometry around irreducible curves

4. The local geometry around reducible curves References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

* Partially supported by a Sloan Foundation Fellowship

1. Introduction

In this paper we study the geometry of the Severi varieties parametrizing plane curves of given degree d and geometric genus g. As an application, we derive a recursive formula for their degrees. This is the formula in Theorem 1.1, which enumerates the number of curves in ℙ^{2} of degree d with δ nodes that pass through the appropriate number of points; that result is actually more general, as we shall explain below.

Such a classical enumerative question has been an object of study by many other people. In 1989, Z. Ran described an inductive approach to the problem (cf. [R]), giving for the first time a recursive formula. More recently, interest in it has been revived by work on quantum cohomology and by the discovery by M. Kontsevich ([KM]) in 1993 of a beautiful recursion solving the problem for curves of genus 0. Another proof in the genus 0 case, using different techniques, was given independently by Ruan and Tian in [RT]. One aspect of the work of Kontsevich and Ruan-Tian is that they relate these numbers to the coefficients in the structure equations of an algebraic object, the quantum cohomology ring. It would be very interesting to see if any similar interpretation could be placed on the degrees of Severi varieties in positive genus. We don’t at present know of any algebraic structure that generates these numbers. However, Ezra Getzler [G] has described a generating function having the degrees of the generalized Severi varieties as coefficients; Getzler observes that the recursion formula in Theorem 1.1 is equivalent to the fact that this function satisfies a certain differential equation.

Our approach is simple. We work over the complex numbers throughout. We denote by ℙ^{N} the projective space of all plane curves of degree d and by V^{d,δ} ⊂ ℙ^{N} the closure of the subset of ℙ^{N} corresponding to curves having exactly δ nodes as singularities. Also, for any point p ∈ ℙ^{2} we let H_{p} ⊂ ℙ^{N} be the hyperplane of curves containing the point p. Our procedure consists in intersecting the variety V^{d,δ} with a succession of hyperplanes of the form H_{p_{i}}, where the points p_{i} are general points on a fixed line L ⊂ ℙ^{2}. At each stage we are able to describe the irreducible components of the intersection; the point is, they all belong to a specific collection of varieties, which we call generalized Severi varieties. These parametrize plane curves of given degree and genus satisfying certain tangency conditions with respect to the line L. More generally, we can express the intersection of any generalized Severi variety with a hyperplane H_{p} corresponding to a general point p ∈ L as a union of generalized Severi varieties of dimension one less; counting multiplicities correctly, this allows us to derive our recursive formula for their degrees.

We would like to thank the referee for a number of helpful comments, and Daishi Watabe for correcting several mistakes in the original version.

1.1. Notation and definitions. We now introduce the notation and precise definitions that will allow us to state our formula.

For any sequence α = (α_{1}, α_{2}, …) of nonnegative integers with all but finitely many α_{i} zero, set

#α = #{i : α_{i} ≠ 0}
|α| = α_{1} + α_{2} + … + α_{n}
Iα = α_{1} + 2α_{2} + … + nα_{n}

and

I^{α} = 1^{α_{1}}2^{α_{2}}3^{α_{3}} · · · .

We denote by lcm(α) the least common multiple of the set {α_{i} ≠ 0}.

We denote by e_{k} the sequence (0, …, 0, 1, 0, …) that is zero except for a 1 in the k^{th} term (so that any sequence α = (α_{1}, α_{2}, …) is expressible as α = ∑ α_{k}e_{k}). By the inequality α ≥ α′ we mean α_{k} ≥ α′_{k} for all k; for such a pair of sequences we set

\binom{α}{α′} = \binom{α_{1}}{α′_{1}}\binom{α_{2}}{α′_{2}}\binom{α_{3}}{α′_{3}} ⋯

We now define the main objects of study, the varieties V^{d,δ}(α, β) parametrizing plane curves of given degree and geometric genus satisfying certain tangency conditions with respect to a line. Fix a line L ⊂ ℙ^{2} and a collection

Ω = {p_{i,j}}_{1≤j≤α_{i}} ⊂ L

of |α| general points on L. For any d, δ, α and β satisfying Iα + Iβ = d, we define the generalized Severi variety V^{d,δ}(α, β)(Ω) to be the closure of the locus of reduced plane curves X of degree d and geometric genus g = \binom{d−1}{2}−δ, not containing L, with (informally) α_{k} “assigned” points of contact of order k and β_{k} “unassigned” points of contact of order k with L. Formally, we require that, if ν : X^{ν} → X is the normalization of X, then there exist |α| points q_{i,j} ∈ X^{ν}, j = 1, …, α_{i}, and |β| points r_{i,j} ∈ X^{ν}, j = 1, …, β_{i}, such that

ν(q_{i,j}) = p_{i,j}

and

ν^{∗}(L) = ∑ i · q_{i,j} + ∑ i · r_{i,j}.

Where the dependency on the points p_{i,j} is not relevant—for example, in discussions of the dimension or degrees of generalized Severi varieties—we will often suppress the Ω.

To take some simple cases, taking α = 0 and β = (d, 0, …) imposes no condition at all, that is, V^{d,δ}((0, 0, …), (d, 0, …)) is simply the closure V^{d,δ} of the locus of plane curves of degree d with δ nodes. Taking α = (1, 0, …) and β = (d − 1, 0, …) we get the closure of the locus of such curves passing through a single fixed point of L; and taking α = 0 and β = (d − 2, 1, 0, …) we get the closure of the locus of such curves tangent to L at a smooth point of the curve.

Note that we do not require X to be irreducible. Classically, the term “Severi variety” means a variety parametrizing irreducible curves of given degree and genus, so we are somewhat at odds with traditional usage here; but we will find it much more convenient, in both the statement and proof of the results below, to include components of V^{d,δ}(α, β) whose general member is reducible.

Let V be a possibly reducible variety. We will say that V has pure dimension if all irreducible components of V have the same dimension. Moreover whenever we make a statement about the general point of V, we mean that the statement holds for a general point of any irreducible component of V.

We will adopt the following convention, we will denote the various types of Severi varieties by the symbol “V ” to which we will add certain decorations, and we will correspondingly use the symbol “N” with same decorations to denote the degree of V as a subvariety of ℙ^{N}; for example, we define N^{d,δ}(α, β) := deg V^{d,δ}(α, β).

1.2. The formula for the degrees of generalized Severi varieties. As a result of the analysis of hyperplane sections of generalized Severi varieties we have the recursive formula

Theorem 1.1. Let N^{d,δ}(α, β) be the degree of V^{d,δ}(α, β). Then

\begin{aligned}N^{d,δ}(α, β) & = ∑_{k:β_{k}>0} k · N^{d,δ}(α + e_{k}, β − e_{k}) \\ & + ∑ I^{β′−β}\binom{α}{α′}\binom{β′}{β}N^{d−1,δ′}(α′, β′) \end{aligned}

where the second sum is taken over all α′, β′ and δ′ ≥ 0 satisfying

\begin{aligned}α′ & ≤ α \\ β′ & ≥ β \\ δ′ & ≤ δ \\ δ − δ′ + |β′ − β| & = d − 1 \end{aligned}

Taking α = 0 and β = (d, 0, …), we get the degree of the closure of the variety of (not necessarily irreducible) plane curves of degree d with δ nodes. We can find the degree of the component parametrizing irreducible such curves—that is, the classical Severi variety—by subtracting off the degrees of the others, which we know recursively. Alternatively, we can give a recursion formula to calculate directly the degrees of the varieties parametrizing irreducible plane curves, and will do so in the last section of this chapter; but this formula is more complicated.

We now illustrate how the above formula works by computing the degree of the Severi variety of quartics with three nodes (we assume known the degrees of the generalized Severi varieties parametrizing cubics satisfying tangency conditions). To shorten notation, we write (d, δ, α, β) for N = N^{d,δ}(α, β), and suppress the zeroes at the end of sequences α and β and the parentheses around sequences α and β of length 1. The result of intersecting the variety V^{4,3}(0, 4) with five successive hyperplanes of the form H_{p} is then the following five equations (we denote the contribution of each component to the degree, where known, in angle brackets).

\begin{aligned}(4, 3, 0, 4) = (4, 3, 1, 3) & = (4, 3, 2, 2) \\ & + (3, 0, 0, 3) ⟨\underline{1}⟩ \end{aligned}

(4, 3, 2, 2) = (4, 3, 3, 1)
+ 3(3, 1, 0, 3) ⟨3 × 12 = \underline{36}⟩
+ 2(3, 0, 1, 2) ⟨2 × 1 = \underline{2}⟩
(4, 3, 3, 1) = (4, 3, 4, 0)
+ 3(3, 2, 0, 3) ⟨3 × 21 = \underline{63}⟩
+ 2(3, 1, 0, (1, 1)) ⟨2 × 36 = \underline{72}⟩
+ 6(3, 1, 1, 2) ⟨6 × 12 = \underline{72}⟩
+ 3(3, 0, 2, 1) ⟨3 × 1 = \underline{3}⟩

and finally

\begin{aligned}(4, 3, 4, 0) & = (3, 3, 0, 3) ⟨\underline{15}⟩ \\ & + 4(3, 2, 1, 3) ⟨4 × 21 = \underline{84}⟩ \\ & + 2(3, 2, 0, (1, 1)) ⟨2 × 30 = \underline{60}⟩ \\ & + 6(3, 1, 2, 1) ⟨6 × 12 = \underline{72}⟩ \\ & + 8(3, 1, 1, (0, 1)) ⟨8 × 16 = \underline{128}⟩ \\ & + 3(3, 1, 0, (0, 0, 1)) ⟨3 × 21 = \underline{63}⟩ \\ & + 4(3, 0, 3, 0) ⟨4 × 1 = \underline{4}⟩ \end{aligned} .

Adding it all up, we find that

\begin{aligned}(4, 3, 4, 0) & = 15 + 84 + 60 + 72 + 128 + 63 + 4 = 426 \\ (4, 3, 3, 1) & = 426 + 63 + 72 + 72 + 3 = 636 \\ (4, 3, 2, 2) & = 636 + 36 + 2 = 674 \end{aligned}

and so

(4, 3, 0, 4) = (4, 3, 1, 3) = 674 + 1 = 675.

Now, V^{4,3}((0), (4)) has two irreducible components of dimension 11, one coincides withe the classical V_{4,3} which parametrizes irreducible quartics with three nodes; the other parametrizes reducible curves that are the union of a line and a cubic and has degrre \binom{11}{2} = 55. and so we conclude that the degree of the classical Severi variety is 620.

R. Vakil has checked the formula in all degrees up to 6, and D. Watabe in all degrees up to 10; and the results agree with those of Vainsencher for δ ≤ 6 and with those of Kontsevich-Manin for g = 0. We list below the numbers obtained (by applying this formula) for the degrees N of the Severi varieties V^{d,δ}(0, d) for d = 5 and 6 and all possible values of δ.

d  δ  g  N
5  0  6  1
1  5  48
2  4  882

3  3  7915
4  2  36975
5  1  90027
6  0  109781
7  -1  65949
8  -2  26136
9  -3  6930
10  -4  945
6  0  10  1
1  9  75
2  8  2370
3  7  41310
4  6  437517
5  5  2931831
6  4  12597900
7  3  34602705
8  2  59809860
9  1  63338881
10  0  40047888
11  -1  15580020
12  -2  4361721
13  -3  918918
14  -4  135135
15  -5  10395

1.3. The main results. As we indicated at the outset, what we actually prove is an equality of cycles rather than of numbers: we show first that for p ∈ L general the intersection V^{d,δ}(α, β) ∩ H_{p} is a union of varieties of the form V^{d′,δ′}(α′, β′), and say which ones occur; and then we will calculate the intersection multiplicity of V^{d,δ}(α, β) and H_{p} along each such variety.

\underline{Remark}. Throughout, we will identify the projective space of plane curves of degree d − 1 with the subspace of plane curves of degree d containing L. Thus, for example, by V^{d−1,δ′}(α′, β′)(Ω′) we mean the closure in ℙ^{N} of the locus of curves X_{0} = X ∪ L where X is a plane curve of degree d − 1, not containing L, having δ′ nodes and α′ assigned and β′ unassigned points of contact with L.

Theorem 1.2. The intersection V^{d,δ}(α, β)(Ω) ∩ H_{p} is contained in a union of varieties (without common components) as follows: a. For each k such that β_{k} > 0, the variety

V^{d,δ}(α + e_{k}, β − e_{k})(Ω ∪ {p_{k,α_{k}+1} = p}) ;

and b. For each α′ ≤ α, β′ ≥ β and δ′ ≤ δ with δ − δ′ + |β′ − β| = d − 1, the union of the

varieties

V^{d−1,δ′}(α′, β′)(Ω′)

where Ω′ ranges over all subsets Ω′ = {p′_{i,j}}_{1≤j≤α′_{i}} ⊂ Ω such that {p′_{i,1}, …, p′_{i,α′_{i}}} ⊂ {p_{i,1}, …, p_{i,α_{i}}} for each i.

\underline{Remarks}. 1. We can express the last condition δ − δ′ + |β′ − β| = d − 1 on δ′ and β′ in terms of the geometric genera g = \binom{d−1}{2} − δ and g′ = \binom{d−2}{2} − δ′: we have

g − g′ = (\binom{d − 1}{2} − δ) − (\binom{d − 2}{2} − δ′)
= d − 2 − (δ − δ′)
= |β′ − β| − 1 .

2. Note that there are a total of \binom{α}{α′} varieties of the form V^{d−1,δ′}(α′, β′) in part (b) of this statement, which accounts for the factor \binom{α}{α′} in the formula in Theorem 1.1. The remaining factors will be intersection multiplicities, as described in Theorem 1.3 below.

3. By the dimension counts of Section 2, all the varieties listed in the statement of Theorem 1.2 have pure dimension dim(V^{d,δ}(α, β)) − 1; so it follows that the intersection V^{d,δ}(α, β)(Ω) ∩ H_{p} will consist of the union of a subset of these. In fact the intersection is equal to the union of all of them, as will follow from the analysis of the local geometry of V^{d,δ}(α, β) given in the proof of Theorem 1.3.

Having described the intersection set-theoretically, we now ask about the local geometry of the larger variety V^{d,δ}(α, β) along each component of the intersection: how many branches it has, and with what multiplicity each intersects the hyperplane H_{p}. The answer to both (and hence the multiplicity with which each component of V^{d,δ}(α, β)∩H_{p} appears in the intersection cycle) is expressed in the following.

Theorem 1.3. a. Let V′ = V^{d,δ}(α + e_{k}, β − e_{k})(Ω ∪ {p}) be as in part (a) of Theorem 1.2. Then

V′ ⊂ V^{d,δ}(α, β)(Ω)∩H_{p}, and at a general point of V′ the variety V^{d,δ}(α, β) is smooth and has intersection multiplicity k with H_{p} along V′.

b. Let V′ = V^{d−1,δ′}(α′, β′)(Ω′) be as in part (b) of Theorem 1.2. At a general point of V′, the variety V^{d,δ}(α, β) will have \binom{β′}{β}I^{β′−β}/lcm(β′ −β) branches, each of which will have intersection multiplicity lcm(β′ − β) with H_{p} along V′.

