THE FORMULA 12 = 10 + 2 × 1 AND ITS GENERALIZATIONS: COUNTING RATIONAL CURVES ON F_{2}

DAN ABRAMOVICH AND AARON BERTRAM

1. The problem

Gromov-Witten invariants are symplectic invariants of compact symplectic manifolds [K-M, R-T]. Thus, they are constant along deformations of compact Kähler manofolds, and in particular along deformations of smooth complex projective varieties. For convex varieties, Gromov-Witten invariants in genus 0 are enumerative, namely they count the number of rational curves satisfying sufficiently general incidence conditions, see [Kon, B-M, F-P, R-T].

In this note we consider an explicit case of a surface - the Hirzebruch surface F_{2} - which is not convex, but deforms algebraically to the convex surface ℙ^{1} × ℙ^{1}. This enables us to compare the Gromov-Witten invariants of F_{2} (calculated as usual on ℙ^{1} × ℙ^{1}) with its enumerative invariants (see [C-H]) explicitly, by following the rational curves on ℙ^{1} × ℙ^{1} to their limits on F_{2}. As a result we get a recursive formula for the enumerative invariants of F_{2} involving those of ℙ^{1} × ℙ^{1}. The formulas are different from those of Caporaso and Harris, but the resulting numbers coincide.

The results of this note, inspired by a talk of L. Caporaso, were worked out in late 1995 and written up in 1998.

2. Preliminaries

We work over ℂ.

2.1. The deformation. We fix once and for all a one-parameter deformation π : X → S of the surface F_{2}. Here S is an affine curve (or the spectrum of a discrete valuation ring), s ∈ S is a closed point, and η ∈ S the generic point (or general point, depending on the reader’s preference). The morphism X → S is flat, projective, of relative dimension 2. We fix isomorphisms F_{2} ⟶^{∼} X_{s} and (ℙ^{1} × ℙ^{1})_{η} ⟶^{∼} X_{η}. An explicit description of such a deformation space can be found in [Kod].

2.2. Divisor classes on F_{2} and ℙ^{1} × ℙ^{1}. The Néron-Severi group of X_{s} = F_{2} is generated by two classes. One class corresponds to the fiber of the ruling, denoted F, and one corresponds to a general section, denoted C_{2}. We denote the exceptional (−2) curve by E - it is linearly equivalent to C_{2} − 2F.

We will only concern ourselves with nef Néron-Severi classes on F_{2}. The associated linear series of these classes have no base points. These classes are of the form a · C_{2} + b · F for a, b ≥ 0.

The Néron-Severi group of X_{η} = (ℙ^{1} × ℙ^{1})_{η} is generated by the rulings, which we may denote by F and C_{0}. Since the surfaces are rational, their Picard groups are isomorphic to their Néron-Severi groups.

2.3. Divisor classes on X. Looking at the total space, the relative Picard group of X → S is a trivial rank-2 local system, where the class F on X_{s} deforms to the class F on X_{η} (and there is no fear of confusion), and C_{2} deforms to C_{0} +F. We denote by L_{a,b} the invertible sheaf on X whose restriction to X_{s} is O_{X_{s}}(a·C_{2} +b·F). On X_{η} it restricts to the line bundle O_{(ℙ^{1}×ℙ^{1})_{η}}(a, a + b). The formation of the locally free sheaf π_{∗}L_{a,b} commutes with base change whenever a, b ≥ 0. Denote by ℙ_{a,b} the associated family of linear series, ℙ_{a,b} = ℙ(π_{∗}L_{a,b})^{∨}.

Date: August 7, 2000.

D.A. was partially supported by NSF grant DMS-9700520 and by an Alfred P. Sloan research fellowship.

A.B. was partially supported by NSF grant DMS-9500964.

s  s η  η F  F + C_{0} C_{2}

Figure 1. The divisor F deforms to F, while C_{2} deforms to F + C_{0}.

2.4. Moduli of stable maps. It follows from [B-M] that for each (a, b) there is a Deligne - Mumford stack \overline{M}_{a,b} = \overline{M}_{0}(X/S, L_{a,b}) parametrizing genus 0 stable maps, mapping to the fibers of X over S, with image class c_{1}(L_{a,b}). This stack admits a coarse moduli space \overline{M}_{a,b}, and by [F-P] there is a projective morphism \overline{M}_{a,b} → S.

