EMBEDDING POINTED CURVES IN K3 SURFACES

BRENDAN HASSETT AND YURI TSCHINKEL

1. Introduction

Let D be a smooth projective curve of genus g over an algebraically closed field of characteristic zero. Let M_{g} denote the moduli stack of such curves. Let (S, h) be a polarized K3 surface of genus g, i.e., h is ample and primitive with h^{2} = 2g − 2. Let F_{g} denote the moduli stack of such surfaces. Now suppose D ⊂ S with [D] = h; let P_{g} denote the moduli space of such pairs (S, D), φ_{g} : P_{g} → M_{g} the forgetful map, and K_{g} ⊂ M_{g} its image. These morphisms have been studied systematically by Mukai; we review this in more detail in Section 2. One of the highlights of this theory is the birationality of φ_{g} when g = 11. It is birational onto its image when g ≥ 13 [CLM93, Th. 5].

We propose variations on this construction. Let F_{Λ} denote a moduli space of lattice polarized K3 surfaces, where Λ ⊃ ⟨h, R⟩ with h a polarization of degree 2g−2 and R an indecomposable (−2)-class (smooth rational curve) with n := h · R > 0. Let P_{Λ} denote the space of pairs (S, D), where S ∈ F_{Λ} and D ∈ |h| is smooth and meets R transversally. Thus we have

\begin{matrix}φ_{Λ} : P_{Λ} & → & M_{g,n}/𝔖_{n} \\ (S, D) & ↦ & (D; D ∩ R) \end{matrix}

as well as the morphism keeping track of the pointed rational curve

\begin{matrix}φ̂_{Λ} : P_{Λ} & → & (M_{g,n} × M_{0,n})/𝔖_{n} \\ (S, D) & ↦ & ((D; D ∩ R), (R; D ∩ R)) \end{matrix}

Suppose that Λ = ⟨h, R⟩ and the source and target moduli spaces have the same dimension; thus 2g + n = 21 in the first case and g + n = 12 in the second. What are deg(φ_{Λ}) and deg(φ̂_{Λ})?

Work of Green and Lazarsfeld [GL87] shows how lattice polarizations control the Brill-Noether properties of curves on K3 surfaces. For example, a smooth curve D ⊂ S admits a g_{d}^{1} with 2d ≤ g + 1 only if there is an elliptic fibration S → ℙ^{1} with D as a multisection of degree d. Hence for suitable lattice polarizations φ_{Λ} maps to Brill-Noether strata of M_{g}. Our variation is relevant to understanding the specialization of Mukai’s theory over these strata.

We summarize our main results. Fix the lattices:

Λ′ = \begin{aligned}& C & R′ \\ C & 12 & 7 \\ R′ & 7 & −2 \end{aligned} , Λ = \begin{aligned}& C & R′ & f \\ C & 12 & 7 & 8 \\ R′ & 7 & −2 & 6 \\ f & 8 & 6 & 4 \end{aligned} , Λ′′ = \begin{aligned}& C & R′ \\ C & 12 & 5 \\ R′ & 5 & −2 \end{aligned}

Date: April 29, 2014.

• The morphism φ_{Λ′} : P_{Λ′} → M_{7,7}/𝔖_{7} has degree 71; see Proposition 6 and Remark 8.

• The restriction φ_{Λ} : P_{Λ} → M_{7,7}/𝔖_{7} is birational onto its image T ⊂ M_{7,7}/𝔖_{7}, the tetragonal curves of genus seven with seven marked points; see Theorem 9.

• The morphism φ_{Λ′′} : P_{Λ′′} → (M_{7,5} × M_{0,5})/𝔖_{5} is birational; see Section 8. Our approach is to specialize Mukai’s construction in genus 11 to the pentagonal

locus (see Theorem 4), degenerate this to a carefully chosen stable curve of genus 11 (see Theorem 5), and then deform a related stable curve of genus 7 together with the ambient K3 surface. The relevant deformation theory is worked out in Section 7; the key is to analyze the ramification behavior of φ_{Λ}. The necessary tangent space computations are in Sections 7.3 and 7.4; we use the unirationality of the relevant Noether-Lefschetz stratum (Proposition 12). The Appendix presents Gromov-Witten computations establishing that deg(φ_{Λ′}) = 71.

Our motivation for considering this geometry comes from arithmetic. The constructions in this paper have strong implications for the structure of spaces of sections for quartic del Pezzo surface fibrations X → ℙ^{1}; fibrations of height twelve admit a family of sections ℙ^{1} → X parametrized by the intermediate Jacobian of X. Details appear in [HT14] and related structural results may be found in [HKT13].

Acknowledgments. Andrew Kresch provided invaluable assistance on this project and in particular, the computations in the enclosed appendix. We benefited from conversations with Shigeru Mukai, Frank-Olaf Schreyer, and Alessandro Verra. We are grateful to the referee for a number of suggestions, including an improvement to the proof of Theorem 4. The first author is supported by NSF grants 0968349, 0901645, and 1148609; the second author is supported by NSF grants 0739380, 0968349, and 1160859.

2. Geometric background

Let (S, h) be a polarized K3 surface of genus g. Consider the moduli space M_{2,h,s}(S) of rank-two stable sheaves E on S with c_{1}(E) = h and χ(E) = 2 + s. This is holomorphic symplectic of dimension h^{2} − 4s + 2 = 2g − 4s, provided this expression is non-negative. When g −1 = 2s the moduli space is again a K3 surface, isogenous to S [Muk87].

Let M_{2,ω}(D) denote the moduli space of semistable rank-two vector bundles on D with canonical determinant. Consider the non-abelian Brill-Noether loci defined by Mukai [Muk01]

M_{2,ω,s}(D) := {E ∈ M_{2,ω}(D) : h^{0}(E) ≥ 2 + s},

which has expected dimension 3g − 3 − \binom{s+3}{2}.

Restricting bundles from S to D yields examples of non-abelian Brill-Noether loci with dimensions frequently larger than expected. Mukai has developed a program, supported by beautiful examples, that seeks to characterize K_{g} ⊂ M_{g} in terms of special non-abelian Brill-Noether loci. We refer the reader to [ABS13] for very recent progress on this program.

A particularly striking result along these lines is:

Theorem 1. [Muk96, Thm. 1] Let D be a generic curve of genus eleven. Then there exist a genus eleven K3 surface (S, h) and an embedding D ↪ S, which are unique up to isomorphisms. Furthermore, we can characterize S as M_{2,h,5}(T), where T = M_{2,ω,5}(D), which is also a genus eleven K3 surface.

Thus φ_{11} : P_{11} → M_{11} is birational and the K3 surface can be recovered via moduli spaces of vector bundles on D. The theorem remains true for the generic hexagonal curves of genus eleven; see [Muk96, Thm. 3]. Hexagonal curves form a divisor in M_{11}, so φ_{11} remains an isomorphism over the generic point of the hexagonal locus. We will analyze what happens over the pentagonal locus.

We shall need a similar result along these lines in genus seven [Muk95]. Let OG(5, 10) ⊂ ℙ^{15} denote the orthogonal Grassmannian, parametrizing five dimensional isotropic subspaces for a ten dimensional non-degenerate quadratic form. Let G denote the corresponding orthogonal group. The intersection of OG(5, 10) with a generic six dimensional subspace Π_{6} ⊂ ℙ^{15} is a canonical curve of genus seven; its intersection with a seven dimensional subspace Π_{7} ⊂ ℙ^{15} is a K3 surface of genus seven. Thus we obtain rational maps

\begin{matrix}Gr(7, 16)/G & −−→ & M_{7} \\ Gr(8, 16)/G & −−→ & F_{7} \\ Fl(7, 8, 16)/G & −−→ & P_{7} \end{matrix}

where Fl(7, 8, 16) ⊂ Gr(7, 16) × Gr(8, 16) is the flag variety. Mukai shows that each of these is birational.

3. Degeneration of Mukai’s construction over the pentagonal locus

Let V ⊂ ℙ^{6} denote the Fano threefold of index two obtained as a generic linear section of the Grassmannian Gr(2, 5).

Proposition 2. Let D be a pentagonal curve of genus eleven, i.e., D admits a basepoint free g_{5}^{1}. Assume D is generic. Consider

D ↪ ℙ^{6},

the embedding induced by the linear series adjoint to the g_{5}^{1}. This admits a canonical factorization

D ⊂ V ⊂ ℙ^{6}.

Proof. Let ϕ_{1} : D → ℙ^{1} be the degree five morphism. Write (ϕ_{1})_{∗}O_{D} = O_{ℙ^{1}} ⊕ F using the trace homomorphism; by relative duality, we have

(ϕ_{1})_{∗}ω_{D} = ω_{ℙ^{1}} ⊕ F^{∨} ⊗ ω_{ℙ^{1}}

and

Γ(D, ω_{D}) = Γ(ℙ^{1}, ω_{ℙ^{1}} ⊕ F^{∨} ⊗ ω_{ℙ^{1}}).

Moreover F^{∨} ⊗ ω_{ℙ^{1}} is globally generated, as the divisors in the basepoint free g_{5}^{1} impose four conditions on the canonical series. Hence the canonical embedding factors

D ↪ ℙ(F ⊗ T_{ℙ^{1}}) → ℙ^{10}.

Our genericity assumption is that the vector bundles above are as ‘balanced’ as possible, i.e.,

E := F ⊗ T_{ℙ^{1}} ≃ O_{ℙ^{1}}(−1) ⊕ O_{ℙ^{1}}(−2)^{⊕3}.

Since E^{∨} ⊗ O_{ℙ^{1}}(−1) is still globally generated, we also have a factorization of the adjoint morphism

D ↪ ℙ(E ⊗ O_{ℙ^{1}}(1)) → ℙ^{6};

the image of the projective bundle is a cone over the Segre threefold ℙ^{1} × ℙ^{2}. The tautological exact sequence on p : ℙ(E) → ℙ^{1} takes the form

0 → O_{ℙ(E)}(−1) → p^{∗}E → Q → 0;

twisting yields

0 → O_{ℙ(E)}(−1) ⊗ p^{∗}O_{ℙ^{1}}(2) → p^{∗}(E ⊗ O_{ℙ^{1}}(2)) → Q′ := Q ⊗ p^{∗}O_{ℙ^{1}}(2) → 0.