The recursive formula for the degrees N^{d,δ}(α, β) of the generalized Severi varieties given in Theorem 1.1 follows directly from these two statements.

The proofs of Theorems 1.2 and 1.3 will be given in Chapters 3 and 4, following some preliminary deformation-theoretic arguments and dimension counts in Section 2. For the most part the proof of Theorem 1.2 follows the lines of that of Proposition 2.5 of [CH]. To prove it, we will use the technique of semistable reduction to analyze a family of curves X ∈ V^{d,δ}(α, β) specializing to a curve X_{0} containing L. This approach yields a number conditions that the curve X_{0} corresponding to a general point of the intersection V^{d,δ}(α, β) ∩ H_{p} must satisfy, some of which are far from obvious from the point of view of the geometry of plane curves alone. We then compare these with the dimension estimates of Chapter 2, using the fact that X_{0} is a general member of a family of dimension dim(V^{d,δ}(α, β)) − 1, to obtain an exact description of the set-theoretic intersection V^{d,δ}(α, β) ∩ H_{p}.

As for Theorem 1.3, this requires a deeper analysis of the local structure of the total space of such a family, based on the deformation theory of the tacnodes of X_{0}. As in the case of Theorem 1.2, this is based on arguments in [CH]; but while the proof of Theorem 1.2 is largely parallel to the corresponding argument of [CH], the argument for Theorem 1.3 requires additional work. Briefly, the proof of Proposition 2.7 in [CH] rests on the description given there of the deformation space of a single tacnode; this suffices for the purposes of that paper. Here we do need to consider degenerations having more than one tacnode, hence we develop an analysis of the geometry of a product of deformation spaces of tacnodes, building on the description given in [CH] of the deformation space of a single tacnode. We should mention that some of the results on deformations of tacnodes have also been obtained by Z. Ran in [R].

1.4. The formula for irreducible curves. We now give a formula for the degrees of the varieties parametrizing irreducible plane curves of given degree and genus satisfying tangency conditions.

Denote by V_{d,δ}(α, β) the union of the components of V^{d,δ}(α, β) whose general point [X] corresponds to an irreducible curve X ⊂ ℙ^{2}, and by N_{d,δ}(α, β) its degree. Now, consider the intersection of a variety V_{d,δ}(α, β) with the hyperplane H_{p}, with p ∈ L, and let [X_{0}] be a general point of a component V of the intersection. If X_{0} = X ∪ L and X has irreducible components X_{1}, …, X_{k} of degrees d_{1}, …, d_{k}, then each component X_{j} will correspond to a general point of a variety V_{d_{j},δ_{j}}(α^{j}, β^{j}). The part of the intersection V_{d,δ}(α, β) ∩ H_{p} corresponding to curves containing L will thus be a Segre image of a product of varieties V_{d_{j},δ_{j}}(α^{j}, β^{j}), and its degree will be the product of the degrees of the factors V_{d_{j},δ_{j}}(α^{j}, β^{j}), times a multinomial coming from the formula for the degrees of Segre images. Denote by γ_{i}^{j} the number of free points of i-fold tangency of X_{j} with L that are new, that is, that are not limits of free points of i-fold tangency of nearby curves X_{λ} with L as [X_{λ}] ∈ V_{d,δ}(α, β) approaches [X_{0}]; so that for example we have

∑ |β^{j}| − ∑ |γ^{j}| = |β| .

We arrive in this way at the formula

N_{d,δ}(α, β) = ∑_{k:β_{k}>0} k · N_{d,δ}(α + e_{k}, β − e_{k})

+ ∑ \frac{1}{σ} \binom{2d + g − 2 + |β|}{2d_{1} + g_{1} − 1 + |β^{1}|, …, 2d_{k} + g_{k} − 1 + |β^{k}|} · \binom{α}{α^{1}, …, α^{k}}·

· ∏_{j=1}^{k} \binom{β^{j}}{β^{j} − γ^{j}} · ∏_{j=1}^{k} I^{γ^{j}} · ∏_{j=1}^{k} N_{d_{j},δ_{j}}(α^{j}, β^{j})

where the second sum is taken over all collections of integers d_{1}, …, d_{k} and δ_{1}, …, δ_{k} and collections of sequences α^{1}, …, α^{k}, β^{1}, …, β^{k} and γ^{1}, …, γ^{k} satisfying

\begin{aligned}α′ = α^{1} + … + α^{k} & ≤ α \\ β′ = β^{1} + … + β^{k} & = β + ∑ |γ^{j}| \\ |γ^{j}| & > 0 \\ d_{1} + … + d_{k} & = d − 1 and \\ δ_{1} + … + δ_{k} & = δ + ∑ |γ^{j}| − ∑_{i<j} d_{i}d_{j} − d + 1 . \end{aligned}

Here by the symbol \binom{n}{a_{1},...,a_{k}} we mean the multinomial coefficient

\binom{n}{a_{1}, …, a_{k}} = \frac{n!}{a_{1}! ⋯ a_{k}!(n − a_{1} − … − a_{k})!}

and correspondingly for a collection of sequences α and α^{1}, …, α^{k} we set

\binom{α}{α^{1}, …, α^{k}} = ∏_{i} \binom{α_{i}}{α_{i}^{1}, …, α_{i}^{k}} .

By g_{j} we mean \binom{d_{j}−1}{2} − δ_{j}.

The symbol σ is 1 except in rare cases. It is the degree of the map from the union of the product of varieties of the form V_{d_{j},δ_{j}}(α^{j}, β^{j}) to its image in ℙ^{N}: given the integers d_{1}, …, d_{k} and δ_{1}, …, δ_{k} and collections of sequences α^{1}, …, α^{k}, β^{1}, …, β^{k} and γ^{1}, …, γ^{k} we define an equivalence relation on the set {1, 2, …, k} by saying i ∼ j if d_{i} = d_{j}, δ_{i} = δ_{j}, α^{i} = α^{j}, β^{i} = β^{j} and γ^{i} = γ^{j} and define σ to be the product of the factorials of the cardinalities of the equivalence classes.

This formula follows from Theorems 1.2 and 1.3 in much the same way as Theorem 1.1. Note that we have here decomposed β into ∑ β^{j} and the difference β′ − β into ∑ γ^{j}, and further specified that |γ^{j}| > 0. This is because (as we will see in Chapter 3) the “new” unassigned points of X ∩ L (that is, the points of X ∩ L that are not limits of points of intersection of nearby curves in V_{d,δ}(α, β) with L) correspond to points of intersection of the normalizations of the components X_{j} with L in the nodal reduction of the family. Since we are only concerned here with curves X arising as limits in families of irreducible curves, their nodal reductions must be connected; and this corresponds to the requirement |γ^{j}| > 0.

Using this formula, D. Watabe has calculated the degrees N_{d,δ}(α, β) for all d ≤ 7; the results agree with known values in case g = 0.

2. Geometry of Severi varieties at a general point

2.1. Statement of results. In this section we will compute the dimensions of generalized Severi varieties V and we will describe the geometry of its general points.

A naive reasoning yields a lower bound for the dimension of V. Namely, if we impose no conditions on the intersections of our curves with the line L, the corresponding locus— the classical Severi variety— has codimension δ in the space ℙ^{N} . Requiring that a curve X have intersection multiplicity i with L at a specified point p_{i,j} is i linear conditions on the coefficients of X, which we would expect to be independent; and if we don’t specify the point, the codimension of the corresponding locus should be one less, that is, i − 1. In sum we have

\begin{aligned}dim(V^{d,δ}(α, β)) & ≥ \binom{d + 2}{2} − 1 − δ − Iα − (Iβ − |β|) \\ & = \binom{d + 1}{2} − δ + |β| . \end{aligned}

Or, in terms of the geometric genus of the curves involved,

\begin{aligned}dim(V^{d,δ}(α, β)) & ≥ 3d + g − 1 − Iα − (Iβ − |β|) \\ & = 2d + g − 1 + |β| . \end{aligned}

We shall prove that equality holds:

Proposition 2.1. V^{d,δ}(α, β) has pure dimension 2d + g − 1 + |β|.

Likewise, there are no surprises when it comes to the geometry of a general member X of a generalized Severi variety V^{d,δ}(α, β). We would expect the curve X to have only nodes as singularities, to be smooth at its points of intersection with L, and so on; and this is indeed the case. We list the relevant facts in the following Proposition.

To do that, fix any curve G ⊂ ℙ^{2} and any finite subset Γ ⊂ ℙ^{2}. Let [X] ∈ V^{d,δ}(α, β) be a general point of a generalized Severi variety, X ⊂ ℙ^{2} the corresponding curve, ν : X^{ν} → X ⊂ ℙ^{2} its normalization. Let {q_{i,j}} and {r_{i,j}} ⊂ X^{ν} be such that ν(q_{i,j}) = p_{i,j} and

ν^{∗}L = ∑ i · q_{i,j} + ∑ i · r_{i,j}

and let s_{i,j} = ν(r_{i,j}) ∈ L be the image of r_{i,j} (that is, {s_{i,j}} ⊂ L will be the “unassigned points” of intersection of X with L). We have then the

Proposition 2.2. a. X has only nodes as singularities. b. X is smooth along X ∩ L. c. The points {q_{i,j}} and {r_{i,j}} ⊂ X^{ν} are all distinct. d. The points {p_{i,j}} and {s_{i,j}} ⊂ L are all distinct. e. X intersects G transversely (in particular, X is smooth along X ∩ G) and is disjoint

from Γ.

To prove these statements, we will need some results about deformations of maps, which will be the object of the next section.

2.2. Deformations of maps. Throughout, we will assume we are working over a field of characteristic zero, and will use the analytic topology where necessary.

We will be concerned with families of maps from a possibly variable smooth domain to a fixed smooth target space. In other words, we will consider a flat, smooth, proper family f : X → B over a smooth connected base B, a smooth variety Y and a morphism ψ : X → B × Y of B-schemes. For each b ∈ B, we let ψ_{b} : X_{b} → Y be the restriction of ψ to the fiber X_{b} of X over b, and

dψ_{b} : T X_{b} ⟶ ψ_{b}^{∗}T Y

the differential of ψ_{b}. We let N_{b} be the normal sheaf of ψ_{b}, that is, the cokernel of the morphism dψ_{b} of sheaves on X_{b}. Equivalently, if we let

dψ : T X ⟶ ψ^{∗}T (B × Y)

be the differential of ψ and N = Coker(dψ) the normal sheaf of ψ, then the normal sheaf N_{b} of ψ_{b} is the restriction of N to the fiber X_{b}, that is, N_{b} = N ⊗ O_{X_{b}} . Note that if ψ_{b} is an immersion, then N_{b} will be locally free; more generally, if ψ_{b} is equidimensional onto its image then the sheaf N_{b} will have a torsion subsheaf supported exactly on the locus where dψ_{b} fails to be an injective bundle map.

We now describe the Kodaira-Spencer map of the family ψ of morphisms. This is a map κ : T_{b}B ⟶ H^{0}(X_{b}, N_{b}) that associates to any tangent vector v ∈ T_{b}B to B a global section σ = κ(v) of the normal sheaf, in such a way that the family is trivial (that is, the family X ≅ B × X_{b} as B-schemes and the morphism ψ = id_{B} × ψ_{b}—if and only if κ(v) = 0 for every v). To define it, let π : B × Y → B be the projection, we have an inclusion of bundles

π^{∗}T_{B} ↪ T_{B×Y} .

We let i : ψ^{∗}π^{∗}T B ↪ ψ^{∗}T (B × Y) be the corresponding inclusion of pullbacks to X, and let κ̃ : ψ^{∗}π^{∗}T B → N be the composition of i with the surjection ψ^{∗}T_{B×Y} → N.

Restricting to X_{b} and taking global sections, we get a map

κ_{b} : T_{b}B ↪ H^{0}(X_{b}, ψ^{∗}π^{∗}T_{B}) ⟶ H^{0}(X_{b}, N_{b})

which we will call the Kodaira-Spencer map of the given family at b. Equivalently, we let κ be the pushforward of κ̃ to B, composed with the inclusion of T_{B} into f_{∗}ψ^{∗}π^{∗}T_{B}: that is,

κ = f_{∗}κ̃ : T_{B} ↪ f_{∗}ψ^{∗}π^{∗}T_{B} ⟶ f_{∗}N .

We will call κ the global Kodaira-Spencer map of the family; the maps κ_{b} are then the composition of the induced maps T_{b}B → (f_{∗}N)_{b} on stalks with the natural maps (f_{∗}N)_{b} → H^{0}(X_{b}, N_{b}).

The standard applications of this construction rest on two facts. The first is that if the family ψ of morphisms is nowhere isotrivial (that is, the restriction of ψ to the subfamily X_{B_{0}} = f^{−1}(B_{0}) ⊂ X is not trivial for any analytic arc B_{0} ⊂ B), then at a general point b ∈ B the map κ_{b} must be injective, so that we have an a priori bound on the dimension of the family:

dim(B) ≤ h^{0}(X_{b}, N_{b}) .

(If, for general b ∈ B, we had Ker(κ_{b}) ≠ 0, we could in an analytic neighborhood of b restrict to a curve whose tangent space was contained in Ker(κ_{b}) at each point.) Secondly, the chern classes of the normal sheaf are in general readily calculated, so that in many cases it may be possible to estimate h^{0}(X_{b}, N_{b}).

In the case of plane curves, for example, if X_{b} is a curve of genus g and ψ_{b} : X_{b} → ℙ^{2} is birational onto a plane curve of degree d, then N_{b} is a rank one sheaf on the curve X_{b}, the degree of whose chern class is

deg(c_{1}(N_{b})) = deg(c_{1}(ψ_{b}^{∗}T_{ℙ^{2}})) − deg(c_{1}(T_{X_{b}}))
= 3d + 2g − 2
> 2g − 2 .

We would thus expect that

dim(B) ≤ h^{0}(X_{b}, N_{b})
= deg(c_{1}(N_{b})) − g + 1
= 3d + g − 1 .

We cannot, however, conclude this yet. The difficulty arises from the possibility that ψ_{b} is not an immersion: if the differential dψ_{b} vanishes at points of X_{b}, the sheaf N_{b} will have torsion there, and in this case the quotient N_{b}/(N_{b})_{tors} (and hence N_{b} itself) may well be special. In such a case, the dimension h^{0}(X_{b}, N_{b}) will indeed be larger than the naive estimate 3d + g − 1 for the dimension of our family, and the method appears to fail.

Happily, there is a standard result, due to Arbarello and Cornalba ([AC]), that deals with this situation. The current version was worked out in conversations with Johan de Jong, to whom we are very grateful.

Let X → B be as before and assume that ψ : X → B × Y is birational onto its image. Then we have Lemma 2.3. If b ∈ B is a general point, then

Im(κ_{b}) ∩ H^{0}(X_{b}, (N_{b})_{tors}) = 0 .

