ENUMERATING RATIONAL CURVES: THE RATIONAL FIBRATION METHOD

Lucia Caporaso and Joe Harris

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. The rational fibration method in general . . . . . . . . . . . . . . . . . . . . . . . 4 3. Plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4. The general recursion for 𝔽_{2} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5. The general recursion for 𝔽_{3} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1. Introduction

1.1. Statement of the problem.

Let S be a smooth projective rational surface and let D be an effective divisor on S. Let V (D) ⊂ |D| be the closure of the locus of irreducible rational curves. For general results about the geometry of V (D), we refer to [H] and to [CH].

If D has nonnegative self-intersection and V (D) is nonempty the dimension of V (D) is known (cf. [K]):

r_{0}(D) := dim V (D) = −(K_{S} · D) − 1.

The problem that we will study here is to compute the degrees

N(D) := deg V (D)

of these varieties as subvarieties of |D| ≅ ℙ^{r}. Alternatively, N(D) is the number of irreducible rational curves in |D| that pass through r_{0}(D) general points of S. If S is the projective plane ℙ^{2} and d = deg D, then one also uses the notation N(d) to denote the number of irreducible, rational curves of degree d passing through 3d − 1 general points.

1.2. Terminology and notation.

We will work over the complex numbers. Throughout, the words “surface” and “curve” will refer to projective varieties.

If D and D′ are effective divisors (or divisor classes) on a surface, we will say that D > D′ if D − D′ is effective and nonzero.

We will denote by 𝔽_{n} the Hirzebruch surface 𝔽_{n} = ℙ(O_{ℙ^{1}} ⊕ O_{ℙ^{1}}(n)). On each 𝔽_{n} with n ≥ 1 there exists a unique curve of negative self intersection, which we will denote by E and refer to as the exceptional curve on 𝔽_{n}. We will denote by F a fiber of the projection 𝔽_{n} → ℙ^{1}; the classes of E and F generate the Picard group of 𝔽_{n}, with intersection pairing given by

E^{2} = −n;  (E · F) = 1  and  F^{2} = 0.

Another useful divisor class is the class of a complementary section, that is, a section C of the ℙ^{1}-bundle 𝔽_{n} → ℙ^{1} disjoint from E. Since (C · E) = 0 and (C · F) = 1, we see that C ≡ E + nF; so the classes C and F also generate the Picard group, with intersection numbers

C^{2} = n;  (C · F) = 1  and  F^{2} = 0.

For any positive integer m, we will denote by V_{m}(D) ⊂ V (D) ⊂ |D| the closure of the locus of irreducible rational curves X having contact of order at least m with E at a smooth point of X. These varieties will also be referred to as Severi varieties. We set N_{m}(D) := deg V_{m}(D).

1.3. Methods and results.

Until very recently, the basic enumerative problem of determining the degrees of Severi varieties was unsolved even in the case of ℙ^{2}. In 1989 Ziv Ran [R] described a recursive procedure for calculating the degrees of the Severi varieties parametrizing plane curves of any degree and genus (see also [R2]). Recently, M. Kontsevich discovered a beautiful and simple recursive formula in the case of rational curves on ℙ^{2} (see [KM] and [RT] for proofs). Kontsevich’s method was based on his description of a compactified moduli space for maps of ℙ^{1} into the surface S = ℙ^{2}; others (e.g., [DI], [KP] and [CM]) were able to use the same method to derive similar formulas in the case of other surfaces S for which a Kontsevich-style moduli space existed, such as S = ℙ^{1} × ℙ^{1}, the ruled surface S = 𝔽_{1} and del Pezzo surfaces.

It was our feeling that the reliance of Kontsevich’s method on the existence of a well-behaved moduli space was not essential. We were especially interested in whether a similar formula might be derived for the Hirzebruch surfaces 𝔽_{n}.

In [CH], we succeeded in recasting the Kontsevich method so as to remove the apparent dependence on the existence of a moduli space: as we set it up, it was necessary only to understand the degenerations of the rational curves in the one-parameter families corresponding to general onedimensional linear sections of V (D). The resulting “cross-ratio method” allowed us to derive a complete recursion for all divisor classes on the ruled surface S = 𝔽_{2}—that is, a formula expressing N(D) in terms of N(D′) for D′ < D—and a closed-form formula for certain divisor classes on the ruled surfaces 𝔽_{n} for any n. (In fact, compactifications of the moduli space of maps ℙ^{1} → S do exist for these surfaces, but they contain in general many components, only one of which parametrizes generically irreducible rational curves and the others of which may have strictly larger dimension. Kontsevich’s method can be carried out in these cases, as was done by Kleiman and Piene [KP]; but at present we do not see how to use the resulting formulas to enumerate irreducible rational curves.)

However, we were unable to go significantly beyond this point: a similarly derived formula in [CH] for the degrees N(D) of Severi varieties V (D) on 𝔽_{n} expresses N(D) not solely in terms of N(D′) for D′ < D, but also in terms of the degrees N_{k}(D′′) of the Severi varieties V_{k}(D′′) parametrizing curves with a point of k-fold tangency with a fixed curve E ⊂ S. For example, if S = 𝔽_{3}, then N(D) is expressed as a function of N(D′) and of N_{2}(D′′), where N_{2}(D′′) is the number of irreducible rational curves in |D′′| that are simply tangent to E and pass through the appropriate number (that is, r_{0}(D′′) − 1) of general points of 𝔽_{3}. A complete recursion in this case would have required a similar analysis of linear sections of the Severi varieties V_{2}(D), which in turn would have have necessitated an analysis of Severi varieties parametrizing curves with more complicated tangency conditions.

In the end, it seems that one way or another we need to deal with the degrees of these “tangential” loci as well. This difficulty led us to the discovery of a computational technique different from and simpler than the cross-ratio method, which we will describe in the present paper. It involves an analysis of the same basic object as the cross-ratio method—that is, the one-parameter family X → Γ of rational curves through r_{0}(D) − 1 general points of S and their limits—but extracts more information from it. It is based on a description of the Néron-Severi group of a minimal desingularization of X (we will therefore refer to it as the “rational fibration method”). The main advantage of this technique for our present purposes is that we are in fact able compute the degrees of the tangential loci involved; at least in all cases that we studied. It also yields other related formulas, such as the number of irreducible rational curves having a node at a given general point p ∈ S and passing through r_{0}(D) − 2 other general points.

1.4. Contents of this paper.

In the following section we will describe the rational fibration method in a general setting. In the succeeding sections we will apply it in the cases S = ℙ^{2}, S = 𝔽_{2} and S = 𝔽_{3}. In the first of these cases, we obtain another (simpler) proof of Kontsevich’s formula, as well as some related formulas derived by Pandharipande [P]. In the second, we will recover the general recursion formula found originally in [CH] for degrees of Severi varieties on 𝔽_{2}. Finally, in the last section we derive a complete set of recursions for 𝔽_{3}, the first case for which the cross-ratio method does not give a complete answer.

We have tried to keep this paper relatively self-contained; in particular it should be intelligible to a reader unfamiliar with [CH]. We will, however, have to appeal to chapter 2 of [CH] for proofs of some of the basic assertions about the local geometry of Severi varieties and the families of curves they parametrize.

2. The rational fibration method in general

2.1. Objects and morphisms.

As we indicated in the introduction, the rational fibration method, like the cross-ratio method, involves studying a suitably general one-parameter family of rational curves. To set it up, first let S be a smooth rational surface and D an effective divisor on S. We will assume that D has nonnegative self-intersection and that the Severi variety V (D) ≠ ∅, so that in particular we have dim V (D) = r_{0}(D) = −(K_{S} ·D)−1. Now, choose r_{0}(D)−1 general points p_{1}, …, p_{r_{0}(D)−1} ∈ S and let Γ ⊂ V (D) be the closure of the locus of points [X] ∈ V (D) corresponding to irreducible rational curves X passing through these points. Equivalently, if for any point p ∈ S we let H_{p} ⊂ ℙ^{r} be the hyperplane of points corresponding to curves passing through p, Γ will be the one-dimensional linear section of V (D)

Γ = V (D) ∩ ( ⋂_{i=1}^{r_{0}(D)−1} H_{p_{i}}).