Note that rank(Q′) = 3, h^{0}(Q′) = 5, Q′ is globally generated, and c_{1}(Q′) = c_{1}(O_{ℙ(E)}(1)) − p^{∗}c_{1}(O_{ℙ^{1}}(1)). Restricting to D ⊂ ℙ(E) gives

c_{1}(Q′|D) = K_{D} − g_{5}^{1},

the adjoint divisor. The classifying map for Q′|D gives a morphism

D → Gr(2, 5)

factoring through a codimension three linear section, which is V. □

Remark 3. We isolate where the generality assumption is used: It is necessary that E not admit any summands of degree ≤ −3, or equivalently,

(ϕ_{1})_{∗}ω_{D} = ω_{ℙ^{1}} ⊕ O_{ℙ^{1}}(1) ⊕ O_{ℙ^{1}}(2)^{⊕3}.

Theorem 4. Let D be a generic pentagonal curve of genus eleven and (ϕ_{1}, ϕ_{2}) : D → ℙ^{1} × V the embedding given by the degree five covering and Proposition 2. We have

• φ_{11}^{−1}(D) ≃ ℙ^{2}, specifically, the K3 surfaces containing D are the codimensiontwo linear sections of ℙ^{1} × V containing D.

• Fix two distinct points d_{1}, d_{2} ∈ D with ϕ_{1}(d_{1}) = ϕ_{1}(d_{2}) =: p; given a conic

Z satisfying

d_{1}, d_{2} ∈ Z ⊂ ϕ_{1}^{−1}(p),

there exists a unique K3 surface S containing D and Z.

The K3 surface S has lattice polarization

(1)  \begin{aligned}& h & E & Z \\ h & 20 & 5 & 2 \\ E & 5 & 0 & 0 \\ Z & 2 & 0 & −2 \end{aligned}

where E is the elliptic fibration inducing the g_{5}^{1} on D ∈ |h|.

Proof. We present the basic geometric set-up. We have

ℙ^{1} × V ⊂ ℙ^{1} × ℙ^{6} ⊂ ℙ^{13}

where the last inclusion is the Segre embedding. Given a flag

ℙ^{10} ⊂ ℙ^{11} ⊂ ℙ^{13},

intersecting with ℙ^{1} × V yields

D ⊂ S ⊂ ℙ^{1} × V,

where D is a canonically embedded pentagonal curve of genus five, and S is a K3 surface with lattice polarization:

\begin{aligned}& h & E \\ h & 20 & 5 \\ E & 5 & 0 \end{aligned}

Fixing D, the K3 surfaces S containing D correspond to the ℙ^{11}’s in ℙ^{13} containing the fixed ℙ^{10}, which are parametrized by ℙ^{2}. This proves the first assertion.

We prove the existence assertion of the second part. The assumption ϕ_{1}(d_{1}) = ϕ_{1}(d_{2}) =: p means that

d_{1}, d_{2} ∈ {p} × V ⊂ ℙ^{1} × V.

Two-dimensional linear sections S containing D and the desired conic correspond to conics Z ⊂ V passing through d_{1}, d_{2} ∈ V. Indeed, S may be recovered from Z:

S = span(D ∪_{d_{1},d_{2}} Z) ∩ (ℙ^{1} × V).

Note that each such S is automatically regular along D. For generic pentagonal D and p ∈ ℙ^{1}, S must be regular everywhere. We see this by a parameter count. Pairs (S, D) where S is a K3 surface with lattice polarization (1) and D ∈ |h| depend on 17 + 11 = 28 parameters. Pairs (D, p) where D admits a degree-five morphism to ℙ^{1} and p ∈ ℙ^{1} depend on 30 − 3 + 1 = 28 parameters. Thus the (D, p) arising from singular K3 surfaces cannot be generic.

The uniqueness assertion boils down to an enumerative problem: How many conics pass through two prescribed generic points d_{1}, d_{2} ∈ V? A conic R ⊂ Gr(2, 5) is a ruling of lines on a quadric surface q_{R} ⊂ ℙ^{4}. The points d_{1}, d_{2} ∈ V ⊂ Gr(2, 5) correspond to a pair of lines which span a ℙ^{3} ⊂ ℙ^{4}; let G ≃ Gr(2, 4) ⊂ Gr(2, 5) denote its Grassmannian of lines. Thus R ⊂ G, i.e., a conic lying on a smooth fourdimensional quadric hypersurface. However, V ⊂ Gr(2, 5) is a codimension-three linear section, hence V ∩ G = R. □

4. Specializing pentagonal curves of genus 11

For our ultimate application to genus 7 curves, we must specialize the construction of Section 3 further:

Theorem 5. Let C_{1} denote a generic curve of genus five, c_{1}, …, c_{7} ∈ C_{1} generic points, and ψ : C_{1} → ℙ^{1} =: R′ a degree four morphism. Let D = C_{1} ∪_{c_{j}=ψ(c_{j})} R′ denote the genus eleven nodal curve obtained by gluing C_{1} and R′, which is automatically pentagonal. Fix an additional generic point c_{0} ∈ C_{1}, and write ψ^{−1}(ψ(c_{0})) = {c_{0}, c′_{1}, c′_{2}, c′_{3}}. Then there exists a unique embedding D ↪ S where S is a K3 surface with lattice polarization

f C_{1} C_{2} R′
f 4 7 1 6
C_{1} 7 8 3 7
C_{2} 1 3 −2 0
R′ 6 7 0 −2

where C_{2} ≃ ℙ^{1} intersects C_{1} at {c′_{1}, c′_{2}, c′_{3}}.

Proof. We claim that the argument of Theorem 4 applies, as D satisfies Remark 3. Indeed, it suffices to show that

h^{0}(ω_{D}(−2g_{5}^{1})) = 3;

if (ϕ_{1})_{∗}ω_{D} failed to have the expected decomposition then

h^{0}(ℙ^{1}, (ϕ_{1})_{∗}ω_{D}) > 3,

a contradiction.

If h^{0}(ω_{D}(−2g_{5}^{1})) > 3 then

m := h^{0}(O_{D}(2g_{5}^{1})) = h^{1}(O_{D}(2g_{5}^{1})) > 3;

clearly 2g_{5}^{1} is basepoint free, so we have a morphism

D → ℙ^{m−1}, m ≥ 4.

The image of R′ under this morphism is a plane conic, hence the images of c_{1}, …, c_{7} are distinct coplanar points. This means that on C_{1}

2g_{4}^{1} − c_{1} − ⋯ − c_{7}, g_{4}^{1} = g_{5}^{1}|C_{1}

is effective, contradicting the genericity of c_{1}, …, c_{7}.

Now we take d_{1} = c_{0} and d_{2} = ψ(c_{0}) ∈ R′ ≃ ℙ^{1}. Thus D is contained in a distinguished surface S containing a rational curve Z ∋ d_{1}, d_{2}. Hence S has lattice polarization

C_{1} R′ E Z
C_{1} 8 7 4 1
R′ 7 −2 1 1
E 4 1 0 0
Z 1 1 0 −2

containing the intersection matrix (1). Using the identifications

h = C_{1} + R′, E = C_{1} + C_{2} − f, Z = C_{1} − f, R′ = R′,

we obtain the desired lattice polarization.

It remains to show that for generic inputs the surface S is in fact smooth. As before, this follows from a parameter count. The data

(C_{1}, c_{0}, c_{1}, …, c_{7})

consist of a genus five curve (12 parameters), a choice of g_{4}^{1} on that curve (1 parameter), and eight generic points, for a total of 21 parameters. On the output side, we have a K3 surface with the prescribed lattice polarization of rank four (16 parameters) and a curve in the linear series |C_{1}| (5 parameters). Thus for generic input data, the resulting surface is necessarily smooth. □

5. Genus 7 K3 surfaces and rational normal septic curves

Consider the moduli space of lattice-polarized K3 surfaces of type

Λ′ := \begin{aligned}& C & R′ \\ C & 12 & 7 \\ R′ & 7 & −2 \end{aligned}

and let P_{Λ′} denote the moduli space of pairs (S, D), where S is such a K3 surface and D ∈ |C| is smooth and meets R′ transversely.

Proposition 6. The forgetting morphism

φ_{Λ′} : P_{Λ′} → M_{7,7}/𝔖_{7}

is generically finite.

Note that the varieties are both of dimension 25. The main ingredient of the proof is:

Lemma 7. Let M_{0,7}(OG(5, 10), 7) be the moduli space of pointed mappings of degree 7ℓ, where ℓ ∈ H_{2}(OG(5, 10), ℤ) is Poincaré dual to the hyperplane class h. Then the evaluation map

ev^{7} : M_{0,7}(OG(5, 10)), 7) → OG(5, 10)^{7}

is dominant.

Proof. Recall that dim(OG(5, 10)) = 10 and the canonical class is

K_{OG(5,10)} = −8h.

The expected dimension of the moduli space is 70, so we expect ev^{7} to be generically finite.

By [dJHS11, 15.7], the evaluation map

ev : M_{0,7}(OG(5, 10)), 1) → OG(5, 10)

is surjective, i.e., pointed lines dominate OG(5, 10). It follows from generic smoothness that H^{1}(N_{ℓ/OG(5,10)}(−1)) = 0 [AK03, p.33] and

N_{ℓ/OG(5,10)} ≃ O_{ℙ^{1}}^{⊕3} ⊕ O_{ℙ^{1}}(1)^{⊕6}.