\underline{Remarks}. 1. If we do not assume the map ψ is birational onto its image, the conclusion of the Lemma may well be false. In fact, it will fail exactly when the map ψ_{b} : X_{b} → Y is multiple-to-one, with constant image but variable branch points.

2. While we will not introduce the definitions needed to make this precise, another way to express this Lemma is to say that “the first-order deformation of the map ψ_{b} corresponding to a torsion section of N_{b} can never be equisingular”. If b ∈ B is general the first-order deformations of ψ_{b} arising from the family ψ : X → B × Y are necessarily equisingular; it follows that they cannot be torsion.

Proof. Note first that, using the analytic topology, it is enough to prove the Lemma in case B is one-dimensional: if we had Im(κ_{b}) ∩ H^{0}(X_{b}, ((N_{b})_{tors}) ≠ 0 at general b ∈ B we could in an analytic neighborhood of b restrict to a curve whose tangent space was contained in (κ_{b})^{−1}(H^{0}(X_{b}, (N_{b})_{tors})) at each point.

We may thus assume that ψ : X → B×Y is a one-parameter family of maps, the image of whose Kodaira-Spencer map κ_{b} at a general point is contained in H^{0}(X_{b}, (N_{b})_{tors}). Let Z = ψ(X) ⊂ B × Y be the image of X, p ∈ X a general point with image ψ(p) = (b, q) ∈ B × Y. We are assuming that for any v ∈ T_{b}B, the image κ_{b}(v) vanishes at p; that is, the tangent space T_{(b,q)}Z is of the form

T_{(b,q)}Z = T_{b}B × Λ_{p}

for some linear subspace Λ_{p} ⊂ T_{q}Y.

Now, let t be a local analytic coordinate on B near b, and (x, y_{1}, …, y_{n}) local coordinates on Y near q such that ψ_{b}^{∗}x is a local coordinate on X_{b} near p (so that the pair (t, x) give local coordinates on the surface X near p). We can write the map ψ locally as

y_{i} = f_{i}(t, x), i = 1, …, n .

The tangent space T_{(b,q)}Z is then the zero locus of the linear forms

dy_{i} − \frac{∂f_{i}}{∂t} dt − \frac{∂f_{i}}{∂x} dx

and that statement that T_{(b,q)}Z = T_{b}B ×Λ_{p} for some linear subspace Λ_{p} ⊂ T_{q}Y says that \frac{∂f_{i}}{∂t} vanishes identically near p. We deduce that the image of ψ_{b} is constant, i.e., that near (b, q) the image Z is equal to the product of a neighborhood of b ∈ B with a neighborhood of p ∈ X_{b}.

This being true for general p ∈ X, it follows that Z = B ×ψ_{b}(X_{b}) everywhere. Finally, since the map ψ is assumed birational, it follows that X is the normalization of Z; thus it is likewise a product, the map ψ = id_{B} × ψ_{b} and the Kodaira-Spencer map identically zero.

We now introduce the map

\overline{κ_{b}} : T_{b}B ⟶ H^{0}(X_{b}, N_{b}/(N_{b})_{tors})

defined to be the composition of κ_{b} with the natural map H^{0}(N_{b}) → H^{0}(X_{b}, N_{b}/(N_{b})_{tors}). Notice that the Lemma implies that such a map is an injection on the subspace Im(κ_{b}).

As an application, we will fix the argument given above for plane curves. In fact, we will prove a slightly more general result. Let Y be a smooth surface and D an effective divisor on Y. Let V be an irreducible component of the Severi variety of curves of given geometric genus g that are linearly equivalent to D. There is a universal family U ⊂ V ×Y of curves over V, consider the normalization U^{ν} of U and let B ⊂ V be an open subset over which U^{ν} is smooth. Let X → B be the restriction of U^{ν} to B and ψ : X → B × Y be as usual. Then we obtain the following well known result (cf. for example [K] and loc. cit.) .

Corollary 2.4. If, for general b ∈ B we have deg_{W} (ψ_{b}^{∗}ω_{Y}) < 0 on every component W of X_{b}, then

dim B ≤ − deg(ψ_{b}^{∗}ω_{Y}) + g − 1 .

In particular, dim V_{d,δ} = 3d + g − 1.

Proof. We have

dim B ≤ dim (Im(κ_{b})) ≤ h^{0}(X_{b}, N_{b}/(N_{b})_{tors})

where the last inequality follows from the previous Lemma.

By the definition of N_{b}, we obtain

deg(c_{1}(N_{b})) = − deg(ψ_{b}^{∗}ω_{Y}) + deg(ω_{X_{b}})

hence, by our hypothesis deg(c_{1}(N_{b})) > deg(ω_{X_{b}}) on each component of X_{b}. We now state for both present and future use the following simple corollary of the Riemann-Roch theorem for curves:

Observation 2.5. Let X be a smooth curve of genus g, and L any line bundle on X of degree d such that L ⊗ ω_{X}^{−1} has positive degree on each component of X. If M is any line bundle on X such that L ⊗ M^{−1} has nonnegative degree on each component of X, then

h^{0}(X, M) ≤ h^{0}(X, L) = d − g + 1 .

If moreover the line bundle L ⊗ ω_{X}^{−1} has degree 2 or more on each component of X, then h^{0}(X, M) = d − g + 1 only if deg M = deg L.

Applying this to X = X_{b} and the line bundles L = ψ_{b}^{∗}ω_{Y}^{−1}⊗ω_{X_{b}} and M = N_{b}/(N_{b})_{tors}, we have

dim B ≤ h^{0}(X_{b}, N_{b}/(N_{b})_{tors})
≤ h^{0}(X_{b}, ψ_{b}^{∗}ω_{Y}^{−1} ⊗ ω_{X_{b}})
= − deg(ψ_{b}^{∗}ω_{Y}) + p_{a}(X_{b}) − 1 .

We need now to consider deformations of a map X_{b} → Y that preserve tangency conditions with a fixed smooth curve G ⊂ Y. There are two cases, depending on whether we require tangency at a fixed point p ∈ G or allow tangency at a variable point.

So, let Y be a smooth surface, G ⊂ Y a curve and p ∈ G a smooth point. Let X → B be as above a smooth family of curves over a reduced base B, ψ : X → B×Y a morphism of B-schemes, and Q ⊂ X a section such that the pullback divisor ψ^{∗}(G) contains the section Q with multiplicity exactly m. Let b ∈ B be a general point and q = X_{b} ∩ Q; suppose ψ(q) = p. Let v ∈ T_{b}B and let σ = κ_{b}(v) ∈ H^{0}(X_{b}, N_{b}) the corresponding first-order deformation, and \overline{σ} = \overline{κ_{b}(v)} ∈ H^{0}(X_{b}, N_{b}/(N_{b})_{tors}). Suppose finally that the differential dψ_{b} vanishes to order l − 1 at q, so that the image ψ_{b}(Δ) of a small neighborhood Δ of q ∈ X_{b} will have multiplicity l at p = ψ(q). We have then the

Lemma 2.6. \overline{σ} vanishes to order at least m − l at q, and cannot vanish to order exactly k for any k with m − l < k < m. Moreover, if we assume that ψ(Q) is a point, \overline{σ} vanishes to order at least m at q.

Proof. It will be sufficient to do this in case B is one-dimensional. Next, since B is reduced and b ∈ B is general, we may assume B smooth at b; so that, restricting to an analytic neighborhood of b ∈ B we may take b the origin in an open subset B of the affine line 𝔸^{1} = Spec k[ϵ]. Finally, since again b ∈ B is general we may assume that the divisor ψ_{b}^{∗}G on X_{b} contains the point q with multiplicity exactly m as well.

Now, choose coordinates (x, y) in an analytic neighborhood of p = ψ(q) so that the curve G is given simply as the zero locus of y. Let then \frac{∂}{∂x} and \frac{∂}{∂y} be the generators of the rank 2 bundle T Y at p; we will abuse notation and write \frac{∂}{∂x} and \frac{∂}{∂y} also for the corresponding sections of ψ_{b}^{∗}T_{Y}.

The first thing we will show is that the image of \frac{∂}{∂x} in N_{b}/(N_{b})_{tors} vanishes to order m − l at q.

We treat the case l < m first for simplicity, and leave the case l = m for later. Let t be an m^{th} root of ψ_{b}^{∗}y in a neighborhood of q ∈ X_{b}, then t will be a local coordinate on X_{b} near q and the map ψ_{b} will be given as

ψ_{b} : t ⟼ (t^{l} + c_{l+1}t^{l+1} + …, t^{m})

so that the differential dψ_{b} is given by

\begin{aligned}dψ_{b} : \frac{∂}{∂t} & ↦ (lt^{l−1} + (l + 1)c_{l+1}t^{l} + …)\frac{∂}{∂x} + mt^{m−1} \frac{∂}{∂y} \\ & = t^{l−1} ((l + (l + 1)c_{l+1}t + …)\frac{∂}{∂x} + mt^{m−l} \frac{∂}{∂y} ) , \end{aligned}

Denote τ(t) := (l + (l + 1)c_{l+1}t + …)\frac{∂}{∂x} + mt^{m−l} \frac{∂}{∂y}; we have that the torsion subsheaf (N_{b})_{tors} ⊂ N_{b} is isomorphic to O_{X_{b}}/𝔪_{q}^{l−1}, generated by the section τ(t). Moreover, the quotient

N_{b}/(N_{b})_{tors} = O_{X_{b}}\{\frac{∂}{∂x}, \frac{∂}{∂y} \}/⟨τ⟩

is generated for example by the image of the section \frac{∂}{∂y}. Note finally that modulo the subsheaf generated by τ,

\frac{∂}{∂x} ∼ \frac{mt^{m−l}}{l + (l + 1)c_{l+1}t + …} · \frac{∂}{∂y}

so that the image of the section \frac{∂}{∂x} in N_{b}/(N_{b})_{tors} vanishes to order exactly m − l at q.

Now, a general deformation ψ of the map ψ_{b} over the base B ⊂ 𝔸_{ϵ}^{1} may be given in terms of coordinates t and ϵ on X near q as

ψ(t, ϵ) = (ϵ; t^{l} + c_{l+1}t^{l+1} + … + ϵ(α_{0} + α_{1}t + …) + (ϵ)^{2}, t^{m} + ϵ(β_{0} + β_{1}t + …) + (ϵ)^{2}) .

The condition that the divisor ψ^{∗}((y)) = mQ near q says that we can take t to be an m^{th} root of the pullback ψ^{∗}y not just on X_{b}, but in a neighborhood of q in X. This means that a deformation satisfying the hypotheses of the lemma may be written as

ψ(t, ϵ) = (ϵ, t^{l} + c_{l+1}t^{l+1} + … + ϵ(α_{0} + α_{1}t + …) + (ϵ)^{2}, t^{m}) .

From the definitions, the image κ_{b}(\frac{∂}{∂ϵ}) ∈ H^{0}(X_{b}, N_{b}) of the tangent vector \frac{∂}{∂ϵ} ∈ T_{b}B under the Kodaira-Spencer map will be given as the image in N_{b} of

σ := κ_{b}(\frac{∂}{∂ϵ}) = (α_{0} + α_{1}t + …)\frac{∂}{∂x} ,

whose image \overline{σ} in N_{b}/(N_{b})_{tors}, as we have seen, vanishes to order at least m − l at q. Moreover, since b ∈ B is general, the differential dψ_{ϵ} will vanish to order l − 1 at X_{ϵ} ∩ Q for all ϵ near b; that is, t^{l−1}|dψ_{ϵ}. This implies that

α_{1} = α_{2} = … = α_{l−1} = 0 ;

or in other words, \overline{σ} cannot vanish to order exactly m − l + 1, …, m − 1 at p. To complete the proof in case m > l, the further condition that ψ(Q) ≡ p ∈ G says that α_{0} = 0, which further implies that \overline{σ} vanishes to order at least m at q.

The case m = l is completely analogous. As before we write the map ψ_{b} as

ψ_{b} : t ⟼ (t^{n} + c_{n+1}t^{n+1} + …, t^{m})

where now n ≥ m. We leave it to the reader to check that the same argument yields that if ψ(Q) ≡ p ∈ G, the section \overline{σ} vanishes to order at least m at q.

2.3. Dimension counts and consequences. We will now use the general theory developed above to establish Propositions 2.1 and 2.2. Proof of Proposition 2.1. To begin with, it follows from the naive dimension count at the beginning that the Severi variety V^{d,δ}(α, β) has dimension at least 2d + g − 1 + |β|.

We thus have to show that dim V^{d,δ}(α, β) ≤ 2d+g−1+|β| everywhere. To do this, let V ⊂ V^{d,δ}(α, β) be an irreducible component, X ⊂ V × ℙ^{2} the universal family curve over V, and X^{ν} the normalization of the total space. We will actually restrict our attention to the open subset of V over which X^{ν} is smooth, which we will still call V.

Let [X] ∈ V be a general point, so that the restriction ν = ψ_{[X]} of ψ to the fiber of X^{ν} over [X] is the normalization ν : X^{ν} → X ⊂ ℙ^{2} of the corresponding curve X ⊂ ℙ^{2}; and let N be the normal sheaf of the map ν; notice that this might appear as an abuse of notation, as we have already used the symbol N with a different meaning, in the previous chapter; we hope that this will not create confusion. By the definition of V^{d,δ}(α, β), we have in an analytic neighborhood of [X] a collection of |α| and |β| sections {Q_{i,j}} and {R_{i,j}} ⊂ X^{ν} such that

ψ(Q_{i,j}) = p_{i,j}

and

ψ^{∗}(L) = ∑ i · Q_{i,j} + ∑ i · R_{i,j} .

Let q_{i,j} = Q_{i,j} ∩ X^{ν} and r_{i,j} = R_{i,j} ∩ X^{ν}. Note that the points {q_{i,j}} are necessarily distinct, since they have distinct images p_{i,j} ∈ L ⊂ ℙ^{2}. We may assume as well that the points {r_{i,j}} are distinct, and disjoint from the {q_{i,j}}: if not, [X] being general in V, V would be as well a component of a Severi variety V^{d,δ}(α′, β′) for some (α′, β′) with |β′| < |β|, which we will show has dimension 2d + g − 1 + |β′| < 2d + g − 1 + |β|.

We need to introduce one more bit of notation. We denote by l_{i,j} − 1 the order of vanishing of the differential dν at the point r_{i,j}. We then let D and D_{0} ∈ Div(X^{ν}) to be the divisors

D = ∑_{1≤j≤α_{i}} i · q_{i,j} + ∑_{1≤j≤β_{i}}(i − 1) · r_{i,j} .

and

D_{0} = ∑_{1≤j≤α_{i}}(l_{i,j} − 1) · r_{i,j} .

Note that D is a divisor of degree

\begin{aligned}deg(D) & = Iα + Iβ − |β| \\ & = d − |β| . \end{aligned}

and that

deg ((ν^{∗}O_{ℙ^{2}}(1))(−D)) ≥ 0 .