Now, let X ⊂ Γ × S → Γ be the family of curves corresponding to Γ ⊂ |D|. Consider the normalization Γ^{ν} → Γ of Γ, and then take the normalization X^{ν} of X ×_{Γ} Γ^{ν} to arrive at a family

X^{ν} ⟶ Γ^{ν}

over a smooth curve Γ^{ν}, whose general fiber is isomorphic to ℙ^{1}.

Next, we apply semi-stable reduction (which we should rather call “nodal reduction”, since our curves have genus zero): after making a base change B → Γ^{ν} and blowing up the total space of the pullback family X^{ν} ×_{Γ^{ν}} B, we arrive at a family f : Y → B whose total space is smooth, whose general fiber is a smooth rational curve and whose special fibers are all nodal curves. In fact, a base change will turn out to be unnecessary in each of the three cases considered below— the minimal desingularization Y of the total space X^{ν} already has this property—but this is not relevant, since even a superfluous base change will not affect the subsequent calculations. We will denote by π : Y → S the composite map

π : Y → X^{ν} ×_{Γ^{ν}} B → X^{ν} → X ⊂ Γ × S → S.

Notice that π is a generically finite map, whose degree is equal to the product of the degree of the map B → Γ^{ν} times the number of irreducible rational curves in the linear series |D| passing through the points p_{1}, …, p_{r_{0}(D)−1} and p (that is, the degree N(D) of the Severi variety V (D)).

Here is a diagram of the basic objects and morphisms we have introduced:

2.2. Outline of the method.

As we indicated, our method involves calculating in the Néron-Severi group of the total space Y of our family. This is motivated by a simple observation: given any two line bundles L and M on S, we have

(π^{∗}L · π^{∗}M) = deg{B → Γ^{ν}} · N(D) · (L · M).

Thus, in order to derive a formula for N(D), we want to compute intersection numbers in the Néron-Severi group of Y. For example, if S is the projective plane ℙ^{2} we can take L = M = O_{ℙ^{2}}(1). Then (π^{∗}L)^{2} = N(D); so if we can compute (π^{∗}L)^{2} we get a formula for N(D).

What makes it possible to perform such calculations is the fact that f expresses Y as the total space of a one-parameter family of generically smooth rational curves, so that to determine the class of a given divisor it is enough to know its degree on each component of each fiber of f. More precisely, the Picard group of Y will be freely generated by the class of a fiber, the class of any section A of f, and the classes of all the irreducible curves contained in fibers of f and disjoint from A. Moreover, in terms of these generators the intersection pairing on Pic(Y) is (except possibly for the self-intersection of A) easy to describe. This means two things: first, we can express a given divisor class as a linear combination of these generators once we know its degree on each component of each reducible fiber and on A; and second, having expressed two divisor classes as linear combinations of these generators, we can readily compute their intersection number.

The method we will apply in each case thus consists of five steps:

• First, we need to describe the reducible fibers of Y → B; that is (given that B will be in practice just the normalization Γ^{ν} of the base of our original family X → Γ), the set of reducible curves in the linear series |D| through the points p_{1}, …, p_{r_{0}(D)−1} that are limits of irreducible rational curves through these points, and the branches of Γ at each one. The characterization of such curves is straightforward in the case of S = ℙ^{2} by simple dimensioncounting. In the case of S = 𝔽_{n} with n ≥ 2 it is less obvious, since in contrast with the case of ℙ^{2} most reducible curves through the points p_{1}, …, p_{r_{0}(D)−1} whose components are all rational are not limits of irreducible rational such curves; the answer is worked out in [CH]. In either case the number of such fibers will be known inductively.

• Secondly, we need to describe the local structure of the family X^{ν} → Γ^{ν} near each reducible fiber; specifically, we need to know whether X^{ν} is smooth or if we have to blow up. This likewise is straightforward in the case of the plane, where in fact X^{ν} is smooth. It is more interesting in the case of the Hirzebruch surfaces 𝔽_{n}, where for n ≥ 3 we see that X^{ν} will indeed have singularities; again, this is worked out in [CH] and we will refer there for the relevant results.

• Third, we choose a basis for the Néron-Severi group of Y, and calculate the intersection pairing on these classes.

• Fourth, since we know the images in S of the components of reducible fibers of f : Y → B, we can calculate the degrees on all such components of the pullback π^{∗}L of any line bundle L on S; and

• Fifth, we are able therefore to express the intersection numbers (π^{∗}L · π^{∗}M) for pairs L, M ∈ PicS of line bundles on S.

Evidently, the particulars of this process will depend on S and D; for the moment we shall just fix some notation and make some preliminary observations. First, for b ∈ B we use the common notation Y_{b} := f^{−1}(b) to denote the fiber of f over b. The class in NS(Y) of such a fiber is denoted by Y.

Secondly, recall that our family parametrizes curves through certain base points. We pick two of them, q and q′, and we denote by A and A′ the corresponding sections of f. The following relations are clear:

Y^{2} = A · A′ = 0 and A · Y = A′ · Y = 1.

Notice also that by symmetry A^{2} = \frac{1}{2}(A − A′)^{2} which will be useful to compute the left hand side. In fact A − A′ is supported on exactly those fibers where q and q′ lie on different components, the number of which we will be able to count.

One further note: the description above of the Néron-Severi group of Y as generated by the classes of A, Y and components of reducible fibers assumes that the base B of the family is connected, which we will not always know in practice. This assumption is not essential, however: in case B has irreducible components B_{1}, …, B_{k} we simply have to replace every multiple of Y in the formulas below by a suitable linear combination of fibers Y_{i} lying over points of B_{i}. As the reader may verify, this does not alter the outcome of the subsequent calculations.

3. Plane curves

Here we study the case S = ℙ^{2}. If D and D_{i} are divisors on ℙ^{2}, we denote their degrees respectively by d and d_{i}. Since a divisor class in the plane is determined by its degree, we will introduce the notation N(d) := N(D).

We have that r_{0}(D) = 3d − 1, so we choose general points p_{1}, …, p_{3d−2} ∈ ℙ^{2}, let Γ ⊂ V (D) be the locus of curves in V (D) containing the points p_{i}, and proceed as described in the preceding section. To describe the resulting family of curves, let Δ be the locus of V (D) parametrizing degenerate curves (that is, curves that are reducible or have singularities other than nodes). Since our curve Γ ⊂ V (D) will intersect Δ only at general points of components of Δ, we may apply the results of [DH] and [H] to conclude the following A. Any fiber X_{γ} of X → Γ is either

1. an irreducible curve with exactly δ = \frac{(d−1)(d−2)}{2} nodes;

2. an irreducible curve with exactly δ − 1 nodes and a cusp;

3. an irreducible curve with exactly δ − 2 nodes and a tacnode;

4. an irreducible curve with exactly δ − 3 nodes and an ordinary triple point; or

5. a curve having exactly two irreducible components X_{1}, X_{2}, of degrees d_{1} and d_{2}, with exactly \frac{(d_{1}−1)(d_{1}−2)}{2} and \frac{(d_{2}−1)(d_{2}−2)}{2} nodes respectively, and intersecting transversally in d_{1}d_{2} points.

B. In cases 1, 3 and 4, the curve Γ is smooth at γ and the family X^{ν} → Γ^{ν} is smooth at the unique point of Γ^{ν} lying over γ. In case 2, Γ has a cusp at γ but the family X^{ν} → Γ^{ν} is still smooth at the unique point of Γ^{ν} lying over γ.