Fix a generic chain of seven lines in OG(5, 10)

C_{0} := ℓ_{1} ∪_{e_{12}} ℓ_{2} ∪_{e_{23}} … ∪_{e_{67}} ℓ_{7},

as well as generic points r_{j}(0) ∈ ℓ_{j}. Consider the following moduli problem: Fix a smooth base scheme B with basepoint 0 and morphisms r_{j} : B → OG(5, 10), j = 1, …, 7 mapping 0 to the point r_{j}(0) specified above. We are interested in subschemes

r_{1}, …, r_{7} ⊂ C ⊂ OG(5, 10) × B
↘ ↓ ↙
B

all flat over B, with the distinguished fiber of C equal to C_{0}. The deformation theory of this Hilbert scheme as a scheme over B [AK03, §6] is governed by the tangent space

Γ(N_{C_{0}/OG(5,10)}(−r_{1}(0) − ⋯ − r_{7}(0)))

and the obstruction space

H^{1}(N_{C_{0}/OG(5,10)}(−r_{1}(0) − ⋯ − r_{7}(0))).

If the latter group is 0, the Hilbert scheme is flat over B of the expected dimension

χ(N_{C_{0}/OG(5,10)}(−r_{1}(0) − ⋯ − r_{7}(0))) = 0.

Since C_{0} is a chain of rational curves, it suffices to exclude higher cohomology on each irreducible component ℓ_{i}. For i = 1, 7 we have

N_{C_{0}/OG(5,10)}|ℓ_{i} = O_{ℙ^{1}}^{3} ⊕ O_{ℙ^{1}}(1)^{5} ⊕ O_{ℙ^{1}}(2)

which has trivial higher cohomology, even after twisting by O_{ℓ_{i}}(−r_{i}(0)). For i = 2, …, 6, we have

N_{C_{0}/OG(5,10)}|ℓ_{i} = O_{ℙ^{1}}^{3} ⊕ O_{ℙ^{1}}(1)^{4} ⊕ O_{ℙ^{1}}(2)^{2},

reflecting the two attaching points on ℓ_{i}. Twisting by O_{ℓ_{i}}(−r_{i}(0)), we have no higher cohomology as well.

Choosing r_{j} : B → OG(5, 10) suitably generic, we conclude there is a zero dimensional collection of rational septic curves passing through seven generic points of OG(5, 10). □

We complete the proof of Proposition 6. Let D denote a generic curve of genus seven and r_{1}, …, r_{7} ∈ D generic points. As we recalled in Section 2, C′ arises as a linear section of OG(5, 10), and Lemma 7 yields a septic rational curve R′ ⊂ OG(5, 10) containing these seven points. The intersection

OG(5, 10) ∩ span(D ∪ R′)

is a K3 surface, with the prescribed lattice polarization.

Remark 8. What is the degree of φ_{Λ′}? The birationality results quoted in Section 2 imply that the degree equals the degree of the generically-finite mapping

ev^{7} : M_{0,7}(OG(5, 10)), 7) → OG(5, 10)^{7}.

See the Appendix for a proof that deg(ev^{7}) = 71.

6. From genus 11 to genus 7

Theorem 9. Let π : C → R′ = ℙ^{1} be a tetragonal curve of genus seven, C ⊂ ℙ^{3} the adjoint embedding as a curve of degree eight, c_{1}, …, c_{7} ∈ C generic points. Then there exists a unique embedding

ϖ : R′ ↪ ℙ^{3}, ϖ(π(c_{j})) = c_{j}

such that there exists a quartic surface S containing both C and R′.

The K3 surfaces in this case have lattice polarization:

(2)  Λ = \begin{aligned}& f & C & R′ \\ f & 4 & 8 & 6 \\ C & 8 & 12 & 7 \\ R′ & 6 & 7 & −2 \end{aligned}

Proof. We regard Theorem 5 as a special case of this, via the specialization

C ⇝ C_{1} ∪ C_{2}.

We can deform the K3 surfaces in Theorem 5 to the K3 surfaces with lattice polarization (2). Under this deformation, R′ deforms to a rational curve in the nearby fibers; smooth rational curves in K3 surfaces always deform provided their divisor classes remain algebraic.

We claim that C_{1} ∪ C_{2} deforms to a generic tetragonal curve of genus seven, with seven generic marked points traced out by R′.

For our purposes, we would like to restrict the curve C to be tetragonal. This is equivalent (see [GL87]) to imposing a lattice polarization of the type

C R′ E
C 12 7 4
R′ 7 −2 a
E 4 a 0

where E is the class of a fiber of an elliptic fibration. Restricting φ_{Λ′} to each such lattice polarization, we obtain a generically finite morphism to the Brill-Noether divisor T ⊂ M_{7,7}/𝔖_{7} corresponding to the tetragonal curves. In our geometric analysis, it will be convenient to use a different basis

C R′  f
C 12 7  8
R′ 7 −2 7 − a
f 8 7 − a 4

where f = C − E. We will restrict our attention to the particular component with a = 1, i.e., R′ is a section of the elliptic fibration inducing the g_{4}^{1} on C. This is P_{Λ}, where Λ is defined in (2).

Theorem 9 asserts that the morphism

φ_{Λ} : P_{Λ} → T

has degree one. Consider the specialization C ⇝ C_{1} ∪ C_{2} as above. After specialization, Theorem 5 guarantees a unique K3 surface containing C_{1} ∪ C_{2}. Thus φ_{Λ} has degree at least one. If the degree were greater than one, then a generic point of T would yield at least two surfaces S and S′, with specializations S_{0} and S′_{0}. These are necessarily K3 surfaces, by the following result about degenerate quartic surfaces:

Lemma 10. Consider the projective space ℙ^{34} parametrizing all quartic surfaces in ℙ^{3}. Let Σ ⊂ ℙ^{34} denote the surfaces with singularities worse than ADE singularities. Then the codimension of Σ is at least four.

Proof. We first address the non-isolated case. The reducible surfaces of this type— unions of two quadric surfaces or a plane and a cubic—have codimension much larger than four. The irreducible surfaces are classified by Urabe [Ura86, §2], building on the work of numerous predecessors over the last century. In each case, the codimension is at least four.

There is also a substantial literature on the classification of isolated singularities of quartic surfaces, e.g., [Sha81, Deg89, IN04]. However, it will be more convenient for us to give a direct argument, rather than refer to the details of a complete classification.

Since ℙ^{3} is homogeneous, it suffices to show that the surfaces with a non ADE singularity at p = [0, 0, 0, 1] have codimension ≥ 3 in the locus of surfaces singular at p. The equations with multiplicity > 2 at p have codimension ≥ 6 and thus can be ignored; the equations of multiplicity two having non-reduced tangent cone at p have codimension 3. The equations with an isolated singularity of multiplicity two at p (and reduced tangent cone) are of ADE type. □

It only remains to exclude ramification at the generic point of the stratum, i.e., that the fibers of φ_{Λ} have zero dimensional tangent space. Section 7 is devoted to proving this. □

7. Deformation computations

7.1. Generalities. We review the formalism of deformations of pairs, following [Kaw78]. For the moment, S denotes a smooth projective variety and D ⊂ S a reduced normal crossings divisor. We have exact sequences

0 → T_{S}(−D) → T_{S} ⟨−D⟩ → T_{D} → 0

and

0 → T_{S} ⟨−D⟩ → T_{S} → N_{D/S} → 0,

where T_{S} ⟨−D⟩ means vector fields on S with logarithmic zeros along D. The tangent space to the deformation space of (S, D) is given by H^{1}(T_{S} ⟨−D⟩). The sequence

Γ(N_{D/S}) → H^{1}(T_{S} ⟨−D⟩) → H^{1}(T_{S})

may be interpreted ‘first-order deformations of (S, D) leaving S unchanged arise from deformations of D ⊂ S’; the sequence

H^{1}(T_{S}(−D)) → H^{1}(T_{S} ⟨−D⟩) → H^{1}(T_{D})

means that we may interpret H^{1}(T_{S}(−D)) as ‘first-order deformations of (S, D) leaving D unchanged’.

Our situation is a bit more complicated as our boundary consists of two log divisors deforming independently. If D = {x = 0} and R′ = {y = 0} meet transversally at x = y = 0 then T_{S} ⟨−D⟩ ⟨−R′⟩ is freely generated by x\frac{∂}{∂x} and y \frac{∂}{∂y} . First-order deformations of (S, D, R′) are given by H^{1}(T_{S} ⟨−D⟩ ⟨−R′⟩).

We analyze the ramification of the forgetting morphism φ from the deformation space of (S, D, R) to the deformation space of (D, D ∩ R′). A slight variation on one of the standard exact sequences above

0 → T_{S} ⟨−R⟩ (−D) → T_{S} ⟨−D⟩ ⟨−R′⟩ → T_{D} ⟨−R′⟩ → 0

induces the differential

dφ : H^{1}(T_{S} ⟨−D⟩ ⟨−R′⟩) → H^{1}(T_{D} ⟨−D ∩ R⟩).

Hence the kernel is given by H^{1}(T_{S} ⟨−R′⟩ (−D)), which is zero precisely when φ is unramified. The exact sequence

0 → T_{S} ⟨−R′⟩ (−D) → T_{S}(−D) → N_{R′/S}(−D ∩ R′) → 0

induces

\begin{matrix}Γ(N_{R′/S}(−D ∩ R′)) → \\ H^{1}(T_{S} ⟨−R′⟩ (−D)) → H^{1}(T_{S}(−D)) →^{h} H^{1}(N_{R′/S}(−D ∩ R′)). \end{matrix}

Thus φ is unramified if Γ(N_{R′/S}(−D ∩ R′)) = 0 and h is injective.

7.2. Our specific situation. We specify what the various objects are: First,

S = {F_{1} = F_{2} = 0} ⊂ ℙ^{1} × ℙ^{3}.

is a smooth complete intersection of forms of bidegree (1, 2). The divisor

D = {L = 0} ⊂ S

is a hyperplane section, i.e., L is of bidegree (1, 1). The divisor R′ ⊂ S is a smooth rational curve of bidegree (1, 6), meeting D transversally, in seven points.

Since deg(N_{R′/S}(−D ∩ R′)) = −2 − 7 it has no global sections; the higher cohomology is computed via Serre duality

H^{1}(N_{R′/S}(−D ∩ R′) = Γ(O_{R′}(D ∩ R′))^{∨} = Γ(O_{ℙ^{1}}(7))^{∨}.