Note also that deg (c_{1}(N_{tors})) ≥ deg(D_{0}) , on every component of X^{ν}, with equality holding if and only if ν is an immersion away from {r_{i,j}}; so that

deg (c_{1}(N /N_{tors})) ≤ deg(c_{1}(N)) − deg(D_{0})

again with equality holding if and only if ν is an immersion away from {r_{i,j}}. Finally, let D_{1} be the effective part of D − D_{0}. Now, applying Lemmas 2.3 and 2.6, we see that

dim V^{d,δ}(α, β) ≤ h^{0}(X^{ν}, (N /N_{tors})(−D_{1})) .

We have

deg ((N /N_{tors})(−D_{1})) ≤ deg(c_{1}(N)) − deg(D)

and since

c_{1}(N) = ν^{∗}O_{ℙ^{2}}(3) ⊗ ω_{X^{ν}}

we see that the line bundle

(c_{1}(N)(−D)) ⊗ ω_{X^{ν}}^{−1} = ((ν^{∗}O_{ℙ^{2}}(1))(−D)) ⊗ ν^{∗}O_{ℙ^{2}}(2)

has strictly positive degree on each component of X^{ν}. We may thus apply the simple Observation 2.5 to the line bundles c_{1}(N)(−D) and (N /N_{tors})(−D_{1}) to conclude that

dim V^{d,δ}(α, β) ≤ h^{0}(X^{ν}, (N /N_{tors})(−D_{1}))
≤ deg (c_{1}(N)(−D)) − g + 1
= (3d + 2g − 2 − deg(D)) − g + 1
= 2d + g − 1 + |β| .

\underline{Remark}. Notice that the argument above implies that the image of the Kodaira Spencer map can be identified as follows:

Im \overline{κ}_{[X]} = H^{0}(X^{ν}, (N /N_{tors})(−D_{1}))

Proof of Proposition 2.2. We start by establishing what is perhaps the subtlest point: that the map ν is indeed an immersion. In fact, much of this has already been accomplished in the proof of Proposition 2.1. Keeping the notations introduced there, we see that since the line bundle (c_{1}(N)(−D) ⊗ ω_{X^{ν}}^{−1} on X^{ν} has degree at least 2 on any component of X^{ν}, we may apply the second part of Observation 2.5 to deduce the equality

(N /N_{tors})(−D_{1}) = c_{1}(N)(−D)

so that D_{1} = D − D_{0} and

N /N_{tors} = c_{1}(N)(−D_{0})

and hence ν is an immersion away from {r_{i,j}}.

To see that ν is an immersion at the point r_{i,j}, we may assume that the component X_{0} of X^{ν} containing r_{i,j} does not map to a line, so that the line bundle (c_{1}(N)(−D) ⊗ ω_{X^{ν}}^{−1} has degree at least 4 on X_{0}. It follows that there exists a section \overline{σ} of c_{1}(N)(−D) = (N /N_{tors})(−D_{1}) vanishing to order exactly 1 at r_{i,j}; and by the previos Remark, this section must be in the image of the Kodaira-Spencer map

κ_{[X]} : T_{[X]}V ⟶ H^{0}(X^{ν}, (N /N_{tors})).

But the multiplicity of r_{i,j} in the divisor D_{1} = D − D_{0} is (i − 1) − (l_{i,j} − 1) = i − l_{i,j}, and it follows that \overline{σ}, viewed as a section of N /(N_{b})_{tors}, vanishes to order exactly i − l_{i,j} + 1 at r_{i,j}. By Lemma 2.6, then, we must have l_{i,j} = 1; that is, ν must be an immersion at r_{i,j}. Since we have already seen that ν is an immersion away from {r_{i,j}}, we conclude that ν is an immersion.

To show that X has only nodes as singularities, we have to show it has no triple points and that no two branches are tangent to each other. For the former, if s, t, u ∈ X^{ν} are points mapping to the same point p ∈ X ⊂ ℙ^{2}, it is enough to show that there exists a section of N (−D) vanishing at s and t but not at u. This follows immediately from Riemann-Roch: if s and t and u all belong to the same component of X^{ν}, that component must map to a plane curve of degree at least 4, so that N ⊗ ω_{X^{ν}}^{−1} will have degree at least 8 there; while if two lie on the same component, N (−D) ⊗ ω_{X^{ν}}^{−1} will have degree at least 6 there. Similarly, for the latter, it is enough to show that if s, t ∈ X^{ν} are points mapping to the same point p ∈ X ⊂ ℙ^{2}, there exists a section of the sheaf N (−D) vanishing at s but not at t, which follows from the same argument.

As for parts c and d of 2.2, we have already seen just from the dimension statement that the points {q_{i,j}} and {r_{i,j}} are all distinct, since otherwise V would be a component of a Severi variety V^{d,δ}(α′, β′) for some (α′, β′) with |β′| < |β|; and the same logic implies that the points {s_{i,j}} are disjoint from the points {p_{i,j}}. To see that the points s_{i,j} are all distinct, on the other hand, it is sufficient to observe that, by the argument of the preceding paragraph, for any (i′, j′) ≠ (i, j) with s_{i′,j′} = s_{i,j}, there is a section of the sheaf N (−D) vanishing at r_{i,j} but not at r_{i′,j′}; by Lemma 2.6 this will correspond to a deformation of X in which s_{i′,j′} moves but s_{i,j} stays still.

Next, given that ν : X^{ν} → X is an immersion, part b follows from d; if ν is one-to-one over points of L then X is smooth along L.

Finally, part e: if a branch of X corresponding to a point s ∈ X^{ν} were tangent to G, it would be enough to show that there exists a section of N (−D) vanishing at s, which we know; and likewise if two points s, t ∈ X^{ν} mapped to the same point p ∈ G, it would be enough to show that there exists a section of N (−D) vanishing at s but not at t, which again we know.

The following restatement of Proposition 2.1 will be useful in the applications in the next section. To set it up, fix a line L ⊂ ℙ^{2} and a finite subset Ω ⊂ L. Let V ⊂ ℙ^{N} be any irreducible, locally closed subset of the space of plane curves of degree d, and [X] ∈ V a general point. Let π : W → X ⊂ ℙ^{2} be any map not constant on any irreducible component of W, whose degree over each irreducible component X_{i} of X is equal to the multiplicity of X_{i} in X (so that in particular the pullback π^{∗}O_{ℙ^{2}}(1) has degree d). Let g be the geometric genus of W, and let e be the cardinality of the intersection # (X ∩ (L \ Ω)). We have then

Corollary 2.7.

dim V ≤ 2d + g − 1 + e;

and if equality holds and # (X ∩ (L \ Ω)) = #π^{−1}(L \ Ω) then V is a dense open subset of a generalized Severi variety V^{d,δ}(α, β).

Proof. This follows readily from (2.1), after a few reductions. To begin with, it is enough to prove this in case W is smooth, since replacing W by its normalization only strengthens the inequality. Secondly, it is enough to do it in case X is irreducible: applying the statement to the inverse image of each component of X in turn and adding the results yields the desired inequality in general. (This second reduction is not really essential, but will allow us to refer to the degree of the map W → X without confusion.)

Now, since W is smooth, the map W → X factors through the normalization X^{ν} → X_{red}, and we claim that it is enough to prove it in case W = X^{ν}. To see this, assume the result proved in case W = X^{ν} and consider what happens if W → X^{ν} is a finite map of degree m > 1. In this case the degree of X_{red} is d/m, and the genus h of X^{ν} is related to the genus g of W by Riemann-Hurwitz:

g ≥ mh − m + 1

Now, applying (2.1) directly to X_{red}, we have

dim(V) ≤ 2\frac{d}{m} + h − 1 + e

≤ 2\frac{d}{m} + \frac{g − 1}{m} + e

< 2d + g − 1 + e

and we have a contradiction.

We may thus assume that W = X^{ν}. Now, suppose first that e = 0. In this case, the statement we want to prove is exactly Proposition 2.1 and we are done. More generally, consider the map ϕ from a neighborhood of [X] ∈ V to |O_{L}(e)| sending a point [W] ∈ V to the reduced intersection W ∩ (L \ Ω). Applying the e = 0 case of the statement of the Corollary to the fiber of ϕ over a general point D ∈ |O_{L}(e)| (replacing Ω by Ω′ = Ω ∪ supp(D)), we conclude that the fibers of ϕ have dimension at most 2d + g − 1, and hence that dim(V) ≤ 2d + g − 1 + e.

Note that, in the case of equality, the map π : W → X is necessarily a birational isomorphism on each component of W.

2.4. Normal sheaves and normal bundles. Before going on, we should say a few words about the relationship between the treatment of Severi varieties given here and other possible approaches. This section will not be logically relevant to the rest of the paper, but should be of use in relating the results here to those of [CH], [DH] and [H].

Briefly, there are two ways of analyzing the deformations of a plane curve X satisfying certain geometric conditions. In the approach taken here, which we may call the “parametric” approach, we look at deformations of the normalization map ν : X^{ν} → X ⊂ ℙ^{2}; so that the tangent space to the space of deformations is a priori a subspace of the space of sections of the normal sheaf N of the map. This has the virtue (at least, it is a virtue in our present circumstances) of incorporating the condition that the geometric genus of X is preserved in the deformations. Moreover, the sheaf N is a sheaf on a smooth curve. On the other hand, it has the defect that, until we know that ν is an immersion, the sheaf N may have torsion.

In the other approach, which we will call the “Cartesian” approach, we look instead at deformations of X as a subscheme of ℙ^{2}; so that the tangent space to the space of deformations is a priori a subspace of the space of sections of the normal bundle N_{X/ℙ^{2}} ≅ O_{X}(d) of the divisor X ⊂ ℙ^{2}. This is in some ways more direct—all we are doing, after all, is practicing the time-honored tradition of varying the coefficients of the defining polynomial of X—and it is in particular useful when we want to intersect our family with other subvarieties of the space ℙ^{N} of plane curves of degree d. But it has the drawback that we have to impose extra conditions to ensure that the geometric genus of X stays constant. These conditions, moreover, sometimes interact badly with conditions such as tangency with a fixed curve.

What is the relationship between the two? In case ν : X^{ν} → X is an immersion, it is reasonably straightforward. To start with, let I ⊂ O_{X} be the conductor ideal of X. This may be characterized in several equivalent ways:

• It is the annihilator of the sheaf ν_{∗}O_{X^{ν}} /O_{X};

• On an affine open subset of X with defining equation f(x, y), it is the ideal of polynomials g(x, y) such that the 1-form

ν^{∗}(\frac{g(x, y)dx}{∂f/∂y})

is regular on X^{ν}; and

• More concretely, in case ν : X^{ν} → X is an immersion, it is the ideal in O_{X} whose restriction to each branch Δ_{i} of X at each point p ∈ X is equal to the restriction to that branch of the ideal of the union of all other branches of X through p. In other words, if p_{i} ∈ X^{ν} is the point lying over p in the branch Δ_{i},

ν^{∗}I = O_{X^{ν}} (− ∑_{i}(∑_{j≠i} mult_{p}(Δ_{i} · Δ_{j})) · p_{i}) .

However we characterize the conductor, it is not hard to see that, in case ν : X^{ν} → X is an immersion, the normal sheaf N of the map ν and the normal bundle N_{X/ℙ^{2}} ≅ O_{X}(d) of the curve X are related by

N = ν^{∗}(I ⊗ N_{X/ℙ^{2}}) .

This is perhaps most easily seen in terms of the last description of the conductor: if the local defining equation f(x, y) of X at a point p ∈ X factors in the completion of the local ring O_{X,p} as

f(x, y) = f_{1}(x, y)f_{2}(x, y) ⋯ f_{n}(x, y)

then a general first-order deformation of the map will simply move each branch, resulting in a curve given by the equation

f_{ϵ}(x, y) = (f_{1}(x, y) + α_{1}ϵ)(f_{2}(x, y) + α_{2}ϵ) ⋯ (f_{n}(x, y) + α_{n}ϵ) .

As a deformation of the map, that is, as a section of N, this will be nonzero at the point of X^{ν} corresponding over the branch Δ_{i} given by f_{i}(x, y) = 0 if and only if the coefficient α_{i} ≠ 0. But the corresponding section of the normal bundle, that is, the restriction to X of the coefficient of ϵ in f_{ϵ}(x, y), on this branch is α_{i} ∏_{j≠i} f_{j}(x, y), which vanishes to order ∑_{j≠i} mult_{p}(Δ_{j} · Δ_{i}).

In any event, the conclusion is that the sections of the N_{X/ℙ^{2}} coming from deformations of the map are simply those lying in the conductor ideal (or, classically, “satisfying the adjoint conditions”, in view of the third characterization above). Moreover, if we impose further conditions of tangency with fixed curves, the allowed deformations of the map correspond to sections of N vanishing to the appropriate order at the points of X^{ν} lying over the points of tangency; and these sections, by the second characterization above, correspond to sections of J ⊗ N_{X/ℙ^{2}} for a unique ideal sheaf J ⊂ I.

We thus have a very useful dictionary between the two languages, at least as long as ν is an immersion. Otherwise the correspondence is more complicated. For example, if [X] is a point on the variety of plane curves of given degree d and genus g, corresponding to a curve with a cusp and δ − 1 = \binom{d−1}{2} − g − 1 nodes, then in a neighborhood of [X] we cannot simultaneously normalize the fibers of the universal family X ⊂ V × ℙ^{2} → V; so deformations of X preserving the geometric genus do not correspond to deformations of the map.

3. Hyperplane sections of Severi varieties: set-theoretic description.

We are now prepared to describe the hyperplane sections of the generalized Severi varieties. In this chapter we will prove Theorem 1.2, showing that the intersection V^{d,δ}(α, β) ∩ H_{p} is indeed a union of generalized Severi varieties of dimension one less, and saying which ones potentially occur. In the following chapter we will prove Theorem 1.3, establishing that all the generalized Severi varieties listed as possible components of V^{d,δ}(α, β) ∩ H_{p} do in fact occur, and describing the multiplicities with which they appear.

3.1. The basic setup. Let V′ be any irreducible component of the intersection V^{d,δ}(α, β)∩ H_{p} and [X_{0}] ∈ V′ a general point. If X_{0} does not contain L it is easy to see that V′ must be a component of one of the generalized Severi varieties listed in the first part of Theorem 1.2, so we will focus on the case L ⊂ X_{0}. We then consider a curve Γ = {[X_{γ}]} ⊂ V^{d,δ}(α, β) passing through the point [X_{0}], and the corresponding family of plane curves X → Γ. Applying a variant of semistable reduction to this family in a neighborhood of [X_{0}] ∈ Γ we arrive at a family Y → B of nodal curves dominating the curves X_{γ} in our family. Analyzing this family, we find a number of geometric conditions that the curve X_{0} must satisfy, which limit the number of its degrees of freedom. Playing these off against the fact that X_{0} is a general member of a variety of dimension dim(V^{d,δ}(α, β)) − 1 we are led to our characterization of V^{d,δ}(α, β) ∩ H_{p}.