C. In case 5 the curve Γ has d_{1}d_{2} smooth branches at γ (corresponding to deformations of X_{γ} smoothing any one of the d_{1}d_{2} nodes of X_{γ} coming from a point p ∈ X_{1} ∩ X_{2} of intersection of X_{1} and X_{2}). At each point of Γ^{ν} lying over γ the fiber of the family X^{ν} → Γ^{ν} has two smooth rational components meeting transversally at one point (more precisely, it is the normalization of X_{γ} at the remaining \frac{(d_{1}−1)(d_{1}−2)}{2} + \frac{(d_{2}−1)(d_{2}−2)}{2} + d_{1}d_{2} − 1 = δ nodes of X_{γ}), and smooth total space.

D. Finally, if X ⊂ ℙ^{2} is any curve of type 1-5 passing through the points p_{1}, …, p_{3d−2}, then conversely [X] ∈ Γ; that is, X is a limit of irreducible rational curves X_{γ} through p_{1}, …, p_{3d−2}.

We see in particular that the total space X^{ν} is smooth and that the fibers of X^{ν} → Γ^{ν} are all nodal, so that no further base changes or blow-ups are necessary; that is, we may take B = Γ^{ν} (as we stated earlier) and Y = X^{ν}. Note also that every reducible fiber of Y has precisely two irreducible components, meeting transversally at one point.

We shall call a reducible fiber of f a fiber of type J and we shall denote by B_{J} the subset of points of B such that the corresponding fiber is of type J, that is, reducible. (This new piece of terminology probably seems pointless, but it will be useful in the sequel). For any b ∈ B_{J}, then we denote the two irreducible components of the fiber over b by J_{1,b} and J_{2,b}. We shall always denote by J_{1,b} the component containing the point q = p_{1}. The picture of Y is thus:

Let D_{i} be the class of π(J_{i,b}). We denote by j(D_{1}, D_{2}) the number of all such fibers, for any given decomposition D = D_{1} + D_{2}. To determine j(D_{1}, D_{2}), note first that if X = X_{1} ∪ X_{2} ∈ Γ is any reducible curve, D_{i} the class of X_{i}, then X_{i} can contain at most r_{0}(D_{i}) of the r_{0}(D) − 1 = 3d − 2 points p_{1}, …, p_{3d−2}. Since

r_{0}(D_{1}) + r_{0}(D_{2}) = r_{0}(D) − 1,

it follows that each component X_{i} must contain exactly r_{0}(D_{i}) of the points p_{1}, …, p_{3d−2}. Thus, to specify such a curve, we have first to choose a decomposition of the set Φ = {p_{1}, …, p_{3d−2}} into disjoint subsets Φ_{1}, Φ_{2} of cardinality r_{0}(D_{1}) and r_{0}(D_{2}) respectively, with the point q = p_{1} ∈ Φ_{1}; and then to choose, for each i, one of the N(D_{i}) curves X_{i} ∈ V (D_{i}) containing Φ_{i}. The number of such curves X is thus

\binom{r_{0}(D) − 2}{r_{0}(D_{1}) − 1} · N(D_{1}) · N(D_{2})

and since we have seen there are (D_{1} · D_{2}) = d_{1}d_{2} points of B lying over each point [X] ∈ Γ corresponding to a curve of this type, we have

j(D_{1}, D_{2}) = N(D_{1})N(D_{2})(D_{1} · D_{2})\binom{r_{0}(D) − 2}{r_{0}(D_{1}) − 1} = N(d_{1})N(d_{2})d_{1}d_{2}\binom{3d − 3}{3d_{1} − 2}.

Note that by a simple dimension count, any of the curves X_{i} ∈ V (D_{i}) passing through r_{0}(D_{i}) of the points p_{1}, …, p_{3d−2} will be irreducible and nodal. By a standard further argument as in Lemma 2.1 of [CH], we see that any pair X_{1}, X_{2} of such curves will intersect transversally, so that the union X = X_{1} ∪ X_{2} will indeed be a curve as described in (5) above.

This completes the first two steps in the general method. Next, we give a basis for the Néron-Severi group NS(Y). We now choose as a system of generators for NS(Y) the class A of the section of f : Y → B coming from the base point q = p_{1} of our family; the class Y of a fiber of the map f : Y → B, and the classes {J_{2,b}}_{b∈B_{J}}. Most of the pairwise intersection numbers of these classes are readily given: we clearly have

\begin{aligned}(A · Y) & = 1 \\ Y^{2} & = 0 \\ (A · J_{2,b}) & = 0 ∀ b \\ (Y · J_{2,b}) & = 0 ∀ b \\ (J_{2,b} · J_{2,b′}) & = 0 ∀ b ≠ b′; and \\ (J_{2,b} · J_{2,b}) & = −1 ∀ b . \end{aligned}

In fact, there is only one intersection number that is not evident: A^{2}. To compute it we choose a base point q′ ≠ q, so that q′ determines a second section A′ of Y ⟶ B disjoint from A. Since the base points p_{1}, …, p_{3d−2} of our family are general points in the plane, by symmetry we have A^{′2} = A^{2}; hence we can write

2A^{2} = (A − A′)^{2}.

To compute the right hand side, let

S_{J} = {b ∈ B_{J} such that q′ ∈ π(J_{2,b})}

be the collection of points in B over which the sections A and A′ meet different components of the fiber; let n_{J} = |S_{J}| be the cardinality of S_{J}. For every b ∉ S_{J}, A and A′ have the same intersection number with each component of the fiber Y_{b}. For b ∈ S_{J}, on the other hand, we have (A · J_{1,b}) = 1 and (A · J_{2,b}) = 0, while (A′ · J_{1,b}) = 0 and (A′ · J_{2,b}) = 1. It follows that the classes

A  and  A′ − ∑_{b∈S_{J}} J_{2,b}

have the same intersection number with every component of every fiber of Y ⟶ B, and so must differ by a multiple of the class Y of a fiber: that is,

A − A′ = − ∑_{b∈S_{J}} J_{2,b} + nY

for some integer n. In fact, n must be equal to n_{J}/2 by symmetry, but that is irrelevant in any case: squaring both sides, we find that

(A − A′)^{2} = ∑_{b∈S_{J}} J_{2,b}^{2} = −n_{J}

and hence

A^{2} = −\frac{n_{J}}{2} .

Thus, it remains only to determine the number n_{J} of reducible fibers of Y ⟶ B lying over curves in our original family in which the points q and q′ lie in different components. We can do this in exactly the same way as we determined the total number of reducible fibers: the only difference is that now we want to count only decompositions Φ = Φ_{1} ∪ Φ_{2} in which q = p_{1} ∈ Φ_{1} and q′ = p_{2} ∈ Φ_{2}. We thus replace the binomial coefficient \binom{r_{0}(D)−2}{r_{0}(D_{1})−1} in the formula for j(D_{1}, D_{2})

above with \binom{r_{0}(D)−3}{r_{0}(D_{1})−1} and sum over all pairs D_{1}, D_{2} with D_{1} + D_{2} = D to obtain

n_{J} = ∑_{D_{1}+D_{2}=D} N(D_{1})N(D_{2})(D_{1} · D_{2})\binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 1}.

This completes the third step of the process.

Now let L ∈ Pic(ℙ^{2}) be any line bundle on the plane, and write the class of its pullback to Y as a general linear combination of our chosen generators

π^{∗}L ≡ c_{a}A + c_{y}Y + ∑_{b∈B_{J}} c_{b}J_{2,b}.

We will denote by J^{L} the third term on the right, that is, we set

J^{L} := ∑_{b∈B_{J}} c_{b}J_{2,b};

this notation is not immediately useful, but will become so in the succeeding calculations.

We now intersect both sides of the above equivalence with each of our chosen generators of NS(Y) to determine the coefficients c_{a}, c_{y} and c_{b}. First, by intersecting both sides with Y we find that

c_{a} = (π^{∗}L · Y) = (L · π_{∗}Y) = (L · D) .

Next we intersect with A: we have

(π^{∗}L · A) = (L · π_{∗}A) = 0

since π is constant on the curve A; and hence

c_{y} = −A^{2}(L · D) = \frac{n_{J}}{2} (L · D) .