We claim that H^{1}(T_{S}(−D)) is eight dimensional.

Remark 11. H^{1}(T_{S}(−D)) is therefore eight dimensional when S is a generic K3 surface of degree 12. This has a nice global interpretation via the Mukai construction [Muk96]: D (resp. S) is a codimension nine (resp. eight) linear section of the orthogonal Grassmannian OG(5, 10), so the K3 surfaces containing a fixed D are parametrized by ℙ^{8}.

To prove the claim, use the normal bundle exact sequence

0 → T_{S}(−D) → T_{ℙ^{1}×ℙ^{3}}(−1, −1)|S → N_{S/ℙ^{1}×ℙ^{3}}(−1, −1) → 0,

the Koszul complex for {F_{1}, F_{2}}

0 → O_{ℙ^{1}×ℙ^{3}}(−2, −4) → O_{ℙ^{1}×ℙ^{3}}(−1, −2)^{⊕2} → O_{ℙ^{1}×ℙ^{3}} → O_{S} → 0

and its twist by T_{ℙ^{1}×ℙ^{3}}(−1, −1). Applying the Kunneth formula to compute the cohomologies of the twists of T_{ℙ^{1}×ℙ^{3}}, we find that

Γ(T_{ℙ^{1}×ℙ^{3}}(−1, −1)|S) = H^{1}(T_{ℙ^{1}×ℙ^{3}}(−1, −1)|S) = 0.

We also find that

Γ(N_{S/ℙ^{1}×ℙ^{3}}(−1, −1)) = Γ(O_{S}(0, 1))^{⊕2} = Γ(O_{ℙ^{3}}(1))^{⊕2},

which is isomorphic to H^{1}(T_{S}(−D)) by the vanishing above.

In concrete terms, the infinitesimal deformations corresponding to H^{1}(T_{S}(−D)) take the form

(3)  S(ϵ) = {F_{1} + ϵLG_{1} = F_{2} + ϵLG_{2} = 0}, G_{1}, G_{2} ∈ Γ(O_{ℙ^{3}}(1)).

The exact sequence above therefore takes the form

0 → H^{1}(T_{S} ⟨−R′⟩ (−D)) → Γ(O_{ℙ^{3}}(1))^{⊕2} →^{h} Γ(O_{ℙ^{1}}(7)),

where h captures the obstructions to deforming the rational curve along an infinitesimal deformation of S. Precisely, suppose we start off with

• (ℙ^{1}, r_{1}, …, r_{7}) a pointed rational curve;

• a K3 surface S = {F_{1} = F_{2} = 0} ⊂ ℙ^{1} × ℙ^{4};

• a nodal hyperplane section curve D ⊂ S with equation L = 0;

• a morphism ι : ℙ^{1} → S of bidegree (1, 6) with d_{j} = ι(r_{j}) ∈ D. Fix ϵ to be a small parameter, or in algebraic terms, a nilpotent with ϵ^{2} = 0. We analyze first-order deformations of

• the K3 surface S(ϵ) as in (3);

• the morphism ι(ϵ) : ℙ^{1} → S_{ϵ} satisfying the constraint

(4)  d_{j} = ι(ϵ)(r_{j}).

Let [t, u] be coordinates on ℙ^{1} and write the seven distinct points [r_{j}, 1], j = 1, …, 7. Let P_{1}, …, P_{7} ∈ k[t, u]_{6} be homogeneous polynomials of degree six such that P_{i}(r_{j}, 1) = δ_{ij}; these form a basis of k[t, u]_{6}. We factor ι as the composition of the 6-uple embedding

\begin{matrix}v : ℙ^{1} & → & ℙ^{6} \\ [t, u] & ↦ & [P_{1}, …, P_{7}] \end{matrix}

and a linear projection given by the 4 × 7 matrix

D = (\begin{matrix}d_{01} & d_{02} & … & d_{07} \\ d_{11} & d_{12} & … & d_{17} \\ d_{21} & d_{22} & … & d_{27} \\ d_{31} & d_{32} & … & d_{37} \end{matrix}) ,

whose jth column consists of the coordinates of d_{j}. The condition that ι(ℙ^{1}) ⊂ S translates into

(5)  F_{m}(t, u; ∑_{j} d_{0j}P_{j}, …, ∑_{j} d_{3j}P_{j}) = 0

for m = 1, 2.

The perturbation of the matrix D entails rescaling each column of D by a multiplicative scalar, to first order. This takes the form

D(ϵ) = (d_{ij}(1 + s_{j}ϵ)) ,

keeping in mind that the case s_{1} = s_{2} = … = s_{7} induces a trivial deformation, i.e., rescaling D by a constant. The condition (4) is automatically satisfied.

We analyze the condition

ι(ϵ)(ℙ^{1}) ⊂ S(ϵ)

to first order. This can be written

\begin{matrix}F_{m}(t, u; ∑_{j} d_{0j}(1 + ϵs_{j})P_{j}, …)+ \\ ϵ(L · G_{m})(t, u; ∑_{j} d_{0j}(1 + ϵs_{j})P_{j}, …) = 0 \end{matrix}

for m = 1, 2. Writing x_{0}, x_{1}, x_{2}, x_{3} for the homogeneous coordinates on ℙ^{3} and extracting the first derivative—the constant term in ϵ is zero by (5)—we obtain

∑_{k=0}^{3} \frac{∂F_{m}}{∂x_{k}} · ∑_{j} d_{0j}s_{j}P_{j} + L · G_{m} = 0.

Here we should regard \frac{∂F_{m}}{∂x_{k}} as a homogeneous form of degree 7 in \{t, u\}, L also of degree 7, and G_{i} of degree 6. Thus we obtain two homogeneous forms of degree 13 in {t, u}, with each coefficient linear in the variables s_{1}, …, s_{7} (and vanishing where s_{1} = ⋯ = s_{7}) and the eight coefficients of G_{1} and G_{2}. Note, however, that

F_{m}(t, u; d_{j}) = F_{m}(t, u; ι(ϵ)(r_{j})) = 0, j = 1, …, 7,

thus F_{m}(t, u; ι(ϵ)([t, u])) is a multiple of ∏_{i=1,...,7}(t − r_{i}u); the same is true for L(ι(ϵ)([t, u])). Dividing out by this, we obtain two homogeneous forms of degree 6 in {t, u}, with each coefficient linear in the variables; thus we obtain 14 linear equations in 14 independent variables. Again, the equations will vanish if s_{1} = … = s_{7} so these depend on only six parameters.

7.3. A unirationality result. To execute the computations outlined in Section 7.2, we must evaluate all the relevant terms in a specific example. In practice, finding such an example is much easier when the underlying parameter spaces are unirational.

Proposition 12. Consider the Hilbert scheme parametrizing the following data: • points r_{1}, …, r_{7} ∈ ℙ^{1}

• a rational curve C_{2} ⊂ ℙ^{1} × ℙ^{3} of bidegree (0, 1) (i.e., a line);

• a hyperplane section {L = 0} ⊂ ℙ^{1} × ℙ^{3} containing C_{2};

• a morphism ι : ℙ^{1} → ℙ^{1}×ℙ^{3} of bidegree (1, 6) with ι({r_{1}, …, r_{7}}) ⊂ {L = 0};

• a K3 surface S ⊂ ℙ^{1} ×ℙ^{3}, given as a complete intersection of forms of degree

degree (1, 2) containing C_{2} and R′ := ι(ℙ^{1}). This space is rational.

Below, let f be induced from the hyperplane class of ℙ^{3} and C_{1} ∪ C_{2} be cut out by L = 0. Note that C_{1} is residual to C_{2} in the hyperplane D = {L = 0}.

Corollary 13. Consider the moduli space of lattice polarized K3 surfaces S of type

\begin{aligned}& f & C_{1} & C_{2} & R′ \\ f & 4 & 7 & 1 & 6 \\ C_{1} & 7 & 8 & 3 & 7 \\ C_{2} & 1 & 3 & −2 & 0 \\ R′ & 6 & 7 & 0 & −2 \end{aligned} ,

equipped with an ordering of the points of C_{1} ∩ R′. This space is unirational.

Proof. The construction is step-by-step: Seven ordered points on ℙ^{1} are parametrized by a rational variety. The lines C_{2} are as well, i.e., the product ℙ^{1} × Gr(2, 4). For each such line, it is a linear condition for a hypersurface {L = 0} to vanish along the line. Once we have C_{2} and ι, the K3 surfaces containing C_{2} ∪ R′ are given by the Grassmannian Gr(2, I_{C_{2}∪R′}(1, 2)).

Thus we only have to worry about the choice of ι. We interpret this as a section of the projection ℙ^{1} × ℙ^{3} → ℙ^{1}, given by a collection of sextic polynomials

[ι_{0}(t, u), …, ι_{3}(t, u)],

parametrized by a dense open subset of the projective space ℙ^{27} on the coefficients. After a linear change of coordinates on ℙ^{3}, we may assume L = tx_{0} − ux_{1}. Then the conditions ι(r_{j}) ∈ {L = 0} take the form

r_{j}ι_{0}(r_{j}, 1) − ι_{1}(r_{j}, 1),

i.e., seven linear equations on the coefficient space. The resulting parameter space is thus rational. □

7.4. Concrete example. We exhibit a specific example over ℚ where the morphism φ_{Λ} is unramified. Its construction closely follows the unirationality proof in Section 7.3.