Although the approach is quite simple, the arguments tend to appear extremely complicated. In fact, there are a priori no restrictions on the number of components, for example, of the special fiber of the family Y → B or their configuration, the notation alone can be very cumbersome. We will therefore proceed in two steps: we will give the analysis first subject to a number of simplifying assumptions, which will make the logic of the argument relatively clear (all of these hypotheses, moreover, will in fact turn out to satisfied in reality). Then we will go back and prove the result without assumptions. Comparing the argument here with the first should make it clear why in fact the assumptions hold: if any of them were indeed violated, we could replace one of the inequalities of the first calculation with a strict inequality, and so arrive at a contradiction.

We start by defining the families X → Γ and Y → B that we will be working with, and establishing the relevant notation. As above, we let [X_{0}] ∈ V′ be a general point of a component V′ of V^{d,δ}(α, β) ∩ H_{p}, and Γ ⊂ V^{d,δ}(α, β) a curve containing [X_{0}]. We will assume that the general point [X_{γ}] of Γ is a general point of V^{d,δ}(α, β); that is, it satisfies the conclusions of Proposition 2.1 above.

Now, let ν : Γ^{ν} → Γ be the normalization of Γ, and choose a point b_{0} ∈ Γ^{ν} lying over [X_{0}]. Let X^{ν} be the normalization of the total space of the pullback X ×_{Γ} Γ^{ν}, so that the family X^{ν} → Γ^{ν} has as general fiber a smooth curve of genus g = \binom{d−1}{2} − δ.

Next, we want to carry out a nodal reduction of the family X^{ν} → Γ^{ν} in a neighborhood of b_{0} to arrive at a family of nodal curves Y → B satisfying the requirements below. (This can be achieved after possibly further base change and blowing up of the nodal reduction.) Let Y^{∗} = f^{−1}(B \ {b_{0}}) be the complement of the special fiber Y_{0} = f^{−1}(b_{0}) of Y → B.

Then we require that

a. The total space Y is smooth. b. The map carrying a general fiber Y_{b} of Y → B to the corresponding plane curve

X_{γ} ⊂ ℙ^{2} extends to a regular morphism

π : Y ⟶ ℙ^{2} .

c. The inverse image π^{−1}(L) ∩ Y^{∗} consists of |α| + |β| disjoint sections {Q_{i,j}^{∗}}_{1≤j≤α_{i}} and {R_{i,j}^{∗}}_{1≤j≤β_{i}}, with π(Q_{i,j}^{∗}) ≡ p_{i,j} and R_{i,j}^{∗} intersecting the general fiber in a point r_{i,j} of multiplicity i in the divisor (π|_{Y_{b}})^{∗}L—that is we have

π^{∗}L ∩ Y^{∗} = ∑ i · Q_{i,j}^{∗} + ∑ i · R_{i,j}^{∗} .

d. The closures Q_{i,j} and R_{i,j} in Y of the sections Q_{i,j}^{∗} and R_{i,j}^{∗} are still disjoint and they

do not pass through any of the singularities of Y_{0}. e. Finally, Y → B is minimal with respect to these properties.

In sum, we have the following diagram of objects and morphisms:

3.2. Some simplifying assumptions and some corollaries. In order to present as clearly as possible the actual picture of the families X → Γ and Y → B, we will first carry out the analysis of the family Y → B under three simplifying assumptions, all of which we will show in the last part of this section do in fact hold. We will also mention some interesting facts that will follow as consequences of the proof of Theorem 1.2 (in particular they will not be assumed in the course of the proof).Since they contribute to the picture of the family Y → B we will state them here.

The first of our assumptions is perhaps the least obvious: it is that Assumption (a). The curve X_{0} contains L with multiplicity 1; that is, X_{0} = X ∪ L, where X is a plane curve of degree d − 1 not containing L.

(We should remark that this seems to be false in slightly more general situations, for example if we consider the variety V of plane curves of given degree and geometric genus having a triple point.)

Now, given (a), we see that the special fiber Y_{0} = f^{−1}(b_{0}) of Y → B will contain a unique component L̃ such that π maps L̃ onto L, and indeed the map π|L̃ : L̃ → L will be an isomorphism. We may then group together the remaining components of Y_{0} into two sets: we will let Y ⊂ Y_{0} be the union of the irreducible components mapping to X on which π is nonconstant, and Z ⊂ Y_{0} the union of the irreducible components of Y_{0} on which π is constant. In these terms, we will assume next that

Assumption (b). The curve Z ⊂ Y_{0} consists of a disjoint union of chains of rational curves joining L̃ to Y.

Assumption (c). The sections {Q_{i,j}}_{1≤j≤α_{i}} and {R_{i,j}}_{1≤j≤β_{i}} are disjoint from Z.

Note that by the last statement, each such section meets either Y or L̃ but not both. To keep track of how many of the sections Q_{i,j} pass through each, we will introduce some more notation. First, we define two further sequences α′ and α′′ with α′ + α′′ = α: we let α′_{i} be the number of the sections {Q_{i,j}}_{j=1,...,α_{i}} passing through Y, and α′′_{i} the number passing through L̃. We will likewise label the sections passing through Y (respectively, L̃) as {Q′_{i,j}}_{1≤j≤α′_{i}} (respectively, {Q′′_{i,j}}_{1≤j≤α′′_{i}}), their points of intersection with Y_{0} as {q′_{i,j}}_{1≤j≤α′_{i}} (respectively, {q′′_{i,j}}_{1≤j≤α′′_{i}}), and their image points in L as the subset Ω′ = {p′_{i,j}}_{1≤j≤α′_{i}} (respectively, {p′′_{i,j}}_{1≤j≤α′′_{i}}).

As for the sections R_{i,j}, the situation is a little different, by virtue of the first of our corollaries:

Consequence 1. Every section R_{i,j} passes through Y.

Notice that this means that β ≤ β′, so that this statement is actually part of Theorem 1.2. Thus, we do not introduce a new set of symbols. Rather, we will provisionally let β_{i}^{0} be the number of the sections {R_{i,j}}_{j=1,...,β_{i}} passing through Y, and suppose (after possibly relabeling) that the sections R_{i,j} passing through Y are {R_{i,j}}_{1≤j≤β_{i}^{0}}. We will let r_{i,j} be the point of intersection of R_{i,j} with Y_{0}, and s_{i,j} = π(r_{i,j}) ∈ L its image point.

Note also that, given (b), we may, at the expense of introducing rational double points into our surface Y, collapse the connected components of Z to points, so that the fiber Y_{0} consists simply of L̃ and Y (we have done this in the diagram below for clarity). We will index the points of intersection of L̃ with Y as follows: for each i, we let r′′_{i,1}, …, r′′_{1,β′′_{i}} be

the points of L̃ ∩ Y appearing with multiplicity i in the divisor π^{∗}L|_{Y}. We will also let s′′_{i,j} = π(r′′_{i,j}) ⊂ L be the images of these points.

In sum, we have the following picture of the family Y → B and its sections:

(Note that we have anticipated in this picture the statement (d) that every section R_{i,j} passes through Y.)

We also want to underline two other corollaries of the proof:

Consequence 2. The curve X is reduced; that is, the map π|_{Y} : Y → X is a birational isomorphism on each component of Y; and

Consequence 3. Y is smooth (though not in general connected).

3.3. Proof of simplified Theorem 1.2 We now prove Theorem 1.2 under assumptions (a), (b) and (c). We do that by comparing two different relations that the arithmetic genus g of Y has to satisfy

To begin with, note that the curves Y and L̃ intersect in |β′′| points, which are nodes of Y_{0}. Since the arithmetic genus of L̃ ≅ L ≅ ℙ^{1} is 0, we have the first relation

g = p_{a}(Y_{0}) = p_{a}(Y) + |β′′| − 1 .

Now we apply Corollary 2.7 to obtain a second relation. We see that the dimension of the family in which X can move is at most 2(d − 1) + g(Y) − 1 + |β′|. But X is a general member of the V′, which has dimension

dim(V′) = dim(V^{d,δ}(α, β)) − 1 = 2d + g − 2 + |β| .

Thus

2d + g − 2 + |β| ≤ 2(d − 1) + g(Y) − 1 + |β′|
≤ 2(d − 1) + p_{a}(Y) − 1 + |β′|
= 2d − 2 + g − |β′′| + |β′|
= 2d − 2 + g + |β^{0}|
≤ 2d + g − 2 + |β| .

We conclude that equality holds throughout, and that V′ is therefore a component of the Severi variety V^{d−1,δ′}(α′, β′)(Ω′) satisying the equality

g′ = g(X) = g − |β′ − β| + 1

or equivalently

δ − δ′ = (\binom{d − 1}{2} − g) − (\binom{d − 2}{2} − g′)
= d − 2 + g′ − g
= d − 1 − |β′ − β| .

Another way to view this situation is via the pullback π^{∗}(L) of L to Y, and in particular its restriction to Y. We have

π^{∗}L = m · L̃ + D + ∑_{1≤j≤α_{i}} i · Q_{i,j} + ∑_{1≤j≤β_{i}} i · R_{i,j}

where D is supported on Z and m is some integer. Restricting to Y we have

π^{∗}L|_{Y} = ∑_{1≤j≤α′_{i}} i · q′_{i,j} + ∑_{1≤j≤β_{i}^{0}} i · r_{i,j} + ∑_{1≤j≤β′′_{i}} i · r′′_{i,j} .

In particular, if we set β′ = β^{0} + β′′, the degree

d − 1 = deg(π^{∗}L|_{Y}) = Iα′ + Iβ′ .

The point is, of the d − 1 points of intersection of X with L, Iα′ will occur at the assigned points p′_{i,j}, while the remaining Iβ′ = Iβ^{0} + Iβ′′ will occur at the |β′| unassigned points s_{i,j} and s′′_{i,j}. The greatest degree of freedom would thus seem to be attained when α′ is as small as possible and β′ = β^{0} + β′′ as large as possible. But β^{0} is bounded above by β, and there is a penalty for taking β′′ large: this will decrease the geometric genus of the curve X, which will drop the dimension of the family which it can move.

Note some additional consequences of this analysis. First, we see from the fact that equality holds in the last of the series of inequalities above that β^{0} = β, so that every section R_{i,j} passes through Y, as stated, in other words, β ≤ β′. Statements (e) and (f) above likewise follow from the proof: the fact that X is reduced is a consquence of the application of Corollary 2.7; while if Y were singular, we would have g(Y) < p_{a}(Y), giving rise to a strict inequality in the second of series of inequalities above. This completes the proof of Theorem 1.2 subject to hypotheses (a)-(c).

3.4. The local picture of the degeneration. Based on the above analysis, we have a complete picture of the behavior of the family of plane curves X_{γ} as they degenerate to X_{0}. Away from L, there is no apparent degeneration; the family is equisingular. We will describe the family near each of the relevant points of L:

• At a point p′′_{i,j}—that is, a point of Ω \ Ω′—the curve X_{0} is smooth, since X does not pass through p′′_{i,j}. Thus, in a neighborhood of p′′_{i,j} we see the curves X_{γ} simply “flatten out” to the line L.

• At a point p′_{i,j} ∈ Ω′, the curve X_{0} has an i^{th} order tacnode, since X is smooth at p′_{i,j} and has contact of order i with L there. On the other hand, the inverse image of p′_{i,j} in Y_{0} has two distinct points, one in Y and one in L̃; in other words, the map Y_{0} → X_{0} factors through the normalization of X_{0} at p_{i,j}. Thus, in an analytic neighborhood U of p′_{i,j} the curves X_{γ} will have two branches, one tending to a neighborhood of p′_{i,j} in L and one to a neighborhood of p′_{i,j} in X. Moreover, these two branches of X_{γ} will have i points of intersection in U, merging to form the one point of intersection multiplicity i; thus, as the curves X_{γ} approach X_{0} we see i nodes of the curves X_{γ} approach the point p′_{i,j} and coalesce to form the tacnode of X_{0}. Note finally that, of the two branches of X_{γ} near p′_{i,j}, it is the one tending to a neighborhood of p′_{i,j} in X that has contact of order i with L at p′_{i,j}.

• The picture at a point s_{i,j} ∈ L—that is, a limit of an unassigned point of intersection of X_{γ} with L—is exactly the same as the picture above near a point p′_{i,j}: the nearby curves X_{γ} will have two branches in a neighborhood of s_{i,j}, one tending to L and one to X; and correspondingly we will see i nodes of the curve X_{γ} merge to form the i^{th}-order tacnode of X_{0} at s_{i,j}. The only difference, in fact, is that where in the preceding case the curves X_{γ} all had a fixed point of contact of order i with L at p′_{i,j}, in this case the curves X_{g} have a point of intersection multiplicity i with L at a variable point tending to s_{i,j}. Note that as in the preceding case, of the two branches of X_{γ} near s_{i,j}, it is the one tending to X that has contact of order i with L at s_{i,j}.

• The picture near the “new” tacnodes of X_{0}—that is, the points s′′_{i,j} of intersection of X with L that are not limits of points of intersection of X_{γ} with L—is the most interesting. Here, the inverse image of s′′_{i,j} in Y_{0} is a single point, which is an ordinary node of Y_{0}. It follows that in an analytic neighborhood of s′′_{i,j} the curves X_{γ} are irreducible, with i−1 nodes tending to the i^{th}-order tacnode s_{i,j} of X_{0}. Local equations for (and another picture of) such a family will be given in section 4.1 below, when we discuss the geometry of deformations of a tacnode.

Note that we can now account for all the nodes of the general curve X_{γ} of our family as X_{γ} tends to X_{0}. First, δ′ of the nodes stay away from L, and become simply nodes of X ⊂ X_{0}. At each point p′_{i,j}, we see i nodes of X_{γ} absorbed into the tacnode formed by X and L; and similarly at s_{i,j}. Finally, we see i − 1 nodes absorbed into each point s′′_{i,j}. In particular, we see once more that the number δ′ of nodes of X is

δ′ = δ − Iα′ − Iβ − (Iβ′′ − |β′′|) = δ − Iα′ − Iβ′ + |β′′| = δ − (d − 1) + |β′ − β| .

We can also use the above analysis to give a complete description of the components of the curve Z. Let us suppose that the curve L̃ appears with multiplicity m in the pullback divisor π^{∗}L. Since the restriction of π^{∗}L to Y has multiplicity i at each point r′′_{i,j}, if i < m it cannot meet L̃ itself: rather, it must meet a component Z_{1} of Z that appears with multiplicity i in π^{∗}L. Since the degree of the restriction π^{∗}L|_{Z_{1}} is zero and Z_{1} has self-intersection −2, we see that Z_{1} must meet another component of Y_{0} that appears with multiplicity 2i in π^{∗}L, and so on. Ultimately, we see that i must divide m, and that Z will include a chain of m/i−1 rational curves joining r′′_{i,j} to L̃, which appear with multiplicities i, 2i, 3i, …, m − 2i, m − i in π^{∗}L.

The picture of the actual fiber Y_{0} of Y → B is thus: where the length of the chain of rational curves joining L̃ to a point r′′_{i,j} ∈ Y is m/i − 1. Note that the multiplicity m with which curve L̃ appears in the pullback divisor π^{∗}L must therefore be a multiple of lcm(β′′). We will see in the following section that for suitably general Γ, we have m = lcm(β′′).