Finally, to determine c_{b} we naturally intersect both sides with the class of J_{2,b}; we find that

c_{b} = −(π^{∗}L · J_{2,b}) = −(L · π_{∗}J_{2,b}) = −(L · D_{2}) .

Thus, in sum,

π^{∗}L ≡ (L · D)A + \frac{n_{J}}{2} (L · D)Y − ∑_{b∈B_{J}}(L · D_{2})J_{2,b} .

For the final step in the process, we evaluate the self-intersection of π^{∗}L: we find

\begin{aligned}(π^{∗}L)^{2} & = \frac{n_{J}}{2} (L · D)^{2} − ∑_{b∈B_{J}}(L · D_{2})^{2} \\ & = ∑_{D_{1}+D_{2}=D}[\frac{1}{2}N(D_{1})N(D_{2})(D_{1} · D_{2})\binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 1}(L · D)^{2} \\ & − N(D_{1})N(D_{2})(D_{1} · D_{2})\binom{r_{0}(D) − 2}{r_{0}(D_{1}) − 1}(L · D_{2})^{2}] \end{aligned} .

Applying this in case L = O_{ℙ^{2}}(1), and recalling that (π^{∗}O_{ℙ^{2}}(1))^{2} = N(d), we have

N(d) = ∑_{d_{1}+d_{2}=d} N(d_{1})N(d_{2}) [\frac{d^{2}d_{1}d_{2}}{2} \binom{3d − 4}{3d_{1} − 2} − d_{1}^{2}d_{2}^{2}\binom{3d − 3}{3d_{1} − 2}] ;

and expanding out d^{2} = (d_{1} + d_{2})^{2} and using the symmetry with respect to d_{1} and d_{2} we get the well known recursive formula of Kontsevich

N(d) = ∑_{d_{1}+d_{2}=d} N(d_{1})N(d_{2}) [d_{1}^{2}d_{2}^{2}\binom{3d − 4}{3d_{1} − 2} − d_{1}d_{2}^{3}\binom{3d − 4}{3d_{1} − 3}] .

\underline{Remark}. In parts (A) and (B) of the statement of results quoted from [H] and [DH], we describe completely all curves X_{γ} in the family X → Γ having other than δ nodes, and the local geometry of X^{ν} → Γ^{ν} along each. This is in fact necessary to describe the Néron-Severi group of Y, since even in those cases where a fiber is irreducible it is a priori possible that X^{ν} will be singular along such a fiber, giving rise to a reducible fiber of Y → B. Looking back over the preceding calculation, though, we see that even if this did happen, it would not affect the outcome of the calculation, as long as the non-nodal singularities did not occur at base points of the family: while the resolution of the singularities of X^{ν} would create additional curves on Y independent in NS(Y), the sections A and A′ and any line bundle pulled back via π from ℙ^{2}, would all have degree 0 on these curves, and so the relations of linear equivalence above would still hold.

Thus it was only necessary to observe that every curve singular at a base point is an irreducible curve with δ nodes. Since this statement will also hold for the families of curves on 𝔽_{n} that we will be considering in the following two sections, we will in the sequel omit the description of the fibers X → Γ other than reducible ones.

As another application, we give a formula for the number N_{2}(d) of plane, irreducible, rational curves X ⊂ ℙ^{2} of degree d passing through 3d − 2 given general points and tangent to a given general line ℓ in the plane. Equivalently, this is the degree of the subvariety V_{2}(D) of V (D) defined as the closure of the locus of irreducible rational curves that are tangent to ℓ in ℙ^{2} at a smooth point of [X] (notice that V_{2}(D) has codimension 1 in V (D)). To calculate this number, let L̃ = π^{−1}(ℓ) ⊂ Y be the preimage of ℓ under π. Then L̃ is an irreducible smooth curve, and the morphism f : Y → B restricts to a finite morphism f̃: L̃ → B of degree d on L̃. Moreover, the set of fibers Y_{b} ⊂ Y of f tangent to L̃—that is, such that the intersection Y_{b} ∩ L̃ has cardinality strictly less than d—corresponds to the set of curves X_{b} in our original family X → Γ tangent to ℓ. Thus, N_{2}(D) is equal to the degree of the ramification divisor of the morphism f̃.

Now, using the adjunction formula, this degree is given by

N_{2}(d) = (π^{∗}O_{ℙ^{2}}(1))^{2} + (ω_{Y/B} · π^{∗}O_{ℙ^{2}}(1)).

where ω_{Y/B} is the relative dualizing sheaf of the family. Since we have already calculated the class of π^{∗}O_{ℙ^{2}}(1) above, it remains only to determine the class of ω_{Y/B} in similar terms, and then we will be able to evaluate this expression. We do this as for π^{∗}O_{ℙ^{2}}(1): we first express ω_{Y/B} as a linear combination of the generators:

ω_{Y/B} ≡ e_{a}A + e_{y}Y + ∑_{b∈B_{J}} e_{b}J_{2,b}

and then intersect both sides with the generators of NS(Y) to determine the coefficients. First, intersecting with Y, we find that

e_{a} = (ω_{Y/B} · Y) = −2

and then intersecting with A and using the fact that (ω_{Y/B} · A) = −A^{2} we find that

e_{y} = A^{2} .

Finally, we have (ω_{Y/B} · J_{2,b}) = −1, ∀b ∈ B_{J}, and it follows that the coefficients e_{b} are all 1. Thus, in sum,

ω_{Y/B} = −2A + A^{2}Y + ∑_{b∈B_{J}} J_{2,b} .

We finally obtain a formula first found by Pandharipande [P]:

\begin{aligned}N_{2}(d) & = N(d) + ∑_{d_{1}+d_{2}=d} N(d_{1})N(d_{2})d_{1}d_{2}^{2}\binom{3d − 4}{3d_{1} − 3} \\ & = ∑_{d_{1}+d_{2}=d} N(d_{1})N(d_{2})d_{1}d_{2} [d_{1}d_{2}\binom{3d − 4}{3d_{1} − 2} − (d_{2}^{2} − d_{2})\binom{3d − 4}{3d_{1} − 3}] \end{aligned} .

This technique can also be used to recover another formula of Pandharipande, for the degree of the closure of the locus of irreducible rational curves of degree d having a cusp. To obtain this, we simply apply Porteous’ formula to the differential

d(f × π) : T_{Y} → (f × π)^{∗}T_{B×ℙ^{2}}

of the map f × π : Y → B × ℙ^{2}; the classes on Y involved have already been calculated. It should also be possible to determine in similar fashion the degrees on Γ of all the divisor classes introduced in [DH], and in particular obtain formulas for the number of irreducible rational curves through 3d − 2 points and having a tacnode, or the number of irreducible rational curves through 3d − 2 points and having a triple point, etc. At this point, however, we conclude our study of the plane and turn to the Hirzebruch surfaces.

3. The general recursion for 𝔽_{2}

Let now S = 𝔽_{2}. Let C, E and F be the curves on 𝔽_{2} described in Section 1. We apply our method exactly as before: for any effective divisor class D on S with V (D) ≠ ∅, we choose r_{0}(D)−1 general points p_{1}, …, p_{r_{0}(D)−1} of S and consider the family X → Γ of curves X ∈ V (D) passing through the points p_{i}; we let X^{ν} → Γ^{ν} and Y → B be derived from this family as in the general set-up.