Since ℙ^{1} ×ℙ^{1} is convex, it follows from [Kon] or [F-P] that when a > 0 and b ≥ 0, the generic fiber (\overline{M}_{a,b})_{η} is smooth, irreducible and of the expected dimension: dim(\overline{M}_{a,b})_{η} = 4a + 2b − 1; in particular, the coarse moduli space (\overline{M}_{a,b})_{η} has only finite quotient singularities. Moreover, it contains a dense open set parametrizing immersions of ℙ^{1} into ℙ^{1} × ℙ^{1} as a nodal curve.

2.5. Gromov-Witten invariants. Fix non-negative integers a and b, to which we have associated an invertible sheaf L_{a,b} and divisor classes aC_{2} + bF on F_{2} and aC_{0} + (a + b)F on ℙ^{1} × ℙ^{1}. The associated Gromov-Witten invariant N(aC_{2} + bF) = N(aC_{0} + (a + b)F) was originally defined as the number of pseudo-holomorphic rational curves, analytic with respect to a generic almost complex structure, on F_{2}. Since ℙ^{1} × ℙ^{1} is convex, its holomorphic structure is sufficiently general for this purpose - thus the Gromov-Witten invariant N(aC_{2}+bF) = N(aC_{0} + (a + b)F) is the number of irreducible nodal rational curves on ℙ^{1} × ℙ^{1} of class aC_{0} + (a + b)F passing through 4a + 2b − 1 general points.

If moreover L_{a,b} is nontrivial, then there is a natural morphism Div : \overline{M}_{a,b} → ℙ_{a,b}, which to a stable map f : C → X associates the unique image divisor. Using this fact, we can define the Gromov-Witten number as the intersection number

N(aC_{2} + bF) = c_{1}(Div^{∗} O_{ℙ_{a,b}}(1))^{4a+2b−1} ∩ [(\overline{M}_{a,b})_{η}].

Remark 2.5.1. In general there is a virtual fundamental class [(\overline{M}_{a,b})^{virt}] which replaces the fundamental class in the non-convex case. The identification of the virtual fundamental class can be subtle.

2.6. Enumerative invariants on F_{2}. Analogously, denote by N_{2}(aC_{2} + bF) the number of irreducible nodal rational curves of class aC_{2} +bF passing through 4a+2b−1 general points on F_{2}. The following question arises naturally:

What is N_{2}(aC_{2} + bF), what is its relationship with N(aC_{2} + bF), and can the relationship be explained via the deformation space X?

A direct recursive formula for N_{2}(aC_{2} + bF) has been given by Caporaso and Harris in [C-H]. For example, N_{2}(2C_{2}) = 10 while N(2C_{2}) = 12. Caporaso and Harris gave an interpretation for the difference N(2C_{2}) − N_{2}(2C_{2}) = 2, but not in terms of a deformation of F_{2}.

3. The formula and its proof

3.1. Statement of enumerative result. Theorem 3.1.1. For a nef divisor class D = aC_{2} + bF we have

N(D) = ∑_{k=0}^{a−1} \binom{b + 2k}{k}N_{2}(D − kE).

Remark 3.1.2. Note that D − kE = (a − k)C_{2} + (b + 2k)F. When D = 2C_{2} we get N(D) = N_{2}(D) + 2 × N_{2}(D −E) = N_{2}(D)+2×N_{2}(C_{2} +2F). Now it is well known (see, e.g., [C-H]) that N(D) = N(2C_{0} +2F) = 12, and N_{2}(C_{2} + 2F) = 1 since C_{2} + 2F has arithmetic genus 0. Thus the number N_{2}(D) = 10 fits in the formula 12 = 10 + 2 × 1, whence the title of this note.

THE FORMULA 12 = 10 + 2 × 1 AND ITS GENERALIZATIONS 3

3.2. Relevant components of (\overline{M}_{a,b})_{s}. Consider the Zariski closure of (\overline{M}_{a,b})_{η} in \overline{M}_{a,b}. This is a flat scheme over S, therefore the degree of the invertible sheaf Div^{∗} O_{ℙ_{a,b}}(1) is constant along the fibers. Calculating this degree on the special fiber boils down to the following issues:

(1) Among the irreducible components of (\overline{M}_{a,b})_{s}, which ones have dimension ≥ 4a + 2b − 1?

(2) Among these irreducible components, for which ones does the image in (ℙ_{a,b})_{s} have dimension ≥ 4a + 2b − 1?

(3) Among the latter components, which ones contain a 4a+2b−1 dimensional subscheme M_{0} of the Zariski closure \overline{(\overline{M}_{a,b})_{η}}, such that the image of M_{0} in (ℙ_{a,b})_{s} has dimension 4a + 2b − 1?