3.5. Verifying the assumptions. To complete the proof of Theorem 1.2, we will now go back and justify the assumptions (a)-(c). This amounts to setting up the same calculation without any of the hypotheses, and then observing that if any of them were violated we would have a strict inequality in one of the series of inequalities above.

To begin with, we extend the definitions of L̃, |β^{0}|, |β′′| and |β′| to the general setting. First, we take L̃ to be simply the union of the components of the fiber Y_{0} dominating L. Then, we define β_{i}^{0} to be the number of the sections {R_{i,j}}_{j=1,...,β_{i}} that meet either Y or a connected component of Z meeting Y —again, if we consider the limits as γ → 0 of the |β| unassigned points of intersection of X_{γ} with L, |β^{0}| will be simply the number that lie on X. Similarly, we let |β′′| be the number of points of Y meeting L̃, plus the number of connected components of Z meeting both L̃ and Y, and set |β′| = |β^{0}| + |β′′|.

A key observation is that no connected component of π^{−1}(L) can be contained in Z, since otherwise we could contract it to obtain an isolated point in the inverse image of L. In other words, every connected component of Z mapping to a point of L must meet L̃. Thus, with these definitions, we have as before

# (X ∩ (L \ Ω)) ≤ |β′| .

Moreover, the genus satisfies

g = p_{a}(Y_{0}) ≥ p_{a}(L̃) + p_{a}(Y) + |β′′| − 1.

Now consider the hypothesis (a) that X_{0} contains L simply. If this were not the case, that is, X_{0} = m · L ∪ X, where X is now a plane curve of degree d − m. Since the map π|L̃ : L̃ → L has degree e, we have to replace the equality p_{a}(L̃) = 0 with the inequality

p_{a}(L̃) ≥ −m + 1

and correspondingly in place of p_{a}(Y) = g − |β′′| + 1 we have

p_{a}(Y) ≤ g − |β′′| + m .

This appears to work against us. But at the same time, the degree of the image X of Y is now d − m rather than d − 1, so that the dimension of the family in which it moves is

2(d − m) + p_{a}(Y) + |β′| − 1 ≤ 2(d − m) + g − |β′′| + |β′| + m − 1
= 2d + g + |β| − m − 1
< 2d + g + |β| − 2

but our hypothesis is that X moves in V′ that has dimension 2d + g + |β| − 2, hence we have a contradiction.

Assumption (b) is more intuitively clear, if slightly more cumbersome to check. The point is, our basic inequality on the genus of Y was based on the fact that

g = p_{a}(Y_{0}) = p_{a}(Y) + p_{a}(L̃) + |β′′| − 1 .

Now, if any connected component of Z met L twice, or met Y twice, or itself had strictly positive arithmetic genus, we would have a strict inequality

g > p_{a}(Y) + p_{a}(L̃) + |β′′| − 1

and so would arrive at a contradiction. Thus each connected component Z_{0} of Z is a tree of rational curves meeting each of L̃ and Y at most once.

To verify (b), then, we simply have to check that every leaf of this tree (that is, every irreducible component W of Z meeting at most one other component of Z) meets either Y or L̃. But, if it did not, by the minimality of Y at least two of the sections Q_{i,j} and R_{i,j} would have to meet it; otherwise we could blow down W in Y and still satisfy all the conditions imposed on Y. Now, clearly no two of the sections Q_{i,j} can meet the same component of Z, since they have distinct images in ℙ^{2}. On the other hand, our basic estimate on the dimension of V′ is based on the fact that the curve X = π(Y) can have at most |β′| = |β| + |β′′| points of intersection with L outside of Ω, corresponding to the points of intersection of Y with the sections R_{i,j} and the points of intersection of Y with the connected components of Z meeting L̃. If a section Q_{i,j} and a section R_{i,j} met the same component of Z, the point s_{i,j} = π(r_{i,j}) would lie in Ω, and there would be strictly fewer than |β′| points of intersection of X with L\Ω, and by Corollary (2.7) the dimension of V′ would be strictly less than 2d + g + |β| − 2. Likewise, if two of the sections R_{i,j} met the same component of Z, two of the points s_{i,j} would coincide, and X would again have strictly fewer than |β′| points of intersection with L outside of Ω.

Finally, the verification of (c) follows the same pattern as that of (b): given that Z consists simply of chains joining L̃ to the points r_{i,j} ∈ Y, if any of the sections Q_{i,j} met Z at all, we would have s_{i,j} ∈ Ω; while if any of the sections R_{i,j} met Z at all, we would have s_{i,j} = s′′_{i′,j′} for some i, j, i′, j′ and once more X would again have strictly fewer than |β′| points of intersection with L outside of Ω.

4. Hyperplane sections of Severi varieties: local geometry

In this section we describe the geometry of V^{d,δ}(α, β) in a neighborhood of a general point of a component of its intersection with a hyperplane H_{p}. As might be expected, this is relatively straightforward for V^{d,δ}(α + e_{k}, β − e_{k}) (the first possibility described in Theorem 1.2), and substantially more complex in the case of a component of V^{d−1,δ′}(α′, β′). In the analysis of the latter case, we will rely heavily on the description given in [CH] of various loci in the versal deformation space of an mth order tacnode. The arguments in this section are based on conversations with Ravi Vakil, and appear as well in [V].

4.1. Deformation spaces of tacnodes. We start by recalling the relevant results from [CH] (the reader is referred to Section 2.4 of [CH] for details). To begin with, let (C, p) be an m^{th} order tacnode, that is, a curve singularity analytically equivalent to the origin in the plane curve given by the equation

y(y − x^{m}) = 0.

The versal deformation space of (C, p) is then the family π : S → Δ, where Δ ≅ 𝔸^{2m−1} with coordinates (a_{m−2}, …, a_{0}, b_{m−1}, …, b_{0}), S is the subscheme of Δ × 𝔸^{2} given by the equation

f(x, y, a_{m−2}, …, a_{0}, b_{m−1}, …, b_{0}) =

= y^{2} + (x^{m} + a_{m−2}x^{m−2} + ⋯ + a_{1}x + a_{0})y + b_{m−1}x^{m−1} + ⋯ + b_{1}x + b_{0} = 0

and π : S → Δ is the projection S ⊂ Δ × 𝔸^{2} → Δ.

As is clear from the description of the generalized Severi varieties in the preceding section, we are primarily interested in two loci in the base Δ of the versal deformation: the locus Δ_{m} ⊂ Δ of points (a, b) ∈ Δ over which the fiber S_{a,b} of S → Δ is reducible— equivalently, the closure of the locus of (a, b) such that S_{a,b} has m nodes—and the closure Δ_{m−1} of the locus of (a, b) such that S_{a,b} has m − 1 nodes. Since the defining equation f(x, y, a, b) for S exhibits S as a double cover of the x-line Δ × 𝔸_{x}^{1} over Δ, we can describe these two loci in terms of the branch divisor of the cover: the discriminant δ = δ_{a,b}(x) of the equation f(x, y, a, b) above as a quadratic polynomial in y is given by

δ_{a,b}(x) = (x^{m} + a_{m−2}x^{m−2} + ⋯ + a_{1}x + a_{0})^{2} − 4(b_{m−1}x^{m−1} + ⋯ + b_{1}x + b_{0}),

and the loci Δ_{m} and Δ_{m−1} ⊂ Δ are the closure of the loci of (a, b) such that δ_{a,b} has m double roots and such that δ_{a,b} has m − 1 double roots, respectively (Δ_{m} could also be characterized as the locus of squares). In particular, we see that Δ_{m} is simply the (m − 1)-plane Δ_{m} ≅ 𝔸^{m−1} ⊂ Δ given by the equations b_{m−1} = ⋯ = b_{1} = b_{0} = 0; and that Δ_{m−1} is an m-dimensional subvariety of Δ containing Δ_{m} (and having multiplicity m at a general point of Δ_{m}).

In these terms, we can now state the results of [CH] that we will use here:

Lemma 4.1. Let m ≥ 2, and let W ⊂ Δ be any smooth, m-dimensional subvariety containing the (m − 1)-plane Δ_{m}, and suppose only that its tangent plane at the origin is not contained in the hyperplane H ⊂ Δ given by b_{0} = 0. Then we have

W ∩ Δ_{m−1} = Δ_{m} ∪ Γ

where Γ is a smooth curve having contact of order exactly m with Δ_{m} at the origin.

There is an alternative way to express this lemma, which is very useful in both its proof and the present appliction; it may also provide more geometric insight. Let Δ̃ be the blow-up of Δ along the (m − 1)-plane Δ_{m}; let E ⊂ Δ̃ be the exceptional divisor and F ≅ ℙ^{m−1} ⊂ E the fiber of E over the origin in Δ. Let Δ̃_{m−1} be the proper transform of Δ_{m−1} in Δ̃. Then we can state the last result as the

Lemma 4.2. The intersection Δ̃_{m−1} ∩ E contains F as a component of multiplicity m. Moreover, Δ_{m−1} is smooth at any point of F not contained in the proper transform H̃ in Δ̃ of the hyperplane H ⊂ Δ given by b_{0} = 0.

To see the equivalence of the two statements, note that by the second the tangent plane to Δ_{m−1} at any point of F not contained in the proper transform of the hyperplane b_{0} = 0 must contain the tangent plane to F and be contained in the tangent plane to E. In particular, the proper transform W̃ of any subvariety W ⊂ Δ satisfying the hypotheses of the first statement must intersect Δ_{m−1} transversely in a smooth curve Γ̃ having intersection multiplicity m with E at its point of intersection with F; and the first statement follows. Conversely, if we apply the first statement just to the m-planes W_{b} in Δ containing Δ_{m}, we see that Δ_{m−1} contains F and (away from the proper transform of the hyperplane b_{0} = 0) is swept out by the smooth curves W̃_{b}; the second statement follows.

It will be helpful also to recall the steps in the proof of these lemmas in [CH]. Briefly, the statement of Lemma 4.1 is first verified by direct calculation in case W is the specific m-plane b_{m−1} = b_{m−2} = ⋯ = b_{1} = 0. Then the homogeneity of F \ (F ∩ H̃) under the action of the automorphism group of the singularity (C, p) allows us to deduce it for W any m-plane containing Δ_{m} and not contained in H. This, as noted, is all we need to deduce the statement of Lemma 4.2, and then Lemma 4.1 in general follows as well.

4.2. Products of deformation spaces of tacnodes. What we have to do now is to use this information to develop an analogous picture in the product of deformation spaces associated to a collection of higher-order tacnodes. For what follows, then, we will let m_{1}, m_{2}, … be any sequence of integers m_{j} ≥ 2, and (C_{j}, p_{j}) be an (m_{j})^{th} order tacnode. We will denote the versal deformation of (C_{j}, p_{j}) by π_{j} : S_{j} → Δ_{j}, and let (a_{j,m_{j}−2}, …, a_{j,0}, b_{j,m_{j}−1}, …, b_{j,0}) be coordinates on Δ_{j} as above. For each j, we will let Δ_{j,m_{j}} and Δ_{j,m_{j}−1} ⊂ Δ_{j} be as above the closures of the loci in Δ_{j} over which the fibers of π_{j} have m_{j} and m_{j} − 1 nodes respectively. Finally, we set

Δ = Δ_{1} × Δ_{2} × ⋯,
Δ_{m} = Δ_{1,m_{1}} × Δ_{2,m_{2}} × ⋯,

and

Δ_{m−1} = Δ_{1,m_{1}−1} × Δ_{2,m_{2}−1} × · · · .

Note that Δ, Δ_{m} and Δ_{m−1} have dimensions ∑ 2m_{j} −1, ∑ m_{j} −1 and ∑ m_{j} respectively.

Our goal now is to describe how a smooth subvariety W ⊂ Δ of dimension ∑(m_{j} − 1)+1, containing Δ_{m}, will intersect Δ_{m−1}, again with some hypothesis on its tangent plane at the origin. Specifically, let H ⊂ Δ be the union of the hyperplanes (b_{j,0} = 0) ⊂ Δ, and suppose that the tangent plane to W is not contained in H. By the dimension count, we would expect W to intersect Δ_{m−1} in the union of Δ_{m} and a residual curve Γ; we will show that this is indeed the case, and that the intersection number of Γ with Δ_{m} at the origin in W is ∏ m_{j}. (What will be different from the single-tacnode case is the local geometry of Γ: as we will see, it may have many branches, each of which may be singular at the origin.) To make the full statement, let λ the the least common multiple of the m_{j}, let μ = ∏ m_{j} and set κ = μ/λ. We will prove the

Lemma 4.3. With the hypotheses above, in an étale neighborbood of the origin in Δ the intersection

W ∩ Δ_{m−1} = Δ_{m} ∪ Γ_{1} ∪ Γ_{2} ∪ … ∪ Γ_{κ}

where Γ_{1}, …, Γ_{κ} ⊂ W are distinct reduced unibranch curves having intersection multiplicity exactly λ with Δ_{m} at the origin. The curves Γ_{α} will all be smooth if λ = m_{j} for some j; otherwise they will all be singular, with multiplicity λ/ max_{j}{m_{j}}.

As before, it will be helpful to express this in terms of the geometry of a blow-up: we let Δ̃ be the blow-up of Δ along the plane Δ_{m}, E ⊂ Δ̃ the exceptional divisor, F ⊂ E the fiber of E over the origin in Δ. Let Δ̃_{m−1} be the proper transform of Δ_{m−1} in Δ̃. Then we can state the last result as the

Lemma 4.4. The intersection Δ̃_{m−1} ∩ E contains F as a component of multiplicity m. Moreover, in an étale neighborhood of any point p ∈ F not contained in the proper transform H̃ of H in Δ̃, Δ̃_{m−1} consists of κ reduced branches, each having multiplicity λ/ max_{j}{m_{j}}, intersection number λ with E along F, and tangent cone at p supported on a linear space contained in E.

Proof. We will prove this in two stages: first, we will verify the statement of Lemma 4.3 directly in case W is any linear subspace of Δ; from this we will deduce the full statement of Lemma 4.4, and hence Lemma 4.3 for arbitrary W. Note that we can not do this as in the single-tacnode case by giving an explicit calculation in the case of a particularly simple plane W and then invoking homogeneity, for one reason: in the single-tacnode case we have an apriori lower bound of m for the intersection multiplicity of Γ with Δ_{m}, so that if we verify that Γ is a smooth curve with contact of order m with Δ_{m} for one plane W, we may deduce it for any plane W′ such that W lies in the closure of the orbit of [W′] ∈ G(m, 2m − 1) under the action of the automorphism group of the deformation. Here we do not have the analogous lower bound (Δ_{m} · Γ) ≥ μ, and so we have to deal first with an arbitrary linear space W.