We first describe the various types of reducible fibers that our family Y ⟶ B has. The following analysis is based on Propositions 2.1, 2.5, 2.6 and 2.7 of [CH]. In particular, the various types of degenerations can be classified as an application of Proposition 2.5, and the singularities of Γ at the points γ ∈ Γ corresponding to each as an application of Proposition 2.6. Moreover, Proposition 2.7 assures us that, just as in the case of ℙ^{2}, the normalization X^{ν} is smooth and so the total space Y coincides with X^{ν}. In particular, we see that no irreducible component of any fiber of Y is mapped to a point by π. With that said, we have the following classification of reducible fibers of Y → B:

Type J. Fibers having two smooth irreducible components J_{1} and J_{2}, meeting transversally at one point, such that π(J_{i}) = D_{i} with D_{i} > 0 and not equal to E. We will always assume that q ∈ J_{1}. For any decomposition D = D_{1} + D_{2} we have that the number j(D_{1}, D_{2}) of fibers of type J such that π(J_{i}) = D_{i} is

j(D_{1}, D_{2}) = N(D_{1})N(D_{2})(D_{1} · D_{2})\binom{r_{0}(D) − 2}{r_{0}(D_{1}) − 1}.

The factor (D_{1} · D_{2}) appears because, just as in the case of ℙ^{2}, if [X] ∈ Γ corresponds to a curve of type J, in the normalization map ν : B ⟶ Γ the fiber over [X] contains exactly (D_{1} · D_{2}) points (here we are using Proposition 2.6 of [CH]).

We let B_{J} be the subset of points b in B whose fiber X_{b} is a curve of type J.

Type G. Fibers having two smooth irreducible components G_{1} and G_{E}, meeting transversally at one point, such that π(G_{E}) = E and π(G_{1}) is simply tangent to E. Clearly q ∈ G_{1}. The total number of such fibers will not matter in the subsequent calculation.

We let B_{G} be the subset of points b in B whose fiber X_{b} is a curve of type G.

Type H. Fibers having three irreducible components H_{1}, H_{2}, H_{E}, such that π(H_{E}) = E and π(H_{i}) = D_{i}, with D_{i} > 0 and D_{1} + D_{2} = D − E (again, we will choose the labelling so that q ∈ H_{1} always). By Proposition 2.6 of [CH], if [X] ∈ Γ is a point corresponding to this type of curve, then the fiber of B over Γ contains exactly (D_{1} · E)(D_{2} · E) points. Hence the total number of fibers of type H that correspond to a given decomposition D = D_{1} + D_{2} + E is

h(D_{1}, D_{2}) = ∑_{D_{1}+D_{2}=D} N(D_{1})N(D_{2})(D_{1} · E)(D_{2} · E)\binom{r_{0}(D) − 2}{r_{0}(D_{1}) − 1}

And just like for the other types, we define B_{H} to be the subset of points of B parametrizing curves of type H.

The picture of Y ⟶ B thus looks like this:

Now we choose the following set of generators for the NéronSeveri group of Y

{A, Y } ∪ {J_{2,b}}_{b∈B_{J}} ∪ {G_{E,b}}_{b∈B_{G}} ∪ {H_{2,b}, H_{E,b}}_{b∈B_{H}} .

The following relations are obvious

G_{E,b}^{2} = J_{2,b}^{2} = H_{2,b}^{2} = −1; H_{E,b}^{2} = −2

and the intersection number of A with any generator other than A and Y is zero.

We compute A^{2} by the same argument as used in the preceding section. We define

S_{H} = {b ∈ B_{H} such that q′ ∈ H_{2,b}}

and we let n_{H} = |S_{J}|. Then we have

n_{H} = ∑_{D_{1}+D_{2}=D−E} N(D_{1})N(D_{2})(D_{1} · E)(D_{2} · E)\binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 1}

Similarly, we let S_{J} and n_{J} to be defined exactly as in the preceding section, and notice that the value of n_{J} is expressed by the same formula that we had in the plane. We obtain

A − A′ = − ∑_{b∈S_{J}} J_{2,b} − ∑_{b∈S_{H}}(H_{E,b} + 2H_{2,b}) + nY

for some integer n. Hence

A^{2} = −\frac{n_{J} + 2n_{H}}{2} .

Let now L ∈ Pic(𝔽_{2}). We want to compute the coefficients of π^{∗}L as a linear combination of the chosen generators of NS(Y). The number of generators being quite large, it is now convenient to use the following notation: if W is any of the chosen generators, we shall denote by {π^{∗}L}_{W} the coefficient of π^{∗}L with respect to W. We shall then write

π^{∗}L = {π^{∗}L}_{A}A + {π^{∗}L}_{Y} Y + G^{L} + J^{L} + H^{L}

where J^{L} is defined just as in the preceding section, and similarly

G^{L} = ∑{π^{∗}L}_{G_{E,b}}G_{E,b} and H^{L} = ∑({π^{∗}L}_{H_{2,b}}H_{2,b} + {π^{∗}L}_{H_{E,b}}H_{E,b}).

We could now easily compute all the missing numbers in term of intersection numbers on 𝔽_{2}; only we don’t really need it. All we need is the expression for π^{∗}C; in fact we shall obtain a formula for N(D) by using the fact that (π^{∗}C)^{2} = 2N(D). The following numbers are obtained in a straightforward way, just as in the case of ℙ^{2}.

{π^{∗}C}_{A} = (C · D) = a.

\{π^{∗}C\}_{Y} = −(C · D)A^{2} = \frac{a(n_{J} + 2n_{H})}{2}.

{π^{∗}C}_{G_{E}} = 0 for any generator of type G_{E}. {π^{∗}C}_{J_{2,b}} = −(C · π(J_{2,b}))

And for any curve H = H_{1} + H_{2} + H_{E} of type H such that π(H_{i}) = D_{i} we have

{π^{∗}C}_{H_{E}} = −(C · D_{2}) and {π^{∗}C}_{H_{2}} = −2(C · D_{2})

In conclusion, we get the same recursive formula that we obtained in [CH]:

Theorem. For any effective divisor D ≠ E on 𝔽_{2} with V (D) ≠ ∅,

\begin{matrix}N(D) = \frac{1}{2} ∑_{D_{1}+D_{2}=D} N(D_{1})N(D_{2})(D_{1} · D_{2}) [\binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 1}(D_{1} · C)(D_{2} · C) − \binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 2}(D_{2} · C)^{2}] \\ + ∑_{D_{1}+D_{2}=D−E} N(D_{1})N(D_{2})(D_{1} · E)(D_{2} · E) [\binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 1}(D_{1} · C)(D_{2} · C) − \binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 2}(D_{2} · C)^{2}] \end{matrix}

4. The general recursion for 𝔽_{3}

Let now S = 𝔽_{3}. and let C, E and F be as in 1.2. Let D be an effective divisor class D on S with V (D) ≠ ∅. We also introduce in this case two additional subvarieties of the linear series |D|: the subvariety V_{2}(D) of V (D) defined to be the closure of the locus of irreducible rational curves X tangent to E at a smooth point of X; and the closure F (D) of the subvariety of V (D) parametrizing irreducible curves having a smooth point of intersection multiplicity 3 with E. Their degrees will be denoted by N_{2}(D) and N_{3}(D) respectively.

Now we proceed as before: we choose r_{0}(D) − 1 general points p_{1}, …, p_{r_{0}(D)−1} of S and consider the family X → Γ of curves X ∈ V (D) passing through the points p_{i}; we let X^{ν} → Γ^{ν} and Y → B be derived from this family as in the general set-up. Our method will again provide us with a recursive formula for the degree of V (D), but there will be now an important difference: the recursion will involve as well the degrees N_{2}(D) and N_{3}(D) of the varieties V_{2}(D) and F (D). More precisely, we are going to obtain three formulas:

(a) A formula expressing N(D) in terms of N(D′) and N_{2}(D′′), where D′ < D and D′′ < D − E;

(b) A formula expressing N_{2}(D) in terms of N(D), N(D′), N_{2}(D′′) and N_{3}(D−E), where D′ < D and D′′ < D − E; and

(c) A formula expressing N_{3}(D − E) in terms of N(D′) and N_{2}(D′′), where D′ < D and D′′ < D − E.

We now describe the various reducible fibers of Y ⟶ B. Again we use the results of [CH], in particular, Proposition 2.5and 2.7 for the geometry of the normalization map B ⟶ Γ and of the total space X^{ν}. By 2.7 we have that X^{ν} is smooth at points lying on fibers corresponding tp types J, G and H below; in other words, no irreducible component of a fiber belonging to one of these types is mapped to a point of 𝔽_{3}.