The relevant data is

• points r_{1} = [0, 1], r_{2} = [1, 1], r_{3} = [−1, 1], r_{4} = [2, 1], r_{5} = [−2, 1], r_{6} =

[1/2, 1], r_{7} = [−1/2, 1] ∈ ℙ^{1};

• a rational curve C_{2} = {u = x_{0} = x_{3} = 0};

• a hyperplane section L = tx_{0} − ux_{1} ⊂ ℙ^{1} × ℙ^{3}, containing C_{2};

• a morphism ι : ℙ^{1} → ℙ^{1} × ℙ^{3} given by

\begin{aligned}x_{0} & = & 4t^{6} + t^{5}u + u^{6} \\ x_{1} & = & t^{6} + 21t^{5}u − 21t^{3}u^{3} + 5tu^{5} \\ x_{2} & = & t^{6} \\ x_{3} & = & t^{6} + t^{5}u + t^{4}u^{2} + t^{3}u^{3} + t^{2}u^{4} + tu^{5} + u^{6}; \end{aligned}

• a K3 surface S ⊂ ℙ^{1} × ℙ^{3}, given as a complete intersection

{F_{1} = F_{2} = 0}

where

F_{1} = u(−36134306460x_{0}^{2} + 1648259021x_{0}x_{1} + 179920405271x_{0}x_{2}
+72385436466x_{0}x_{3} + 49839426x_{1}^{2} − 3784378416x_{1}x_{2}
−2345703360x_{1}x_{3} − 181391061852x_{2}^{2} − 225811403454x_{2}x_{3}
−36251130006x_{3}^{2}) + t(−13678895854x_{0}^{2} + 671675907x_{0}x_{1}
+56417839468x_{0}x_{2} − 8926222977x_{0}x_{3} + 26209164072x_{3}^{2})

and

F_{2} = u(3638964x_{0}^{2} − 1272831x_{0}x_{1} + 29670963x_{0}x_{2}
−13458270x_{0}x_{3} − 22974x_{1}^{2} + 3555552x_{1}x_{2}
+1904792x_{1}x_{3} − 114701748x_{2}^{2} − 4837990x_{2}x_{3}
+9819306x_{3}^{2}) + t(12731586x_{0}^{2} − 575505x_{0}x_{1}
−52280172x_{0}x_{2} − 22071733x_{0}x_{3} + 96004264x_{2}x_{3}).

The linear projection matrix is given by

D = (\begin{matrix}1 & 6 & 4 & 289 & 225 & 35/32 & 33/32 \\ 0 & 6 & −4 & 578 & −450 & 35/64 & −33/64 \\ 0 & 1 & 1 & 64 & 64 & 1/64 & 1/64 \\ 1 & 7 & 1 & 127 & 43 & 127/64 & 43/64 \end{matrix}) .

For the tangent space computation, write

\begin{aligned}G_{1} & = & g_{10}x_{0} + g_{11}x_{1} + g_{12}x_{2} + g_{13}x_{3} \\ G_{2} & = & g_{20}x_{0} + g_{21}x_{1} + g_{22}x_{2} + g_{23}x_{3} \end{aligned}

and extract the terms linear in ϵ in

F_{m}(t, u; ∑_{j} d_{0j}(1 + ϵs_{j})P_{j}, …) + ϵ(L · G_{m})(t, u; ∑_{j} d_{0j}(1 + ϵs_{j})P_{j}, …) = 0.

This yields two homogeneous forms in s and u of degree 13, with coefficients linear in the g_{ik} and s_{j}, each divisible by

∏_{j}(t − r_{j}u) = t(t − u)(t + u)(t − 2u)(t + 2u)(t − u/2)(t + u/2).

Dividing out, we obtain two forms in s and u of degree 6 with coefficients linear in the g_{ik} and s_{j}. We will not reproduce these coefficients here, but after Gaussian elimination we are left with the system

\begin{matrix}g_{10} = g_{11} = g_{12} = g_{13} = g_{20} = g_{21} = g_{22} = g_{23} = 0 \\ s_{1} = s_{2} = s_{3} = s_{4} = s_{5} = s_{6} = s_{7}. \end{matrix}

Thus the space of infinitesimal deformations of S containing our curves is trivial, hence our morphism φ_{Λ} is unramified at this point.

8. A related construction

Consider the moduli space F_{Λ′′} of lattice-polarized K3 surfaces of type

(6)  Λ′′ := \begin{aligned}& C & R′ \\ C & 12 & 5 \\ R′ & 5 & −2 \end{aligned}

and the moduli space P_{Λ′′} parametrizing pairs (S, D) with S ∈ F_{Λ′′} and D ∈ |C| smooth and transverse to R′, which has dimension 25. We consider

φ_{Λ′′} : P_{Λ′′} → (M_{7,5} × M_{0,5})/𝔖_{5}

to a variety of dimension 18 + 5 + 2 = 25.

Proposition 14. φ_{Λ′′} is birational.

Proof. We construct the inverse mapping. Fix (C, c_{1}, …, c_{5}) a generic five-pointed curve of genus seven and (ℙ^{1}, a_{1}, …, a_{5}) a five-pointed curve of genus zero. Realize C as a linear section in OG(5, 10). We seek a curve R′ arising as a rational normal quintic curve ϖ : ℙ^{1} → OG(5, 10) with ϖ(a_{i}) = c_{i}, i = 1, …, 5. The K3 surface S would then arise as the intersection

span(C ∪ R′) ∩ OG(5, 10).

Figure 1. A stable five-pointed curve

The construction of ϖ boils down to an enumerative computation by Andrew Kresch:

Lemma 15. Given generic points Λ_{1}, ⋯, Λ_{5} ∈ OG(5, 10), and generic points a_{1}, …, a_{5} ∈ ℙ^{1}, there exists a unique morphism ϖ : ℙ^{1} → OG(5, 10) with image of degree five such that ϖ(a_{i}) = Λ_{i} for i = 1, …, 5.

We have the tautological diagram

\overline{M}_{0,5}(OG(5, 10), 5) →^{ev^{5}} OG(5, 10)^{5}

(ϕ ↓) / (\overline{M}_{0,5})

where the ϕ is the forgetting morphism and ev^{5} is evaluation at all five marked points. We seek to show that

deg(ev_{1}^{∗}[pt] ⋯ ev_{5}^{∗}[pt] · ϕ^{∗}(pt)) = 1.

We specialize the point in \overline{M}_{0,5} to the configuration of Figure 1.

We assert that

(1) there is no line on OG(5, 10) through two general points

(2) the closure of the union of all conics through two general points is a Schubert variety parametrizing spaces containing a particular one-dimensional isotropic vector space Ξ of the ambient ten dimensional space.

(3) there is no conic through three general points. For the first statement, note that the space of lines is parametrized by three dimensional isotropic subspaces of the ambient ten dimensional space. These are contained in the intersection of all the maximal isotropic subspaces parametrized by the line [LM03, Example 4.12]. Thus the space of lines has dimension 15, which is incompatible with any two points containing a line.

For the second statement, consider the conics containing Λ_{1} and Λ_{2}; set Ξ := Λ_{1} ∩ Λ_{2}, which is one dimensional, and note that Ξ^{⊥} = Λ_{1} + Λ_{2}. The Λ ∈ OG(5, 10) satisfying

Ξ ⊂ Λ ⊂ Ξ^{⊥}

are parametrized by OG(4, 8). Recall that OG(4, 8) is just a quadric sixfold by triality; the conics through two generic points are parametrized by the ℙ^{5} parametrizing planes containing those two points. Altogether, these trace out the full OG(4, 8) mentioned above.

The last statement follows immediately, as the union of the conics through two points is a proper subvariety of OG(5, 10).

These observations imply that the ‘end components’ of our stable curve map to conics and the ‘middle component’ maps to a line. So we are reduced to:

Given

Ξ = Λ_{1} ∩ Λ_{2}, Ξ′ = Λ_{4} ∩ Λ_{5}

how many lines on OG(5, 10) are incident to • the Schubert variety of Λ ∈ OG(5, 10) satisfying Ξ ⊂ Λ;

• the Schubert variety of Λ ∈ OG(5, 10) satisfying Ξ′ ⊂ Λ;

• the point Λ_{3}? We know that lines on OG(5, 10) are in bijection with isotropic three dimensional

spaces, so for an isotropic three dimensional W we must have:

(a) W can be extended to a maximal isotropic space containing Ξ;

(b) W can be extended to a maximal isotropic space containing Ξ′;

(c) W ⊂ Λ_{3}. We deduce from (a) that W ⊂ Ξ^{⊥} and from (b) that W ⊂ (Ξ′)^{⊥}. It follows that

W ⊂ Ξ^{⊥} ∩ (Ξ′)^{⊥} ∩ Λ_{3}.

Since Λ_{3} is general, this intersection is already three dimensional so that W is uniquely determined and the answer to our enumerative problem is 1. □

References

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Appendix

71 rational septics through 7 general points on OG(5, 10)

Overview and generalities

The orthogonal Grassmannian OG(n, 2n) is, by definition, one component of the space of n-dimensional subspaces of V = ℂ^{2n}, isotropic for a given nondegenerate symmetric bilinear form on V. By convention we fix the standard bilinear form, for which ⟨v, w⟩ = ∑_{i=1}^{2n} v_{i}w_{2n+1−i}, and we take OG = OG(n, 2n) to be the component containing ⟨e_{1}, …, e_{n}⟩. Then OG is a homogeneous projective variety of dimension n(n − 1)/2, with deg c_{1}(OG) = 2(n − 1). It follows that the space of rational degree d curves has dimension n(n − 1)/2 − 3 + 2(n − 1)d, so for n = 5 and d = 7 this is 63 = 7·9 and we expect a finite number of rational septics to pass through 7 general points.

It is known (cf. [FP97, LM11]) that the number of rational curves on OG satisfying incidence conditions imposed by Schubert varieties of codimension ≥ 2 in general position is equal to the corresponding Gromov-Witten invariant:

# \{ \begin{matrix}degree d rational curves through \\ general translates of X_{λ^{1}}, …, X_{λ^{m}} \end{matrix} \} = I_{d}([X_{λ^{1}}], …, [X_{λ^{m}}])

for |λ^{i}| ≥ 2, ∑ |λ^{i}| = n(n − 1)/2 − 3 + 2(n − 1)d + m, where I_{d}([X_{λ^{1}}], …, [X_{λ^{m}}]) denotes the Gromov-Witten invariant

∫_{\overline{M}_{0,m}(OG,d)} ev_{1}^{∗}[X_{λ^{1}}] ⋯ ev_{m}^{∗}[X_{λ^{m}}].