So: let W ⊂ Δ be any plane of dimension ∑(m_{j} − 1) + 1 containing Δ_{m} and not contained in H; W will be spanned by Δ_{m} and one additional vector v ∈ Δ. Let t be a nonzero linear function on W vanishing on the hyperplane Δ_{m}. Let ρ_{j} : Δ → Δ_{j} be the projection, and v_{j} = ρ_{j}(v), so that the image W_{j} = ρ_{j}(W) ⊂ Δ_{j} of W in Δ_{j} will be the plane spanned by the subspace Δ_{j,m_{j}} and the vector v_{j}. Since by hypothesis v_{j} does not lie in the hyperplane b_{j,0} = 0, by Lemma 2.10 of [CH], we may write the intersection W_{j} ∩ Δ_{j,m_{j}} as the union of Δ_{j,m_{j}} and a smooth curve Γ_{j} having contact of order m_{j} with Δ_{j,m_{j}} at the origin. It follows that in some étale neighborhood of the origin in W_{j} we may choose coordinates (x_{j,0}, x_{j,1}, …, x_{j,m−2}, t_{j}) so that

(ρ_{j})^{∗}t_{j} = t;

and the hyperplane Δ_{j,m_{j}} is given by t_{j} = 0 and the curve Γ_{j} by the equations

x_{j,1} = x_{j,2} = ⋯ = x_{j,m−2} = 0

and

(t_{j})^{m_{j}} = x_{j,0}.

We may then take the collection of functions {y_{j,i} = (ρ_{j})^{∗}x_{j,i}}_{1≤i≤m_{j}−2} and t as local coordinates in an étale neighborhood of the origin in W, in terms of which the the hyperplane Δ_{m} is given by t = 0 and the residual intersection Γ of W with Δ_{m−1} by the equations

y_{j,i} = 0 , ∀ j and i : 1 ≤ i ≤ m_{j} − 2

and

t^{m_{j}} = y_{j,0} , ∀ j .

Γ is a curve, since specifying the value of the coordinate t at a point of Γ determines the value of the coordinates y_{j,0} (and hence all the coordinates y_{j,i}) up to a choice of an (m_{j})^{th} root. More explicitly, suppose that we choose for each j an (m_{j})^{th} root ζ_{j}. Then we may parametrize a branch Γ_{ζ} of the curve Γ by

t = z^{λ},

y_{j,0} = \frac{z^{λ/m_{j}}}{ζ_{j}}

and of course y_{j,i} = 0 ∀ i > 0. This parametrization is one-to-one, since the powers of z appearing have no common factor; the multiplicity of the image at the origin is the smallest power of z appearing, which is λ/ max{m_{j}}; and since the pullback of t is z^{λ}, the intersection multiplicity of the image with the hyperplane Δ_{m} ⊂ W defined by t = 0 is λ. Moreover, all branches of Γ are parametrized in this fashion; and two collections of roots ζ_{j} and η_{j} will give rise to the same branch if, and only if, for some λ^{th} root of unity ϵ we have

ζ_{j} = ϵ^{m_{j}}η_{j}

for all j. Since ζ_{j} = ϵ^{m_{j}}ζ_{j} for all j only if ϵ = 1, the number of such branches is the number μ of collections of roots ζ_{j} divided by λ, that is, κ. Thus the statement of Lemma 4.3 is established for any linear space W.

The remainder of the argument for Lemmas 4.3 and 4.4 is straightforward: exactly as before, the statement of Lemma 4.3 for a linear space W (satisfying the hypotheses of the Lemma) implies the statement of Lemma 4.4, which in turn implies 4.3 in general. To carry this out, let U = ℙ(Δ/Δ_{m}) \ ℙ(H) be the complement of the projectivizations of the hyperplanes b_{j,0} = 0 in the projectivization of the quotient Δ/Δ_{m}, so that we have a morphism

τ : V = Δ̃ \ H̃ ⟶ U

expressing the complement V of the proper transform H̃ of H in the blow up Δ̃ as a projective bundle (with fiber dimension ∑ m_{j} − 1) over U. Let Δ̃_{m−1}^{0} = Δ̃_{m−1} ∩ V be the intersection of the proper transform Δ̃_{m−1} with this open subset of Δ̃, and let σ : Δ̃_{m−1}^{0} → U be the restriction of τ to Δ̃_{m−1}^{0}. The statement of Lemma 4.3 for linear spaces W says precisely that the fibers of σ are curves consisting of κ reduced branches, each having multiplicity λ/ max_{j}{m_{j}}, intersection number λ with E at its unique point p of intersection with F, and tangent cone at p supported on a linear space contained in the tangent space to E. It follows that exactly the same is true of Δ̃_{m−1} in a neighborhood of any point p ∈ F not in H̃: it has κ reduced branches, each having multiplicity λ/ max_{j}{m_{j}} and intersection multiplicity λ with E along F, and tangent cone supported on a linear space contained in the tangent space to E. In other words, we have proved Lemma 4.4; and as before Lemma 4.3 follows for an arbitrary smooth m-dimensional subvariety W ⊂ Δ satisfying the hypotheses of the Lemma.

4.3. The local geometry around irreducible curves. In the remaining two parts of this section we will complete the proof of Theorem 1.3 by analyzing the geometry of a generalized Severi variety V^{d,δ}(α, β) in a neighborhood of a general point [X_{0}] of one of the generalized Severi varieties V′ listed in Theorem 1.2. We will start in this subsection with the (relatively) simple case of a general point of V^{d,δ}(α + e_{k}, β − e_{k})(Ω ∪ {p}); in the following one we will apply the preceding results to carry out the analysis at a general point of V^{d−1,δ′}(α′, β′)(Ω′).

So: assume that β_{k} > 0, and let [X_{0}] be a general point of V′ = V^{d,δ}(α + e_{k}, β − e_{k})(Ω ∪ {p}). We have then the

Proposition 4.5. The variety V^{d,δ}(α, β) contains [X_{0}]; it is smooth there and has intersection multiplicity k with H_{p} along V′.

Proof. This follows directly from an analogous statement about the linear series of divisors on L ≅ ℙ^{1}. To set this up, consider the rational map

π : |O_{ℙ^{2}}(d)| ≅ ℙ^{N} ⟶ |O_{L}(d)| ≅ ℙ^{d}

given by restriction (note that this is a linear projection, with vertex the subspace in ℙ^{N} of curves containing L). Inside the target space |O_{L}(d)|, we consider three loci: we let H = {D : D − p ≥ 0} be the hyperplane of divisors containing the point p; and we set

Φ = {D ∈ |O_{L}(d)| : D = ∑_{1≤j≤α_{i}} i · p_{i,j} + ∑_{1≤j≤β_{i}} i · p′_{i,j} for some p′_{i,j} ∈ L}

and similarly, setting β′ = β − e_{k},

Ψ = {D ∈ |O_{L}(d)| : D = k · p + ∑_{1≤j≤α_{i}} i · p_{i,j} + ∑_{1≤j≤β′_{i}} i · p′_{i,j} for some p′_{i,j} ∈ L} .

Now, let V = V^{d,δ} ⊂ ℙ^{N} be the ordinary Severi variety. we have

H_{p} = π^{−1}(H) ;

and if we let

σ = π|_{V} : V ⟶ |O_{L}(d)|

be the restriction of π to V,

V^{d,δ}(α, β) = σ^{−1}(Φ)

and

V^{d,δ}(α + e_{k}, β − e_{k}) = σ^{−1}(Ψ) .

Proposition 4.5 will thus follow from the combination of the two Lemmas

Lemma 4.6. The differential dσ of σ : V → |O_{L}(d)| is surjective at [X].

and

Lemma 4.7. In a neighborhood of the point D_{0} = X · L ∈ |O_{L}(d)|, the variety Φ is smooth and has intersection multiplicity k with the hyperplane H along Ψ.

Proof of Lemma 4.6. Since the map π : |O_{ℙ^{2}}(d)| ≅ ℙ^{N} → |O_{L}(d)| ≅ ℙ^{d} is a linear projection, we need only check that the projective tangent plane P T_{[X]}V ⊂ ℙ^{N} to V at [X] intersects the vertex L+|O_{ℙ^{2}}(d−1)| ≅ ℙ^{N−d−1} of π transversely, that is, in codimension d + 1 in P T_{[X]}V. But now the projective tangent space P T_{[X]}V is simply the linear series of curves of degree d passing through the δ nodes of X, and its intersection with the vertex the linear series of curves of degree d − 1 passing through the nodes; and since the nodes impose independent conditions on curves of degree at least d−2 these will have dimensions \frac{d(d+3)}{2} − δ and \frac{(d−1)(d+2)}{2} − δ = \frac{d(d+3)}{2} − δ − (d + 1) respectively.

Proof of Lemma 4.7. We can do this simply in coordinates: let x be an affine coordinate on L ≅ ℙ^{1} such that the point p is given by x = 0; let the point p_{i,j} have coordinate λ_{i,j} and suppose that the β′_{i} points other than p and {p_{i,j}} at which the divisor D_{0} has multiplicity i have coordinates μ_{i,j} (note that by Lemma 2.2 the λ_{i,j} and the μ_{i,j} are all distinct). Then we can parametrize a neighborhood of [D_{0}] in Φ by

(ϵ, ϵ_{i,j}) ⟼ [f(x)] = [(x − ϵ)^{k} ∏(x − λ_{i,j})^{i} ∏(x − μ_{i,j} − ϵ_{i,j})^{i}]

from which we see in particular that Φ is smooth at the point [D_{0}]. Now, writing a point [f(x)] ∈ |O_{L}(d)| as

f(x) = x^{d} + b_{d−1}x^{d−1} + … + b_{1}x + b_{0}

the defining equation of the hyperplane H ⊂ |O_{L}(d)| is simply b_{0} = 0, which pulls back via this parametrization to ϵ^{k} times a polynomial in the ϵ_{i,j} nonzero at the origin; it follows that in a neighborhood of [D_{0}], the divisor cut on Φ by H is simply k times Ψ.

4.4. The local geometry around reducible curves. It remains to describe the local geometry of a generalized Severi variety V^{d,δ}(α, β) in a neighborhood of a point [X_{0}] , where X_{0} = X ∪ L and [X] is a general point of a generalized Severi variety V′ = V^{d−1,δ′}(α′, β′)(Ω′). Thus, we will suppose that Ω′_{i} = {p′_{i,j}}_{1≤j≤α′_{i}} is any subset of cardinality |α′| of Ω_{i} = {p_{i,j}}_{1≤j≤α_{i}} such that {p′_{i,1}, …, p′_{i,α′_{i}}} ⊂ {p_{i,1}, …, p_{i,α_{i}}} for each i; and that there are points {q′_{i,j}}_{1≤j≤α′_{i}} and {r′_{i,j}}_{1≤j≤β′_{i}} in the normalization ν : X̃ → X ⊂ ℙ^{2} such that ν(q′_{i,j}) = p′_{i,j} and the pullback ν^{∗}(L) = ∑ iq_{i,j} + ∑ ir_{i,j}. With this said, our basic result is the

Proposition 4.8. In a neighborhood of [X_{0}], the variety V^{d,δ}(α, β) will have

\binom{β′}{β}\frac{I^{β′−β}}{lcm(β′ − β)}

branches, each of which will have intersection multiplicity lcm(β′ − β) with H_{p} along V′.

Proof. As in the case of Lemma 4.5, we want to deduce this from a local calculation, in this case, Lemma 4.3. Before we can do this, we have to specify which of the points r_{i,j} ∈ X̃ will be limits of points of unassigned tangency on nearby curves in the family; each such specification will determine a collection of branches of V^{d,δ}(α, β). So: to start with, choose any subset Λ = {r_{i,j}}_{1≤j≤β_{i}} of the set {r′_{i,j}}_{1≤j≤β′_{i}} such that {r_{i,1}, …, r_{i,β_{i}}} ⊂ {r′_{i,1}, …, r′_{i,β′_{i}}} for each i. By way of notation, let β′′ = β′ − β, and label the complement of the subset {r_{i,j}} ⊂ {r′_{i,j}} as {r′′_{i,j}}_{1≤j≤β′′_{i}}; and let s_{i,j} ⊂ L ⊂ ℙ^{2} (respectively, s′_{i,j}, s′′_{i,j}) be the images of the point r_{i,j} (respectively, r′_{i,j}, r′′_{i,j}). Similarly, set α′′ = α − α′, and label the complement of the subset {p′_{i,j}} ⊂ {p_{i,j}} as {p′′_{i,j}}_{1≤j≤α′′_{i}}.

Now, in an analytic neighborhood of the point [X_{0}] = [X + L] ∈ ℙ^{N}, we will define the relaxed local Severi variety W_{Λ} to be the closure of the locus of curves X_{t} satisfying the following six conditions:

i) X_{t} preserves the δ′ nodes of X; that is, for every node of X_{0} away from L, X_{t} will have a node nearby.

ii) At each point p′′_{i,j}, X_{t} has contact of order i with L. iii) In a neighborhood of each point p′_{i,j}, X_{t} has i nodes iv) In a neighborhood of each point s_{i,j}, X_{t} has i nodes

To specify the remaining two conditions we need to make one remark. Conditions iii) requires that, in an étale or analytic neighborhood of a point p′_{i,j}, the deformation X_{t} of X will be reducible; that is, it will continue to have two branches, one a deformation of a neighborhood of the point p′_{i,j} in L and the other a deformation of a neighborhood of p′_{i,j} in X. Similarly, a deformation X_{t} of X_{0} satisfying condition iv) will have two branches near each point s_{i,j}, deformations of the two branches of X_{0} at s_{i,j}. In these terms, we make the further requirements that:

v) In a neighborhood of each point p′_{i,j}, the branch of X_{t} that is a deformation of a neighborhood of p′_{i,j} in X has contact of order i with L at p′_{i,j}

vi) In a neighborhood of each point s_{i,j}, the branch of X_{t} that is a deformation of a neighborhood of s_{i,j} in X has a point of contact of order i with L.

\underline{Remarks}. 1. We are being colloquial here in the definition of the relaxed Severi variety, using terms like “nearby” and “in a neighborhood of” each point p′_{i,j} or s_{i,j}. This is to avoid introducing yet more notation. The definition may be made precise, for example, by specifying in ℙ^{2} disjoint analytic neighborhoods U_{i,j}, V_{i,j} and W_{1}, …, W_{δ′} of the points p′_{i,j} and s_{i,j}, and the nodes u_{1}, …, u_{δ′} of X; or by considering deformations {X_{t}} of X_{0} having nodes at deformations {u_{i}(t)} of the points u_{i}, etc.

2. We remark again that the relaxed local Severi variety W_{Λ} depends on the choice of subset \{r_{i,j}\} ⊂ \{r′_{i,j}\}; there are thus \binom{β′}{β} such varieties W_{Λ} in a neighborhood of [X_{0}].