Type J. (This is the exact analog of the type J for 𝔽_{2}) Fibers having two smooth irreducible components J_{1} and J_{2}, meeting transversally at one point, such that π(J_{i}) = D_{i} with D_{i} > 0 and not equal to E. We will always assume that q ∈ J_{1}. For any decomposition D = D_{1} + D_{2} we have that the number j(D_{1}, D_{2}) of fibers of type J such that π(J_{i}) = D_{i} is

j(D_{1}, D_{2}) = N(D_{1})N(D_{2})(D_{1} · D_{2})\binom{r_{0}(D) − 2}{r_{0}(D_{1}) − 1}.

We have the coefficient (D_{1} · D_{2}) because, just as in the case of ℙ^{2}, if [X] ∈ Γ corresponds to a curve of type J, in the normalization map ν : B ⟶ Γ the fiber over [X] contains exactly (D_{1} · D_{2}) points.

We let B_{J} be the subset of points b in B whose fiber X_{b} is a curve of type J.

Type G. Fibers having two smooth irreducible components G_{1} and G_{E}, meeting transversally at one point, such that π(G_{E}) = E and π(G_{1}) has a smooth point of contact of order 3 with E. Clearly q ∈ G_{1}. The total number of such fibers is N_{3}(D).

We let B_{G} be the subset of points b in B whose fiber X_{b} is a curve of type G.

Type K. Fibers having four irreducible components K_{1}, K_{E}, K_{0} and K_{2} forming a chain in the given order, that is K_{1} ∩K_{E} = K_{0} ∩K_{E} = K_{0} ∩K_{2} = 1 so that K_{1}^{2} = K_{2}^{2} = −1 and K_{E}^{2} = K_{0}^{2} = −2. As usual, we have that q ∈ π(K_{1}). Moreover, π(K_{E}) = E and π(K_{1}) is tangent to E; π(K_{0}) is a point of E (namely, the point E ∩π(K_{2})), in fact the exceptional curve K_{0} arises from the fact that the surface X^{ν} is singular at the point corresponding to E ∩ π(K_{2}) (cf. Proposition 2.7 in [CH]). Let B_{K} be the subset of B whose corresponding fiber is a curve of type K. Finally, for any given decomposition D = E + D_{1} + D_{2} we have that the number k(D_{1}, D_{2}) of corresponding fibers of type K is given by

k(D_{1}, D_{2}) = N_{2}(D_{1})N(D_{2})(E · D_{2})\binom{r_{0}(D) − 2}{r_{0}(D_{1}) − 2}

Type K′. These are just like the fibers of type K with the only difference that the point q belongs to the curve that is not tangent to E, that is, we have now π(K_{2}) tangent to E. We denote the irreducible components of such a fiber K′_{1}, K′_{0}, K′_{E} and K′_{2}, forming a chain in the given order, so that

(K′_{1} · K′_{0}) = (K′_{0} · K′_{E}) = (K′_{E} · K_{2}) = 1

and

(K′_{1})^{2} = (K′_{2})^{2} = −1 and (K′_{E})^{2} = (K′_{0})^{2} = −2.

Moreover, π(K′_{E}) = E and π(K′_{0}) is a point of E (namely, the point E ∩ π(K′_{1})). We define as usual B_{K′} to be the subset of B whose corresponding fiber is a curve of type K′. Finally we see that the number k′(D_{1}, D_{2}) of such fibers is

k′(D_{1}, D_{2}) = N(D_{1})N_{2}(D_{2})(D_{1} · E)\binom{r_{0}(D) − 2}{r_{0}(D_{1}) − 1}

Type H. Fibers having four irreducible components H_{1}, H_{2}, H_{3} and H_{E}, such that π(H_{E}) = E and π(H_{i}) = D_{i}, with D_{i} > 0 and D_{1} + D_{2} + D_{3} = D − E. In Proposition 2.6 of [CH] we proved that if [X] ∈ Γ is a point corresponding to this type of curve, then the fiber of B over Γ contains exactly (D_{1} ·E)(D_{2} ·E)(D_{3} ·E) points. Hence the total number of fibers of type H that correspond to a given decomposition D = D_{1} + D_{2} + D_{3} + E is given by

h(D_{1}, D_{2}, D_{3}) = N(D_{1})N(D_{2})N(D_{3})(D_{1} · E)(D_{2} · E)(D_{3} · E)\binom{r_{0}(D) − 2}{r_{0}(D_{1}) − 2, r_{0}(D_{2})}

And as usual, we define B_{H} to be the subset of points of B corresponding to curves of type H.

Here is a picture displaying the various types of reducible fibers in our family:

Now we choose the following set of generators for the Néron-Severi group of Y:

{A, Y } ∪ {J_{2,b}}_{b∈B_{J}} ∪ {G_{E,b}}_{b∈B_{G}} ∪ {K_{2,b}, K_{E,b}, K_{0,b}}_{b∈B_{K}} ∪
∪{K′_{2,b}, K′_{E,b}, K′_{0,b}}_{b∈B_{K′}} ∪ {H_{3,b}, H_{2,b}, H_{E,b}}_{b∈B_{H}} .

The following relations are obvious:

G_{E,b}^{2} = J_{2,b}^{2} = K_{2,b}^{2} = (K′_{2,b})^{2} = K_{1,b}^{2} = H_{2,b}^{2} = −1;
K_{E,b}^{2} = K_{0,b}^{2} = (K′_{E,b})^{2} = (K′_{0,b})^{2} = −2;
H_{E,b}^{2} = −3

and the intersection number of A with any generator other than A and Y is zero.

It will also be convenient to have a symbol denoting the class in NS(Y) of all generators of the same type. Therefore we introduce the classes

G_{E} := ∑_{b∈B_{G}} G_{E,b}, J_{2} := ∑_{b∈B_{J}} J_{2,b},

K_{E} := ∑_{b∈B_{K}} K_{E,b}, K_{0} := ∑_{b∈B_{K}} K_{0,b}, K_{2} := ∑_{b∈B_{K}} K_{2,b}

K′_{E} := ∑_{b∈B_{K′}} K′_{E,b}, K′_{0} := ∑_{b∈B_{K′}} K′_{0,b}, K′_{2} := ∑_{b∈B_{K′}} K′_{2,b}

and

H_{E} := ∑_{b∈B_{H}} H_{E,b}, H_{3} := ∑_{b∈B_{H}} H_{3,b}, H_{2} := ∑_{b∈B_{H}} H_{2,b}

We now use this notation immediately to write the class of the relative dualizing sheaf ω_{Y/B} of the family f : Y ⟶ B. We have

ω_{Y/B} = −2A + A^{2}Y + G_{E} + J_{2} + K_{E} + 2K_{0} + 3K_{2} + K′_{0} + 2K′_{E} + 3K′_{2} + H_{E} + 2H_{3} + 2H_{2}

Now we compute A^{2}. Let q′ be a base point different from q, then q′ determines a section A′ such that (A · A′) = 0. As before, we get 2A^{2} = (A − A′)^{2} and we can compute the right hand side by expressing the difference A − A′ as a linear combination of components of fibers. So, let n_{J} be the number of fibers of type J such that q′ lies on a different component than q. We have

n_{J} = ∑_{D_{1}+D_{2}=D} N(D_{1})N(D_{2})(D_{1} · D_{2})\binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 1}

Now let n_{K} be the number of fibers of type K such that q′ lies on a different component than q; we have

n_{K} = ∑_{D_{1}+D_{2}=D−E} N_{2}(D_{1})N(D_{2})(E · D_{2})\binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 2}.