The space \overline{M}_{0,m}(OG, d) is Kontsevich’s moduli space of stable maps of genus zero m-marked curves to OG in degree d (see [KM94]), and it comes with m evaluation maps (at the marked points) ev_{1}, …, ev_{m} to OG. The Schubert varieties in OG are denoted X_{λ}, indexed by strict partitions λ whose parts are < n. (A partition is called strict if has no repeated parts.) The codimension of X_{λ} is equal to |λ|, the sum of the parts of λ. The article [KT04], which includes a determination of the m = 3 invariants, gives a geometric description of OG including the Schubert varieties.

Line numbers

The space of lines on OG is known; see [LM03, Example 4.12]. It is itself a projective homogeneous variety, the space OG(n − 2, 2n) of isotropic (n − 2)-dimensional subspaces of V = ℂ^{2n}. Therefore the computation of the I_{1}([X_{λ^{1}}], …, [X_{λ^{m}}]) reduces to the problem of computing intersection numbers on this homogeneous variety.

There is a well-developed theory using divided difference operators on polynomials for performing computations in the cohomology rings of projective homogeneous varieties of linear algebraic groups, due to Bernstein, Gelfand, and Gelfand [BGG73] and Demazure [Dem74]. In the setting of the orthogonal flag variety OF(2n), parametrizing a space in OG together with a complete flag of subspaces, this has been worked out explicitly by Billey and Haiman [BH95]. It leads to an explicit formula for the Gromov-Witten invariants counting lines on OG satisfying incidence conditions with respect to Schubert varieties in general position. The formula uses the Schur P -polynomials P_{λ} = P_{λ}(X) indexed by strict partitions λ, which form a ℚ-basis for the ring ℚ[p_{1}, p_{3}, …] generated by the odd power sums p_{k} = p_{k}(X) = x_{1}^{k} + x_{2}^{k} + … (cf. Proposition 3.1 of op. cit.). Following op. cit., to these we associate polynomials in z_{1}, …, z_{n}, which we will denote by P_{λ}(z_{1}, …, z_{n}), by sending p_{k}(X) to −(1/2)(z_{1}^{k} + ⋯ + z_{n}^{k}).

Proposition. Introduce the divided difference operators on ℚ[z_{1}, …, z_{n}]:

\begin{aligned}∂_{i}f & = \frac{f(z_{1},...,z_{n})−f(z_{1},...,z_{i+1},z_{i},...,z_{n})}{z_{i}−z_{i+1}}, \\ ∂ˆ1f & = \frac{f(z_{1},...,z_{n})−f(−z_{2},−z_{1},z_{3},...,z_{n})}{−z_{1}−z_{2}}, \end{aligned}

and for i ≤ j let ∂_{i...j} denote ∂_{i}∂_{i+1} … ∂_{j} and let ∂_{j...i} denote ∂_{j} … ∂_{i}. Then for any m and λ^{1}, …, λ^{m} satisfying |λ^{i}| ≥ 2, ∑ |λ^{i}| = n(n − 1)/2 − 3 + 2(n − 1) + m, if we set

F = ∏_{i=1}^{m} ∂ˆ1P_{λ^{i}}(z_{1}, …, z_{n})

with the above convention on P -polynomials, then we have

I_{1}([X_{λ^{1}}], …, [X_{λ^{m}}]) =
\begin{cases}∂_{2...n−1}∂_{1...n−2}∂ˆ1 ⋯ ∂_{2...3}∂_{1...2}∂ˆ1∂_{n−2...1}∂_{n−1...2}F & if n is even, \\ ∂_{2...n−1}∂ˆ1∂_{2...n−2}∂_{1...n−3}∂ˆ1 ⋯ ∂_{2...3}∂_{1...2}∂ˆ1∂_{n−2...1}∂_{n−1...2}F & if n is odd, \end{cases}

where the ⋯ stand for compositions of operators in which the upper limits of the indices are successively decreased by 2.

Proof. According to Theorem 4 of op. cit., if we work with countably many z variables and follow the above convention for associating a symmetric polynomial in these to a P -polynomial P_{λ} = P_{λ}(X), then ∂¯1P_{λ} represents the cycle class of the space of lines incident to X_{λ}, and the displayed composition of divided operators sends the polynomial representing the class of a point on the space of lines on OG to 1. So the proposition follows from the observation that the computation may be performed in the polynomial ring ℚ[z_{1}, …, z_{n}]. □

When n = 5, there are 1071 Gromov-Witten numbers I_{1}([X_{λ^{1}}], …, [X_{λ^{m}}]), which we take as known in what follows.

Example. One of these numbers counts the number of lines incident to 15 general translates of X_{2} (the codimension-2 Schubert variety of spaces in OG meeting a given isotropic 3-dimensional space nontrivially). We have P_{2}(X) = p_{1}^{2}(X) sent to (1/4)(z_{1} + ⋯ + z_{5})^{2}, which upon applying ∂ˆ1 yields −z_{3} − z_{4} − z_{5}. We evaluate

∂_{2}∂_{3}∂_{4}∂ˆ1∂_{2}∂_{3}∂_{1}∂_{2}∂ˆ1∂_{3}∂_{2}∂_{1}∂_{4}∂_{3}∂_{2}(−z_{3} − z_{4} − z_{5})^{15}

and find

I_{1}([X_{2}], …, [X_{2}]) = 240240.

τ_{4}  τ_{31}  τ_{41}  τ_{32}  τ_{42}  τ_{321} τ_{43} τ_{421}
τ_{1} τ_{41} τ_{41} + τ_{32}  τ_{42}  τ_{42} + τ_{321} τ_{43} + τ_{421} τ_{421} τ_{431} τ_{431}
τ_{2} τ_{42} 2τ_{42} + τ_{321} τ_{43} + τ_{421} τ_{43} + 2τ_{421} 2τ_{431}  τ_{431} τ_{432} τ_{432}
τ_{3} τ_{43} τ_{43} + 2τ_{421}  τ_{431}  2τ_{431}  τ_{432}  τ_{432} 0 τ_{4321}
τ_{21} τ_{421} τ_{43} + τ_{421}  τ_{431}  τ_{431}  τ_{432}  0 τ_{4321} 0
τ_{4} 0  τ_{431}  0  τ_{432}  0  τ_{4321} 0  0
τ_{31} τ_{431}  2τ_{431}  τ_{432}  τ_{432}  τ_{4321}  0  0  0

Table 1. Portion of multiplication table for H^{∗}(OG(5, 10))

We list a few more such numbers:

(7)  \begin{aligned}I_{1}([X_{2}], [X_{3}], [X_{421}], [X_{421}]) & = 2, & I_{1}([X_{2}], [X_{21}], [X_{421}], [X_{421}]) & = 2, \\ I_{1}([X_{2}], [X_{42}], [X_{4321}]) & = 1, & I_{1}([X_{2}], [X_{321}], [X_{4321}]) & = 0, \\ I_{1}([X_{2}], [X_{421}], [X_{432}]) & = 1, & I_{1}([X_{3}], [X_{421}], [X_{431}]) & = 1, \\ I_{1}([X_{21}], [X_{421}], [X_{431}]) & = 1, & I_{1}([X_{4}], [X_{421}], [X_{421}]) & = 0, \\ I_{1}([X_{31}], [X_{421}], [X_{421}]) & = 1, & I_{1}([X_{43}], [X_{4321}]) & = 1, \\ I_{1}([X_{421}], [X_{4321}]) & = 0, & I_{1}([X_{431}], [X_{432}]) & = 1. \end{aligned}

Conic numbers

The associativity relations of quantum cohomology (also known as WDVV equations) are a system of polynomial relations in Gromov-Witten invariants, which can be used to deduce new invariants from known ones. We recall the statement, as formulated in [KM94, Eqn. (3.3)], for the case of OG. First, the Poincaré duality involution λ ↦ λ^{∨} on the set of partitions indexing the Schubert classes of OG (basis of the classical cohomology ring of OG), is such that the set of parts of λ^{∨} is the complement in {1, …, n − 1} of the set of parts of λ. We have focused on Gromov-Witten invariants involving Schubert classes of codimension ≥ 2 above, because the ones with fundamental or divisor classes reduce to these by the following identities:

I_{0}([X_{λ}], [X_{μ}], [X_{ν}]) = ∫_{OG}[X_{λ}] · [X_{μ}] · [X_{ν}],
I_{0}([X_{λ^{1}}], …, [X_{λ^{m}}]) = 0 for m ≠ 3,

and for d ≥ 1,

I_{d}([X_{λ^{1}}], …, [X_{λ^{m}}], [X_{1}]) = dI_{d}([X_{λ^{1}}], …, [X_{λ^{m}}]).

Since it is needed for the discussion that follows, we record in Table 1 a portion of the multiplication table for the Schubert classes τ_{λ} = [X_{λ}] in the classical cohomology ring. (One can produce this, e.g., using the Pieri formula of [HB86].)

Given d ≥ 1, m ≥ 4 and λ^{1}, …, λ^{m} satisfying

|λ^{i}| ≥ 1, ∑ |λ^{i}| = n(n − 1)/2 − 4 + 2d(n − 1) + m,

the corresponding associativity relation reads

\begin{matrix}∑_{d′,μ,A} I_{d′}(τ_{λ^{i_{1}}} , …, τ_{λ^{ia}} , τ_{λ^{m−3}}, τ_{λ^{m−2}}, τ_{μ})I_{d−d′}(τ_{λ^{j_{1}}} , …, τ_{λ^{j_{b}}} , τ_{λ^{m−1}}, τ_{λ^{m}}, τ_{μ^{∨}}) \\ = ∑_{d′,μ,A} I_{d′}(τ_{λ^{i_{1}}} , …, τ_{λ^{ia}} , τ_{λ^{m−3}}, τ_{λ^{m}}, τ_{μ})I_{d−d′}(τ_{λ^{j_{1}}} , …, τ_{λ^{j_{b}}} , τ_{λ^{m−2}}, τ_{λ^{m−1}}, τ_{μ^{∨}}), \end{matrix} (8)

where the first, respectively second sum is over integers 0 ≤ d′ ≤ d, strict partitions μ with parts less than n, and subsets A ⊂ {1, …, m − 4} such that

(9)  ∑_{i∈A∪{m−3,m−2}} |λ^{i}| + |μ| = n(n − 1)/2 + 2d′(n − 1) + a,

respectively the same condition with m − 2 replaced by m. In the equations (8)–(9) a, respectively b denotes the cardinality of A, respectively B := {1, …, m − 4} ∖ A, and we write A = {i_{1}, …, i_{a}} and B = {j_{1}, …, j_{b}}.