Note that we are at this point making no requirements about the deformations X_{t} in a neighborhood of a point s′′_{i,j}, even though we have seen that a family of curves X_{t} in V^{d,δ}(α, β) tending to X_{0} will have i − 1 nodes tending to each point s′′_{i,j} (hence the name “relaxed”). Thus, in particular, a general point [X_{t}] ∈ W_{Λ} will correspond to a curve X_{t} with only δ′′ = δ − (Iβ′′ − |β′′|) nodes—in other words, W_{Λ} will be an open subset of the variety V^{d,δ′′}(α, β). In fact, our strategy is exactly this: to consider first the conditions a curve X_{t} must satisfy away from the points s′′_{i,j} in order to belong to a component of V^{d,δ}(α, β) containing [X_{0}] in its closure; and then secondly the conditions around the points s′′_{i,j}.

The point is, the conditions on the curve X_{t} at the points p′_{i,j} and s′_{i,j} and the nodes u_{1}, …, u_{δ′} of X are all essentially linear conditions, and well behaved. Thus, omitting any requirements on the behavior of the curves X_{t} around s′′_{i,j} will result in a parameter space W_{Λ} that is smooth with identifiable tangent space at the point [X_{0}]. Once we have described this space, we will then consider the map ϕ_{Λ} from W_{Λ} to the product Δ of the deformation spaces of the tacnodes of X_{0} at the points s′′_{i,j}. In a neighborhood of [X_{0}], the Severi variety V^{d,δ}(α, β) will be the union, over all Λ, of the closures of the inverse images ϕ_{Λ}^{−1}(Δ_{m−1} \ Δ_{m}). Once we have shown that for each Λ the image ϕ_{Λ}(W_{Λ}) satisfies the hypotheses of Lemma 4.3, then, Theorem 1.3 will follow.

The first step in carrying out this plan is thus the identification of the tangent space to W_{Λ} at [X_{0}] (from which it will follow that W_{Λ} is indeed smooth at [X_{0}], once we estimate its dimension). This tangent space, viewed as a subspace of the tangent space T_{[X_{0}]}ℙ^{N} = H^{0}(X_{0}, O(d)), is the subspace H^{0}(X_{0}, I(d)) determined by an ideal sheaf I ⊂ O_{X_{0}}, which we will describe in the following Lemma.

To do this, we have to introduce some local ideals. Specifically, for each i and j with 1 ≤ j ≤ α′_{i}, we let I′_{i,j} ⊂ O_{X_{0}} be the sheaf of regular functions in a neighborhood of p′_{i,j} ∈ X_{0} whose restriction to L ⊂ X_{0} vanishes to order i at p′_{i,j} and whose restriction to X ⊂ X_{0} vanishes to order 2i at p′_{i,j}. Similarly, for each i and j with 1 ≤ j ≤ β_{i}, we let I_{i,j} ⊂ O_{X_{0}} be the sheaf of regular functions in a neighborhood of s_{i,j} ∈ X_{0} whose restriction to L ⊂ X_{0} vanishes to order i at s_{i,j} and whose restriction to X ⊂ X_{0} vanishes to order 2i − 1 at s_{i,j}. We have then the

Lemma 4.9. The variety W_{Λ} is smooth at [X_{0}], and its tangent space is given by the linear series

T_{[X_{0}]}W_{Λ} = H^{0}(X_{0}, I(d)) ⊂ T_{[X_{0}]}ℙ^{N} = H^{0}(X_{0}, O(d))

where the ideal sheaf I is the product

I = ∏_{i=1}^{δ′} 𝔪_{u_{i}} · ∏_{1≤j≤α′′_{i}} 𝔪_{p′′_{i,j}}^{i} · ∏_{1≤j≤α′_{i}} I′_{i,j} · ∏_{1≤j≤β_{i}} I_{i,j}

Proof. We will first show that the tangent space to W_{Λ} at [X_{0}] is contained in the series H^{0}(X_{0}, I(d)). We will then argue that the ideal I imposes independent conditions on the series H^{0}(X_{0}, O(d)), so that we can calculate the dimension h^{0}(X_{0}, I(d)); comparing this with the actual dimension of W_{Λ} we deduce the smoothness of W_{Λ} at [X_{0}] and the identification of the tangent space with H^{0}(X_{0}, I(d)).

For both parts, it will be useful to introduce a partial normalization X̃_{0} of X_{0}: specifically, we let μ : X̃_{0} → X_{0} be the normalization of X_{0} at the points p′_{i,j} and s_{i,j}, and the nodes u_{1}, …, u_{δ′} of X, but not at the points s′′_{i,j}. Note that the normalization X̃ of X is actually a closed subscheme of X̃_{0}. We will abuse notation slightly and denote by q′_{i,j} and r_{i,j} the points of X̃ ⊂ X̃_{0} lying over p′_{i,j} and s_{i,j}. We will also denote by q′′_{i,j} ∈ X̃_{0} the (unique) point of X̃_{0} lying over p′′_{i,j}.

Note that since I is contained in the conductor J ⊂ O_{X_{0}} of the map X̃_{0} → X_{0}, the pullback μ^{∗}I is locally free on X̃_{0}; specifically, it is the sheaf

μ^{∗}I = μ^{∗}J ⊗ OX̃_{0} (− ∑ i · q′_{i,j} − ∑ i · q′′_{i,j} − ∑(i − 1) · r_{i,j}) .

Also, because I is contained in the conductor J of the map X̃_{0} → X_{0}, the space of sections H^{0}(X̃_{0}, μ^{∗}I(d)) = μ^{∗}H^{0}(X_{0}, I(d)).

Now, to prove the inclusion T_{[X_{0}]}W_{Λ} ⊂ H^{0}(X_{0}, I(d)), we observe that by conditions i), iii) and iv) in the definition of W_{Λ} any deformation of X_{0} in W_{Λ} arises from a deformation of the composite map \overline{μ} : X̃_{0} → X_{0} → ℙ^{2}. Thus, any tangent vector to W_{Λ} at [X_{0}] must lie in the subspace

H^{0}(X_{0}, J (d)) ⊂ H^{0}(X_{0}, O(d))

which we may identify in turn with H^{0}(X̃_{0}, μ^{∗}J (d)). Now, as we observed in section 2 above, we may further identify μ^{∗}J (d) with the normal sheaf N = N\overline{μ} of the map \overline{μ}; and in terms of these identifications conditions ii), v) and vi) of the definition of W_{Λ} amount to the assertion that the tangent space to W_{Λ} at [X_{0}] satisfies

T_{[X_{0}]}W_{Λ} ⊂ H^{0}(X̃_{0}, N(− ∑ i · q′_{i,j} − ∑ i · q′′_{i,j} − ∑(i − 1) · r_{i,j}))
= H^{0}(X̃_{0}, μ^{∗}J (d))(− ∑ i · q′_{i,j} − ∑ i · q′′_{i,j} − ∑(i − 1) · r_{i,j}))
= H^{0}(X̃_{0}, μ^{∗}I(d))
= H^{0}(X_{0}, I(d))

For the second part, to estimate of the dimension h^{0}(X_{0}, I(d)), we will equate this with the dimension h^{0}(X̃_{0}, μ^{∗}I(d)) of the space of sections of the pullback, and apply Riemann-Roch on X̃_{0}. A key fact is that the line bundle μ^{∗}I(d) is nonspecial. In the sequel we will need as well the fact that for p ∈ L general the bundle μ^{∗}I(d)(−p) is nonspecial as well; we will state these as the

Lemma 4.10. For p ∈ L general,

h^{1}(X̃_{0}, μ^{∗}I(d)) = h^{1}(X̃_{0}, μ^{∗}I(d)(−p)) = 0 .

Proof. First, note that the dualizing sheaf of X̃_{0} is given by

ωX̃_{0} = μ^{∗}(J (d − 3)).

Thus, we may write

μ^{∗}I(d) = μ^{∗}O(3) ⊗ ωX̃_{0} (− ∑ i · q′_{i,j} − ∑ i · q′′_{i,j} − ∑(i − 1) · r_{i,j}) .

and correspondingly

ωX̃_{0} ⊗ (μ^{∗}I(d))^{−1} = μ^{∗}O(−3) (∑ i · q′_{i,j} + ∑ i · q′′_{i,j} + ∑(i − 1) · r_{i,j}) .

Now, the restriction of this line bundle to X̃ has degree

deg (ωX̃_{0} ⊗ (μ^{∗}I(d))^{−1} ⊗ OX̃) = −3(d − 1) + Iα′ + Iβ
≤ −3(d − 1) + Iα′ + Iβ′
= −2(d − 1)
< 0

and so every global section σ ∈ H^{0}(ωX̃_{0} ⊗ (μ^{∗}I(d))^{−1}) must vanish identically on X̃. The restriction of σ to the component L̃ of X̃_{0} lying over L is then a section of the bundle

ωX̃_{0} ⊗ (μ^{∗}I(d))^{−1} ⊗ IX̃ ⊗ OL̃

and since the restriction IX̃ ⊗ OL̃ to L̃ of the ideal sheaf of X̃ has degree −Iβ′′, we have

deg (ωX̃_{0} ⊗ (μ^{∗}I(d))^{−1} ⊗ IX̃ ⊗ OL̃) = Iα′′ − Iβ′′ − 3
= Iα − Iα′ − (Iβ′ − Iβ) − 3
= (Iα + Iβ) − (Iα′ + Iβ′) − 3
= d − (d − 1) − 3
= −2 .

Thus σ must vanish identically on L̃ as well, and hence

h^{1}(X̃_{0}, μ^{∗}I(d)) = h^{0}(X̃_{0}, ωX̃_{0} ⊗ (μ^{∗}I(d))^{−1}) = 0 .

Finally, if we had started with μ^{∗}I(d)(−p) in place of μ^{∗}I(d), we would have wound up with

deg (ωX̃_{0} ⊗ (μ^{∗}I(d)(−p))^{−1} ⊗ IX̃ ⊗ OL̃) = −1

and we would conclude as before that the line bundle μ^{∗}I(d)(−p) is nonspecial.

We may now apply the Riemann-Roch formula on X̃_{0} to complete the proof of Lemma 4.9. By Lemma 4.10, we have

\begin{aligned}h^{0}(X_{0}, I(d)) & = h^{0}(X̃_{0}, μ^{∗}I(d)) \\ & = deg(μ^{∗}I(d)) − p_{a}(X̃_{0}) + 1 \end{aligned}

Since

\begin{aligned}deg(μ^{∗}I(d)) & = deg (μ^{∗}O(3) ⊗ ωX̃_{0}(− ∑ i · q′_{i,j} − ∑ i · q′′_{i,j} − ∑(i − 1) · r_{i,j})) \\ & = 3d + 2p_{a}(X̃_{0}) − 2 − Iα − (Iβ − |β|) \end{aligned}

we can rewrite this as

h^{0}(X_{0}, I(d)) = 3d + p_{a}(X̃_{0}) − 1 − Iα − (Iβ − |β|) .

Now, the arithmetic genus of X̃_{0} is simply the genus of a general member X_{t} of W; thus

p_{a}(X̃_{0}) = \binom{d − 1}{2} − δ′′.

Equivalently, we could arrive at this by observing that the arithmetic genus of X̃_{0} is simply the genus of X̃, plus the degree of the intersection of X̃ ⊂ X̃_{0} with the component L̃ of X̃_{0} lying over the line L ⊂ ℙ^{2}, minus 1; thus

\begin{aligned}p_{a}(X̃_{0}) & = \binom{d − 2}{2} − δ′ + Iβ′′ − 1 \\ & = \binom{d − 2}{2} − δ + d − 1 − |β′ − β| + Iβ′′ − 1 \\ & = \binom{d − 1}{2} − δ + Iβ′′ − |β′′| \\ & = \binom{d − 1}{2} − δ′′ \end{aligned} .

Either way, we have

dim(T_{[X_{0}]}W_{Λ}) ≤ h^{0}(X_{0}, I(d))

= 3d + p_{a}(X̃_{0}) − 1 − Iα − (Iβ − |β|)

= 3d + \binom{d − 1}{2} − δ′′ − 1 − Iα − (Iβ − |β|)

= \binom{d + 2}{2} − 1 − δ′′ − Iα − (Iβ − |β|)

= dim(V^{d,δ′′}(α, β))

= dim(W_{Λ}) ;

so W_{Λ} must be smooth, with tangent space equal to H^{0}(X_{0}, I(d)).

All that remains to complete the proof of Proposition 4.8 is to consider the map ϕ_{Λ} from W_{Λ} to the product Δ of the deformation spaces of the tacnodes of X_{0} at the points s′′_{i,j}. Note first that the image of ϕ_{Λ} contains the locus Δ_{m} ⊂ Δ, and that the inverse image W_{0} = (ϕ_{Λ})^{−1}(Δ_{m}) of this locus is simply the set of points [X_{t}] ∈ W_{Λ} corresponding to curves X_{t} containing L. This has codimension at most one in W_{Λ}: if we choose any point p ∈ L not among the points p_{ij} or s_{i,j}, then any curve X_{t} in W_{Λ} containing p will have a total of

Iα + Iβ + 1 = d + 1

points of intersection with L, and so will contain L.

On the other hand, under the differential

dϕ_{Λ} : T_{[X_{0}]}W_{Λ} ⟶ T_{0}Δ

of the map ϕ_{Λ}, the inverse image of the tangent space T_{0}Δ_{m} ⊂ T_{0}Δ is simply the subspace of H^{0}(X_{0}, I(d)) of sections vanishing on L. Now, the restriction of I(d) to L has degree d − Iα − Iβ = 0—that is, it is trivial—and by Lemma 4.10 not every global section of I(d) vanishes on L. Thus

dim((dϕ_{Λ})^{−1}(T_{0}Δ_{m})) = dim(ϕ^{−1}(Δ_{m}) = dim(W_{Λ}) − 1.

We may conclude that the image of ϕ_{Λ} is smooth of dimension

dim(ϕ_{Λ}(W_{Λ})) = dim(Δ_{m}) + 1

with tangent space the image of dϕ_{Λ}. Since again the linear system H^{0}(X_{0}, I(d)) has no base points on L, the image W = ϕ_{Λ}(W_{Λ}) satisfies the hypotheses of Lemma 4.3. Thus we may apply Lemma 4.3 to conclude that the closure of the inverse image ϕ_{Λ}^{−1}(Δ_{m−1} \ Δ_{m}) will have κ = I^{β′−β}/lcm(β′ − β) reduced branches, each having intersection multiplicity lcm(β′ − β) with W_{0} and hence with the hyperplane H_{p}. Since in a neighborhood of [X_{0}] the Severi variety

V^{d,δ}(α, β) = ⋃_{Λ} \overline{ϕ_{Λ}^{−1}(Δ_{m−1} \ Δ_{m})}

we conclude finally that near [X_{0}], V^{d,δ}(α, β) will have \binom{β′}{β}I^{β′−β}/lcm(β′ − β) branches, each of which will have intersection multiplicity lcm(β′ − β) with H_{p} along V′. This completes the proof of Proposition 4.8 and thereby of Theorem 1.3

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Lucia Caporaso Mathematics department, Harvard University, Cambridge MA 02138, USA and Dipartimento di Matematica, Università di Roma “Tor Vergata”, Roma 00133, Italy caporaso@abel.harvard.edu

Joe Harris Mathematics department, Harvard University, Cambridge MA 02138, USA harris@abel.harvard.edu

January 14, 1997