We define n_{K′} analogously, and obtain

n_{K′} = ∑_{D_{1}+D_{2}=D−E} N(D_{1})N_{2}(D_{2})(D_{1} · E)\binom{r_{0}(D) − 3}{r_{0}(D_{2}) − 2};

and similarly n_{H}, for which we have

n_{H} = ∑_{D_{1}+D_{2}+D_{3}=D−E} N(D_{1})N(D_{2})N(D_{3})(D_{1} · E)(D_{2} · E)(D_{3} · E)\binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 1, r_{0}(D_{2}) − 1}

(where we will denote by \binom{n}{a, b} the multinomial n!/a!b!(n − a − b)!).

Let S_{J} be the subset of B consisting of those points b such that Y_{b} is a fiber of type J for which q and q′ lie on different components. Obviously S_{J} contains n_{J} points. If b ∈ S_{J} we write Y_{b} = J_{1,b} + J_{2,b} (hence q ∈ J_{1,b} and q′ ∈ J_{2,b}). In a completely analogous fashion we define S_{K}, S_{K′} and S_{H}. If b ∈ S_{K} then we write Y_{b} = K_{1,b} + K_{E,b} + K_{0,b} + K_{2,b} and similarly if b is in S_{K′} or S_{H}. We therefore have that if b ∈ S_{K} (respectively, b ∈ S_{K′} and b ∈ S_{H}), then q′ lies on K_{2,b} (respectively on K′_{2,b} and H_{2,b}). Now we have

\begin{aligned}A′ − A = & ∑_{b∈S_{J}} J_{1,b} + ∑_{b∈S_{K}}(3K_{1,b} + 2K_{E,b} + K_{0,b}) + ∑_{b∈S_{K′}}(3K′_{1,b} + 2K′_{0,b} + K′_{E,b})+ \\ & + ∑_{b∈S_{H}}(H_{1,b} − H_{2,b}) + nY. \end{aligned}

where n is some integer that is irrelevant for our computation. Finally we obtain

A^{2} = \frac{−n_{J} − 2n_{H} − 6n_{K}}{2}

Now, for any L ∈ Pic(𝔽_{3}), we have

{π^{∗}L}_{A} = (L · D)

{π^{∗}L}_{Y} = −(L · D)A^{2}

and

{π^{∗}L}_{G_{E,b}} = −(L · E)

for any b in B_{G}. These are obtained, in the given order, from the products (π^{∗}L · Y) = (L · D) , (π^{∗}L · A) = 0 and (π^{∗}L · G_{E,b}) = (L · E)

Let us fix a fiber of type J which we write as J_{1,b} + J_{2,b} as usual; let D_{2} be the class in 𝔽_{3} of π(J_{2,b}). From the product (π^{∗}L · J_{2,b}) = (L · D_{2}) we see that

{π^{∗}L}_{J_{2,b}} = −(L · D_{2})

Fix now a fiber of type K, which we shall write as K_{1,b} + K_{E,b} + K_{0,b} + K_{2,b}, such that the image in 𝔽_{3} has corresponding divisor classes D_{1} for K_{1,b} and D_{2} for K_{2,b}. The relation (π^{∗}L · K_{1,b}) = (L · D_{1}) implies

{π^{∗}L}_{K_{E,b}} = (L · D_{1}) − (L · D);

the above formula together with (π^{∗}L · K_{E,b}) = (L · E) gives

{π^{∗}L}_{K_{0,b}} = 2(L · D_{1}) − 2(L · D) + (L · E);

and the two previous formulas combined with (π^{∗}L · K_{0}) = 0 gives

{π^{∗}L}_{K_{2,b}} = 3(L · D_{1}) − 3(L · D) + 2(L · E).

With completely analogous notation and procedure, for a fixed fiber of type K′ we have

{π^{∗}L}_{K′_{0,b}} = (L · D_{1}) − (L · D)

{π^{∗}L}_{K′_{E,b}} = 2(L · D_{1}) − 2(L · D)

{π^{∗}L}_{K′_{2,b}} = 3(L · D_{1}) − 3(L · D) + (L · E)

Finally, fix a fiber of type H such that the class in 𝔽_{3} corresponding to H_{i,b} is D_{i}; again the same procedure yields

{π^{∗}L}_{H_{E,b}} = (L · D_{1}) − (L · D)

{π^{∗}L}_{H_{3,b}} = (L · D_{1}) − (L · D) − (L · D_{3})

and

{π^{∗}L}_{H_{2,b}} = (L · D_{1}) − (L · D) − (L · D_{2}).

We shall also use the following short notation:

π^{∗}L = (L · D)A + (L · D)(\frac{−n_{J} − n_{H} − 6n_{K}}{2} )Y − (L · E)G_{E} + J^{L} + K^{L} + K^{′L} + H^{L}  (∗).

Now we want to compute the intersection product on Y of the pull-back of two line bundles L and M on 𝔽_{3}. We easily have

\begin{aligned}(π^{∗}L · π^{∗}M) = & − (L · D)(M · D)A^{2} − (L · E)(M · E)f(D − E) \\ & + (J^{L} · J^{M}) + (K^{L} · K^{M}) + (K^{′L} · K^{′M}) + (H^{L} · H^{M}). \end{aligned}

And now a completely straightforward computation yields

(J^{L} · J^{M}) = − ∑_{D_{1}+D_{2}=D} j(D_{1}, D_{2})(L · D_{2})(M · D_{2}),

\begin{aligned}(K^{L} · K^{M}) = − ∑_{D_{1}+D_{2}=D−E} & k(D_{1}, D_{2})((L · E)((M · D_{1}) − (M · D))+ \\ & − (L · D_{2})(3(M · D_{2}) + (M · E))), \end{aligned}

\begin{aligned}(K^{′L} · K^{′M}) = − ∑_{D_{1}+D_{2}=D−E} & k′(D_{1}, D_{2})((L · E)(2(M · D_{1}) − 2(M · D))+ \\ & − (L · D_{2})(3(M · D_{2}) + 2(M · E))) \end{aligned}

and

\begin{matrix}(H^{L} · H^{M}) = −\frac{1}{2} ∑_{D_{1}+D_{2}+D_{3}=D−E} h(D_{1}, D_{2}, D_{3}) & (−(L · D)(M · D) + (L · D)(M · D_{1}) \\ + (L · D_{1})(M · D) − (L · D_{1})(M · D_{1}) \\ − (L · D_{2})(M · D_{2}) − (L · D_{3})(M · D_{3})) \end{matrix} .

In the last formula, we divide by 2 because D_{2} and D_{3} are not distinguished from one another. Now we are ready to write down the three formulas that we mentioned at the beginning of

this chapter. Before we carry out the computation, we can explain briefly the procedure. We have to look at the relation (∗) and keep track of the Severi degrees on which the characteristic numbers depend.

(a) The first that relation we shall use is

(π^{∗}C · π^{∗}C) = 3N(D)

This will give a formula expressing

N(D) in terms of N(D′) and N_{2}(D′′) with D′ < D and D′′ < D − E.

This is clear; since (C · E) = 0 if we apply (LM) to L = M = O_{S}(C) the Severi degree N_{3}(D − E) disappears.

(b) Now we need a formula for N_{2}(D). We will imitate what we did to compute the degree of the variety of rational curves tangent to a fixed line in the plane. We define Ẽ to be the class of the irreducible component of π^{−1}(E) that dominates B. Then we have

Ẽ = π^{∗}E − 3G_{E} − 2K_{E} − K_{0} − 2K′_{E} − K′_{0} − H_{E}.

This is obtained as follows: for the coefficient of G_{E} we notice that for any b ∈ B_{G} we have (Ẽ · G_{E,b}) = 0, while on the other hand (π^{∗}E · G_{E,b}) = −3. The same procedure yields the remaining terms.

N_{2}(D) = Ẽ^{2} + (Ẽ · ω_{Y/B}).

This will give a recursion expressing

N_{2}(D) in terms of N(D), N(D′), N_{3}(D − E), and N_{2}(D′′) with D′ < D and D′′ < D − E.