(4) What are the degrees of these subschemes M_{0} with respect to Div^{∗} O_{ℙ_{a,b}}(1)?

An irreducible component which contains a 4a + 2b − 1 dimensional subscheme M_{0} of the Zariski closure \overline{(\overline{M}_{a,b})_{η}} whose image in (ℙ_{a,b})_{s} has dimension 4a+2b−1, is called relevant. Ideally, one would hope that every relevant irreducible component M has dimension exactly 4a + 2b − 1, is reduced, and has an easily computable degree; in particular M = M_{0}. The following propositions state that this is indeed the case: Proposition 3.2.1. Let M ⊂ (\overline{M}_{a,b})_{s} be a relevant irreducible component. Then the general point of M corresponds to a genus 0 stable map f : C → F_{2} of the following form:

(1) the curve C decomposes into irreducible components C = C′ ∪ E_{1} ∪ … ∪ E_{k} for some 0 ≤ k ≤ a − 1;

(2) the morphism f maps each of the components E_{i} isomorphically to the exceptional (−2) curve E;

(3) the morphism f maps C′ birationally onto a nodal rational curve of class D − kE in F_{2} meeting E transversally;

(4) the intersection C′ ∩ E_{i} consists of a single node. The intersection E_{i} ∩ E_{j} is empty for i ≠ j Remark 3.2.2. We call such a stable maps an antenna. The component C′ is its spine and the components E_{i} are the cross-pieces.

E C′ E_{1} E_{2} E_{3}  C′

Figure 2. An antenna.

Proposition 3.2.3. For each integer 0 ≤ k ≤ a − 1 there is a unique relevant irreducible component M_{k}(D) ⊂ (\overline{M}_{a,b})_{s} whose generic point corresponds to an antenna with k cross-pieces. This component M_{k}(D) is reduced, and has degree \binom{b+2k}{k}N_{2}(D − kE) with respect to Div^{∗} O_{ℙ_{a,b}}(1).

3.3. Relevant components: sufficiency of conditions. Let f : C → F_{2} be an antenna with k crosspieces. The morphism f^{∗}Ω_{F_{2}}^{1} → Ω_{C}^{1} is surjective. Its kernel is the conormal sheaf N^{∗} which is invertible. The infinitesimal deformation space of f (keeping the target F_{2} fixed) has tangent space Hom(N^{∗}, O_{C}) = H^{0}(C, N), and the obstructions lie in Ext^{1}(N^{∗}, O_{C}) = H^{1}(C, N). It is easy to see that deg_{E_{i}} N = −1 and deg_{C′} N = 4a + 2b − 2 + k. A simple calculation reveals that H^{1}(C, N) = 0, namely there are no obstructions, and dim Hom(N^{∗}, O_{C}) = 4a + 2b − 1.

In particular, since deg_{E_{i}} N = −1 every section of N vanishes on each E_{i}. Therefore any deformation of f in F_{2} remains an antenna with k cross-pieces. Since there are no obstructions, the space of stable maps (\overline{M}_{a,b})_{s} is smooth at f, with dimension 4a + 2b − 1. For the same reason (see [Kol]) f deforms freely in the family X → S, therefore any point in (\overline{M}_{a,b})_{s} in a neighborhood of f deforms to (\overline{M}_{a,b})_{η}. This shows that f belongs to a relevant component of (\overline{M}_{a,b})_{s}.

An easy monodromy and degeneration argument following [Har] or [C-H] shows that the space of antennas with k cross-pieces is irreducible. The morphism Div : M_{k}(D) → ℙ_{a,b} has degree \binom{b+2k}{k} onto the image, since for a general antenna with k cross-pieces there are b + 2k intersection points of f(C′) with E, and the antenna is formed by choosing k of these points as attaching points with E_{i}. Also, the explicit description shows that the degree of the image Div(M_{k}(D)) coincides with the degree of Div(M_{0}(D − kE)), which is by definition the enumerative invariant N_{2}(D − kE).

Thus, in order to complete the proof of the propositions and the theorem, it suffices to show that there are no other relevant components.

3.4. Relevant components: necessity of conditions. Consider a relevant component M, containing a subscheme M_{0} of dimension 4a + 2b − 1 in the closure of (\overline{M}_{a,b})_{η}, and a general point on M_{0} corresponding to a stable map f : C → F_{2}. Let C_{E} = f^{−1}E and C′ = \overline{C C_{E}}. The curve C′ consists of the components of C whose image does not lie in E.