In case d = 2 in (8) we observe the following: (i) all terms with d′ = 1, and hence d − d′ = 1, are known by the previous section; (ii) terms with d′ = 0 contribute

(10)  ∑_{μ} ( ∫_{OG} τ_{λ^{m−3}}τ_{λ^{m−2}}τ_{μ^{∨}})I_{2}(τ_{λ^{1}}, …, τ_{λ^{m−4}}, τ_{λ^{m−1}}, τ_{λ^{m}}, τ_{μ})

to the left-hand side and

(11)  ∑_{μ} ( ∫_{OG} τ_{λ^{m−3}}τ_{λ^{m}}τ_{μ^{∨}})I_{2}(τ_{λ^{1}}, …, τ_{λ^{m−4}}, τ_{λ^{m−2}}, τ_{λ^{m−1}}, τ_{μ})

to the right-hand side; (iii) terms with d′ = 2 contribute

(12)  ∑_{μ} ( ∫_{OG} τ_{λ^{m−1}}τ_{λ^{m}}τ_{μ^{∨}})I_{2}(τ_{λ^{1}}, …, τ_{λ^{m−4}}, τ_{λ^{m−3}}, τ_{λ^{m−2}}, τ_{μ})

to the left-hand side and

(13)  ∑_{μ} ( ∫_{OG} τ_{λ^{m−2}}τ_{λ^{m−1}}τ_{μ^{∨}})I_{2}(τ_{λ^{1}}, …, τ_{λ^{m−4}}, τ_{λ^{m−3}}, τ_{λ^{m}}, τ_{μ})

to the right-hand side.

Now it is clear that the associativity relations determine many of the Gromov-Witten numbers I_{2}(τ_{λ^{1}}, …, τ_{λ^{m}}). We spell out the cases of interest, and for each case we will subsequently take the corresponding Gromov-Witten numbers as known. Notice that we always take d = 2 in the following applications of (8).

Case 1. Two point conditions: I_{2}(…, τ_{4321}, τ_{4321}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 432, |λ^{m−1}| ≥ 2, λ^{m} = 4321. Then (11)–(13) vanish, while (10) contributes I_{2}(τ_{λ^{1}}, …, τ_{λ^{m−4}}, τ_{λ^{m−1}}, τ_{4321}, τ_{4321}).

Case 2. Point and line conditions: I_{2}(…, τ_{432}, τ_{4321}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 431, |λ^{m−1}| ≥ 2, λ^{m} = 4321. Then (11)–(12) vanish, (13) either vanishes or is known by Case 1, and (10) contributes I_{2}(τ_{λ^{1}}, …, τ_{λ^{m−4}}, τ_{λ^{m−1}}, τ_{432}, τ_{4321}).

Case 3. Point and plane: I_{2}(…, τ_{431}, τ_{4321}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 421, |λ^{m−1}| ≥ 2, λ^{m} = 4321. Then (11)–(12) vanish, (13) either vanishes or is known by previous cases, and (10) contributes I_{2}(τ_{λ^{1}}, …, τ_{λ^{m−4}}, τ_{λ^{m−1}}, τ_{431}, τ_{4321}).

Case 4. Point and X_{421}: I_{2}(…, τ_{421}, τ_{4321}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 321, |λ^{m−1}| ≥ 2, λ^{m} = 4321, and proceed as in Case 3.

Case 5. Point and X_{43}: I_{2}(…, τ_{43}, τ_{4321}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 42, |λ^{m−1}| ≥ 2, λ^{m} = 4321. Then (11)–(12) vanish, (13) either vanishes or is known by previous cases, and (10) contributes I_{2}(τ_{λ^{1}}, …, τ_{λ^{m−4}}, τ_{λ^{m−1}}, τ_{43}, τ_{4321}) + I_{2}(τ_{λ^{1}}, …, τ_{λ^{m−4}}, τ_{λ^{m−1}}, τ_{421}, τ_{4321}).

Case 6. Point and X_{42}: I_{2}(…, τ_{42}, τ_{4321}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 41, |λ^{m−1}| ≥ 2, λ^{m} = 4321, and proceed as in Case 3.

Case 7. Point and X_{321}: I_{2}(…, τ_{321}, τ_{4321}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 32, |λ^{m−1}| ≥ 2, λ^{m} = 4321, and proceed as in Case 5.

Case 8. Point and X_{41}: I_{2}(…, τ_{41}, τ_{4321}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 4, |λ^{m−1}| ≥ 2, λ^{m} = 4321, and proceed as in Case 3.

Case 9. Point and X_{32}: I_{2}(…, τ_{32}, τ_{4321}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 31, |λ^{m−1}| ≥ 2, λ^{m} = 4321, and proceed as in Case 5.

Case 10. Two line conditions: I_{2}(…, τ_{432}, τ_{432}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 431, |λ^{m−1}| ≥ 2, λ^{m} = 432. Then (12) vanishes, (13) vanishes or is known by Case 2, (11) contributes I_{2}(τ_{λ^{1}}, …, τ_{λ^{m−4}}, τ_{λ^{m−1}}, τ_{431}, τ_{4321}), and (10) contributes I_{2}(τ_{λ^{1}}, …, τ_{λ^{m−4}}, τ_{λ^{m−1}}, τ_{432}, τ_{432}).

Case 11. Line and plane: I_{2}(…, τ_{431}, τ_{432}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 421, |λ^{m−1}| ≥ 2, λ^{m} = 432, and proceed as in Case 10.

Case 12. Line and X_{421}: I_{2}(…, τ_{421}, τ_{432}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 321, |λ^{m−1}| ≥ 2, λ^{m} = 432, and proceed as in Case 10.

Case 13. Line and X_{43}: I_{2}(…, τ_{43}, τ_{432}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 42, |λ^{m−1}| ≥ 2, λ^{m} = 432. Then (12) vanishes, (13) vanishes or is known by previous cases, (11) contributes I_{2}(τ_{λ^{1}}, …, τ_{λ^{m−4}}, τ_{λ^{m−1}}, τ_{42}, τ_{4321}), and (10) contributes I_{2}(τ_{λ^{1}}, …, τ_{λ^{m−4}}, τ_{λ^{m−1}}, τ_{43}, τ_{432}) + I_{2}(τ_{λ^{1}}, …, τ_{λ^{m−4}}, τ_{λ^{m−1}}, τ_{421}, τ_{432}).

Case 14. Line and X_{42}: I_{2}(…, τ_{42}, τ_{432}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 41, |λ^{m−1}| ≥ 2, λ^{m} = 432, and proceed as in Case 10.

Case 15. Line and X_{321}: I_{2}(…, τ_{321}, τ_{432}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 32, |λ^{m−1}| ≥ 2, λ^{m} = 432, and proceed as in Case 13.

Case 16. Two plane conditions: I_{2}(…, τ_{431}, τ_{431}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 421, |λ^{m−1}| ≥ 2, λ^{m} = 431. Then (12) vanishes or is known by Case 4,

(11) is known by Case 12, (13) vanishes or is known by previous cases, and (10) contributes I_{2}(τ_{λ^{1}}, …, τ_{λ^{m−4}}, τ_{λ^{m−1}}, τ_{431}, τ_{431}).

Case 17. Plane and X_{421}: I_{2}(…, τ_{421}, τ_{431}). We apply (8) with λ^{m−3} = 1, λ^{m−2} = 321, |λ^{m−1}| ≥ 2, λ^{m} = 431. Then (11) is known by Case 15, (12) and (13) vanish or are known by previous cases, and (10) contributes I_{2}(τ_{λ^{1}}, …, τ_{λ^{m−4}}, τ_{λ^{m−1}}, τ_{421}, τ_{431}).

We list a few of the conic numbers:

(14)  \begin{aligned}I_{2}(τ_{2}, τ_{421}, τ_{431}, τ_{4321}) & = 3, & I_{2}(τ_{3}, τ_{421}, τ_{431}, τ_{432}) & = 5, \\ I_{2}(τ_{21}, τ_{421}, τ_{431}, τ_{432}) & = 4, & I_{2}(τ_{421}, τ_{432}, τ_{4321}) & = 1, \\ I_{2}(τ_{431}, τ_{431}, τ_{4321}) & = 1, & I_{2}(τ_{431}, τ_{432}, τ_{432}) & = 2, \end{aligned}

In total, Cases 1 through 17 determine 1459 conic numbers.