(c) The third and last step will be to find a formula for N_{3}(D − E). This will be done by using

(π^{∗}F · π^{∗}F) = 0

which, as one can imply using (∗), will give

N_{3}(D − E) in terms of N(D′) and N_{2}(D′′) with D′ < D and D′′ < D − E.

\underline{Example.} If D = 2C we have on one hand that (π^{∗}C · π^{∗}C) = 3N(2C), and on the other hand our formulas give

\begin{matrix}(π^{∗}C · π^{∗}C) = 3[−12 & A^{2} − 3j(C, C) − k(C + 2F, F) − 25k′(F, C + 2F) \\ − 14h(F, F, C + F) − 2h(C + F, F, F)] \end{matrix}

Here are the relevant numbers for the case D = 2C.

j(C, C) = 105 k(C + 2F, F) = 14 k′(F, C + 2F) = 2 h(F, F, C + F) = 7 h(C + F, F, F) = 21 n_{J} = 60 n_{K} = n_{K′} = 2 n_{H} = 13

so that

A^{2} = 49

and we can conclude that N(2C) = 69.

We will now state our main result for 𝔽_{3}:

Theorem. Let D ∈ Pic(𝔽_{3}). Let N(D) be the number of irreducible rational curves in |D| that pass through r_{0}(D) general points. Then

\begin{matrix}N(D) = \\ = \frac{1}{3} ∑_{D_{1}+D_{2}=D} N(D_{1})N(D_{2})(D_{1} · D_{2})[\binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 1}(D_{1} · C)(D_{2} · C) − \binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 2}(D_{2} · C)^{2}]+ \\ + ∑_{D_{1}+D_{2}=D−E} N_{2}(D_{1})N(D_{2})(E · D_{2})[\binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 2}(D_{1} · C)(D_{2} · C) − \binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 3}(D_{2} · C)^{2}] \\ + ∑_{D_{1}+D_{2}=D−E} N(D_{1})N_{2}(D_{2})(E · D_{1})[\binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 1}(D_{1} · C)(D_{2} · C) − \binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 2}(D_{2} · C)^{2}] \\ + \frac{1}{3} ∑_{D_{1}+D_{2}+D_{3}=D−E} N(D_{1})N(D_{2})N(D_{3})(E · D_{1})(E · D_{2})(E · D_{3})· \\ ·[\binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 1, r_{0}(D_{2}) − 1}[2(C · D_{1})(C · D_{2}) + (C · D_{1})(C · D_{3}) + (C · D_{2})(C · D_{3}) − (C · D_{3})^{2}]− \\ − \binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 2, r_{0}(D_{2})}[(C · D_{2})^{2} + (C · D_{3})^{2} + (C · D_{2})(C · D_{3})]] \end{matrix} .

Proof. We just have to compute. Applying (*) to π^{∗}C gives

(π^{∗}C)^{2} = − (C · D)^{2}A^{2} − ∑_{D_{1}+D_{2}=D} j(D_{1}, D_{2})(C · D_{2})^{2}−

− ∑_{D_{1}+D_{2}=D−E} 3k(D_{1}, D_{2})(C · D_{2})^{2}−

− ∑_{D_{1}+D_{2}=D−E} 3k′(D_{1}, D_{2})(C · D_{2})^{2}−

− ∑_{D_{1}+D_{2}+D_{3}=D−E} h(D_{1}, D_{2}, D_{3})[(C · D_{2})^{2} + (C · D_{3})^{2} + (C · D_{2})(C · D_{3})]

This gives

\begin{matrix}N(D) = \\ = \frac{1}{3} ∑_{D_{1}+D_{2}=D} N(D_{1})N(D_{2})(D_{1} · D_{2})[\binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 1}(D_{1} · C)(D_{2} · C) − \binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 2}(D_{2} · C)^{2}]+ \\ + ∑_{D_{1}+D_{2}=D−E} N_{2}(D_{1})N(D_{2})(E · D_{2})[2\binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 2}(D_{1} · C)(D_{2} · C)− \\ − \binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 1}(D_{1} · C)^{2} − \binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 3}(D_{2} · C)^{2}]+ \\ + \frac{1}{3} ∑_{D_{1}+D_{2}+D_{3}=D−E} N(D_{1})N(D_{2})N(D_{3})(E · D_{1})(E · D_{2})(E · D_{3})[(C · D)^{2}\binom{r_{0}(D) − 3}{r_{0}(D_{1}) − 1, r_{0}(D_{2}) − 1}− \\ − ((C · D_{2})^{2} + (C · D_{3})^{2} + (C · D_{2})(C · D_{3}))\binom{r_{0}(D) − 2}{r_{0}(D_{1}) − 1, r_{0}(D_{2})}] \end{matrix} .

And this concludes the proof.

We will write as well the formulas for the degrees of the other loci that we need. The first formula is obtained by

N_{2}(D) = Ẽ^{2} + (Ẽ · ω_{Y/B}),

which gives

N_{2}(D) = − 3N(D) + 9N_{3}(D − E) + (E · D)A^{2}+

+ ∑_{D_{1}+D_{2}=D} j(D_{1}, D_{2})(E · D_{2})+

+ ∑_{D_{1}+D_{2}=D−E} 6(k(D_{1}, D_{2}) + k′(D_{1}, D_{2}))+

+ ∑_{D_{1}+D_{2}+D_{3}=D−E} h(D_{1}, D_{2}, D_{3})[2(E · D_{2}) + 2(E · D_{3}) − 1].

Finally, the degree of the Severi variety parametrizing rational curves having a point of contact of order at least 3 with E is obtained by π^{∗}F^{2} = 0, which translates into

N_{3}(D − E) =

− (F · D)^{2}A^{2}−

− ∑_{D_{1}+D_{2}=D} j(D_{1}, D_{2})(F · D_{2})^{2}−

− ∑_{D_{1}+D_{2}=D−E} k(D_{1}, D_{2})[1 + 2(F · D_{2}) + 3(F · D_{2})^{2}]−

− ∑_{D_{1}+D_{2}=D−E} k′(D_{1}, D_{2})[2 + 4(F · D_{2}) + 3(F · D_{2})^{2}]−

− ∑_{D_{1}+D_{2}+D_{3}=D−E} h(D_{1}, D_{2}, D_{3})[(F · D)^{2} − 2(F · D)(F · D_{1}) + (F · D_{1})^{2} + (F · D_{2})^{2} + (F · D_{3})^{2}]

References

[CH] L. Caporaso and J. Harris, Parameter spaces for curves on surfaces and enumeration of rational curves 1995, Preprint.

[CM] B. Crauder and R. Miranda, Quantum cohomology of rational surfaces. The Moduli Space of Curves, Progress in mathematics 129, Birkhauser (1995) pp. 33-80.

[DH] S. Diaz and J. Harris, Geometry of the Severi variety. Trans.A.M.S. 309 (1988), pp. 1-34.

[DI] P. Di Francesco and C. Itzykson, Quantum intersection rings. The Moduli Space of Curves, Progress in mathematics 129, Birkhauser (1995) pp. 81-148.

[H] J. Harris, On the Severi problem. Invent. Math. 84 (1986), pp. 445-461.

[K] J. Kollár, Rational Curves on Algebraic Varieties, Springer 1996.

[KP] S. Kleiman and R. Piene, Private communication.

[KM] M. Kontsevich and Y. Manin, Gromov-Witten classes, quantum cohomology and enumerative geometry.

[P] R. Pandharipande, Intersections of ℚ-divisors on Kontsevich’s moduli space \overline{M}_{0,n}(ℙ^{r}, d) and enumerative geometry. 1995, Preprint.

[R] Z. Ran, Enumerative geometry of singular plane curves. Invent. Math. 97 (1989), pp. 447-465.

[R2] Z. Ran, On the quantum cohomology of the plane old and new. (1995) Preprint, alg-geom 9508011.

[RT] Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. Diff. Geom. 42 No. 2 (1995) pp. 259-367).

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