An easy dimension count, using the fact that Div(M) has dimension 4a+2b−1, shows that C′ is irreducible. This means that there is a decomposition into connected curves C = C′ ∪ E_{1} … ∪ E_{l}, where E_{i} map generically finitely to E with degree d_{i}, and C′ maps to a general irreducible nodal rational curve of class D − kE where k = ∑_{i=1}^{l} d_{i}. The curves E_{i} are trees of rational curves, each meeting C′ at a single node. The picture is similar to that of an antenna, but where cross-pieces can be more elaborate.

E E_{1} E_{2} E_{3}  C′ C′

Figure 3. A generalized antenna.

We need to show that d_{i} = 1 for all i. This will be derived, in a manner similar to that in [C-H], from the fact that f deforms to X_{η}.

For any choice of a fiber F on F_{2}, the space X contains a Cartier divisor H such that H_{η} = C_{0} and H_{s} = E + F. Indeed, the linear series of C_{0}, which has dimension 1, has as its limit a subscheme of the linear series of E + F, which also has dimension 1, therefore every element E + F is in this limit.

Looking at our proposed general stable map f : C → F_{2}, note that f(C′) meets E + F transversely in (D − kE) · (E + F) = a + b + k points.

Consider now a finite surjective base change S′ → S, a deformation C̃ → S′ of C and a morphism f̃: C̃ → X_{S′} (where X_{S′} = X ×_{S} S′). Assume f̃ is a general deformation of f into X, namely that C̃_{η′} is a smooth rational curve immersed by f̃_{η′} in X_{η′}. Consider f̃^{∗}H. This is a Cartier divisor on C̃. Its restriction to C̃_{η′} has degree D · H_{s} = a + b. Thus we can write f̃^{∗}H = H_{h} + ∑_{i=1}^{l} E′_{i}, where H_{h} → S′ is finite of degree a + b, and E′_{i} are Weil divisors supported along E_{i}. In particular, the total number of points in C′ ∩ f̃^{∗}H, counted without multiplicities, is at most a + b + l. Since l ≤ k an equality must hold, in particular d_{i} = 1 for all i. □

s  ηC_{0} FE  H_{h}

Figure 4. The divisor H (left) and its pullback (right).

Remark 3.4.1. The virtual fundamental class [(\overline{M}_{a,b})^{virt}] does have nontrivial support on all the components of (\overline{M}_{a,b})_{s} described above, even when d_{i} > 1. Indeed the virtual fundamental class is the direct image of the fundamental class of the closure of (\overline{M}_{a,b})_{η}. What we have shown above is that the components where d_{i} > 1 do not contribute to Gromov-Witten invariants.

References

[B-M] Kai Behrend and Yuri I. Manin, Stacks of stable maps and Gromov-Witten invariants, Duke Math. J. 85 (1996), no. 1,

1–60.

[C-H] Lucia Caporaso and Joe Harris, Parameter spaces for curves on surfaces and enumeration of rational curves, Com-

positio Math. 113 (1998), no. 2, 155–208.

[F-P] William Fulton and Rahul Pandharipande, Notes on stable maps and quantum cohomology, in Algebraic geometry—

Santa Cruz 1995, 45–96, Proc. Sympos. Pure Math., Part 2, Amer. Math. Soc., Providence, RI, 1997.

[Har] Joe Harris, On the Severi problem, Invent. Math. 84 (1986), no. 3, 445–461.

[Kod] Kunihiko Kodaira, Complex manifolds and deformation of complex structures, Translated from the Japanese by Kazuo Akao, Springer, New York, 1986.

[Kol] János Kollár, Rational curves on algebraic varieties, Springer, Berlin, 1996.

[Kon] Maxim Kontsevich, Enumeration of rational curves via torus actions, in The moduli space of curves (Texel Island, 1994), 335–368, Progr. Math., 129, Birkhäuser, Boston, Boston, MA, 1995.

[K-M] Maxim Kontsevich and Yuri I. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry,

Comm. Math. Phys. 164 (1994), no. 3, 525–562.

[R-T] Yongbin Ruan and Gang Tian, A mathematical theory of quantum cohomology, J. Differential Geom. 42 (1995), no. 2, 259–367.

Department of Mathematics, Boston University, 111 Cummington, Boston, MA 02215, USA

   E-mail address: abrmovic@math.bu.edu

Department of Mathematics, University of Utah, Salt Lake City, Utah, 84112, USA

   E-mail address: bertram@math.utah.edu