I_{3}(τ_{2}, τ_{2}, τ_{2}, τ_{2}, τ_{4321}, τ_{4321}, τ_{4321}) = 81,  I_{3}(τ_{2}, τ_{2}, τ_{2}, τ_{3}, τ_{432}, τ_{4321}, τ_{4321}) = 216,
I_{3}(τ_{2}, τ_{2}, τ_{2}, τ_{21}, τ_{432}, τ_{4321}, τ_{4321}) = 135,  I_{3}(τ_{2}, τ_{2}, τ_{3}, τ_{4321}, τ_{4321}, τ_{4321}) = 18,
I_{3}(τ_{2}, τ_{2}, τ_{21}, τ_{4321}, τ_{4321}, τ_{4321}) = 9,  I_{3}(τ_{2}, τ_{2}, τ_{4}, τ_{432}, τ_{4321}, τ_{4321}) = 24,
I_{3}(τ_{2}, τ_{2}, τ_{31}, τ_{432}, τ_{4321}, τ_{4321}) = 42,  I_{3}(τ_{2}, τ_{3}, τ_{3}, τ_{432}, τ_{4321}, τ_{4321}) = 40,
I_{3}(τ_{2}, τ_{3}, τ_{21}, τ_{432}, τ_{4321}, τ_{4321}) = 26,  I_{3}(τ_{2}, τ_{21}, τ_{21}, τ_{432}, τ_{4321}, τ_{4321}) = 16,
I_{3}(τ_{2}, τ_{4}, τ_{4321}, τ_{4321}, τ_{4321}) = 3,  I_{3}(τ_{2}, τ_{31}, τ_{4321}, τ_{4321}, τ_{4321}) = 3,
I_{3}(τ_{2}, τ_{41}, τ_{432}, τ_{4321}, τ_{4321}) = 7,  I_{3}(τ_{2}, τ_{32}, τ_{432}, τ_{4321}, τ_{4321}) = 6,
I_{3}(τ_{3}, τ_{3}, τ_{3}, τ_{431}, τ_{4321}, τ_{4321}) = 52,  I_{3}(τ_{3}, τ_{3}, τ_{21}, τ_{431}, τ_{4321}, τ_{4321}) = 36,
I_{3}(τ_{3}, τ_{3}, τ_{4321}, τ_{4321}, τ_{4321}) = 4,  I_{3}(τ_{3}, τ_{21}, τ_{21}, τ_{431}, τ_{4321}, τ_{4321}) = 25,
I_{3}(τ_{3}, τ_{21}, τ_{4321}, τ_{4321}, τ_{4321}) = 2,  I_{3}(τ_{3}, τ_{4}, τ_{432}, τ_{4321}, τ_{4321}) = 4,
I_{3}(τ_{3}, τ_{31}, τ_{432}, τ_{4321}, τ_{4321}) = 8,  I_{3}(τ_{3}, τ_{41}, τ_{431}, τ_{4321}, τ_{4321}) = 7,
I_{3}(τ_{3}, τ_{32}, τ_{431}, τ_{4321}, τ_{4321}) = 10,  I_{3}(τ_{21}, τ_{21}, τ_{21}, τ_{431}, τ_{4321}, τ_{4321}) = 17,
I_{3}(τ_{21}, τ_{21}, τ_{4321}, τ_{4321}, τ_{4321}) = 1,  I_{3}(τ_{21}, τ_{4}, τ_{432}, τ_{4321}, τ_{4321}) = 3,
I_{3}(τ_{21}, τ_{31}, τ_{432}, τ_{4321}, τ_{4321}) = 5,  I_{3}(τ_{21}, τ_{41}, τ_{431}, τ_{4321}, τ_{4321}) = 5,
I_{3}(τ_{21}, τ_{32}, τ_{431}, τ_{4321}, τ_{4321}) = 7,  I_{3}(τ_{41}, τ_{4321}, τ_{4321}, τ_{4321}) = 1,
I_{3}(τ_{32}, τ_{4321}, τ_{4321}, τ_{4321}) = 0,  I_{3}(τ_{42}, τ_{432}, τ_{4321}, τ_{4321}) = 2,
I_{3}(τ_{321}, τ_{432}, τ_{4321}, τ_{4321}) = 0,  I_{3}(τ_{43}, τ_{431}, τ_{4321}, τ_{4321}) = 1,
I_{3}(τ_{421}, τ_{431}, τ_{4321}, τ_{4321}) = 2.

Table 2. Degree 3, Case 1 numbers

Example. The number I_{2}(τ_{2}, τ_{421}, τ_{431}, τ_{4321}) falls under Case 3. We have m = 5, λ^{2} = 1, λ^{3} = 421, λ^{5} = 4321, and either λ^{4} = 2, hence λ^{1} = 421 with (8) giving

I_{2}(τ_{2},τ_{421}, τ_{431}, τ_{4321}) + I_{1}(τ_{1}, τ_{4}, τ_{421}, τ_{421})I_{1}(τ_{2}, τ_{321}, τ_{4321})
+ I_{1}(τ_{1}, τ_{31}, τ_{421}, τ_{421})I_{1}(τ_{2}, τ_{42}, τ_{4321})
= I_{2}(τ_{1}, τ_{421}, τ_{432}, τ_{4321}) + I_{1}(τ_{1}, τ_{43}, τ_{4321})I_{1}(τ_{2}, τ_{21}, τ_{421}, τ_{421})

+ I_{1}(τ_{1}, τ_{421}, τ_{4321})I_{1}(τ_{2}, τ_{3}, τ_{421}, τ_{421}) + I_{1}(τ_{1}, τ_{1}, τ_{421}, τ_{4321})I_{1}(τ_{2}, τ_{421}, τ_{432}),

or λ^{4} = 421, hence λ^{1} = 2 and (8) giving

I_{2}(τ_{2},τ_{421}, τ_{431}, τ_{4321}) + I_{1}(τ_{1}, τ_{2}, τ_{421}, τ_{432})I_{1}(τ_{1}, τ_{421}, τ_{4321})

= I_{1}(τ_{1}, τ_{43}, τ_{4321})I_{1}(τ_{2}, τ_{21}, τ_{421}, τ_{421}) + I_{1}(τ_{1}, τ_{421}, τ_{4321})I_{1}(τ_{2}, τ_{3}, τ_{421}, τ_{421}) + I_{1}(τ_{1}, τ_{2}, τ_{42}, τ_{4321})I_{1}(τ_{31}, τ_{421}, τ_{421}) + I_{1}(τ_{1}, τ_{2}, τ_{321}, τ_{4321})I_{1}(τ_{4}, τ_{421}, τ_{421}).

Either way, we obtain I_{2}(τ_{2}, τ_{421}, τ_{431}, τ_{4321}) = 3. One way requires the Case 2 number I_{2}(τ_{421}, τ_{432}, τ_{4321}). The needed line numbers appear in (7).

Higher degree numbers

The associativity relations also determine many higher-degree Gromov-Witten numbers. For instance, taking d = 3 we may apply (8) with λ^{m−3} = 1, λ^{m−2} = 432, |λ^{m−1}| ≥ 2, λ^{m} = 4321 just as in Case 1 above, and obtain many Gromov-Witten numbers I_{3}(…, τ_{4321}, τ_{4321}). However, since we obtained only some of the degree 2 Gromov-Witten numbers in the previous section, we need to check that the contributions with d′ = 2 or d − d′ = 2 involve only degree 2 Gromov-Witten

EMBEDDING POINTED CURVES IN K3 SURFACES 25

I_{3}(τ_{2}, τ_{2}, τ_{3}, τ_{3}, τ_{432}, τ_{432}, τ_{4321}) = 548,  I_{3}(τ_{2}, τ_{2}, τ_{3}, τ_{21}, τ_{432}, τ_{432}, τ_{4321}) = 379,
I_{3}(τ_{2}, τ_{2}, τ_{21}, τ_{21}, τ_{432}, τ_{432}, τ_{4321}) = 260,  I_{3}(τ_{2}, τ_{2}, τ_{41}, τ_{432}, τ_{432}, τ_{4321}) = 80,
I_{3}(τ_{2}, τ_{2}, τ_{32}, τ_{432}, τ_{432}, τ_{4321}) = 105,  I_{3}(τ_{2}, τ_{3}, τ_{3}, τ_{3}, τ_{431}, τ_{432}, τ_{4321}) = 753,
I_{3}(τ_{2}, τ_{3}, τ_{3}, τ_{21}, τ_{431}, τ_{432}, τ_{4321}) = 531, I_{3}(τ_{2}, τ_{3}, τ_{21}, τ_{21}, τ_{431}, τ_{432}, τ_{4321}) = 377,
I_{3}(τ_{2}, τ_{3}, τ_{4}, τ_{432}, τ_{432}, τ_{4321}) = 47,  I_{3}(τ_{2}, τ_{3}, τ_{31}, τ_{432}, τ_{432}, τ_{4321}) = 109,
I_{3}(τ_{2}, τ_{3}, τ_{41}, τ_{431}, τ_{432}, τ_{4321}) = 96,  I_{3}(τ_{2}, τ_{3}, τ_{32}, τ_{431}, τ_{432}, τ_{4321}) = 139,
I_{3}(τ_{2}, τ_{21}, τ_{21}, τ_{21}, τ_{431}, τ_{432}, τ_{4321}) = 270,  I_{3}(τ_{2}, τ_{21}, τ_{4}, τ_{432}, τ_{432}, τ_{4321}) = 33,
I_{3}(τ_{2}, τ_{21}, τ_{31}, τ_{432}, τ_{432}, τ_{4321}) = 76,  I_{3}(τ_{2}, τ_{21}, τ_{41}, τ_{431}, τ_{432}, τ_{4321}) = 66,
I_{3}(τ_{2}, τ_{21}, τ_{32}, τ_{431}, τ_{432}, τ_{4321}) = 103,  I_{3}(τ_{2}, τ_{42}, τ_{432}, τ_{432}, τ_{4321}) = 22,
I_{3}(τ_{2}, τ_{321}, τ_{432}, τ_{432}, τ_{4321}) = 9,  I_{3}(τ_{2}, τ_{43}, τ_{431}, τ_{432}, τ_{4321}) = 19,
I_{3}(τ_{2}, τ_{421}, τ_{431}, τ_{432}, τ_{4321}) = 21,  I_{3}(τ_{3}, τ_{3}, τ_{3}, τ_{432}, τ_{432}, τ_{4321}) = 92,
I_{3}(τ_{3}, τ_{3}, τ_{21}, τ_{432}, τ_{432}, τ_{4321}) = 64,  I_{3}(τ_{3}, τ_{3}, τ_{4}, τ_{431}, τ_{432}, τ_{4321}) = 59,
I_{3}(τ_{3}, τ_{3}, τ_{31}, τ_{431}, τ_{432}, τ_{4321}) = 142,  I_{3}(τ_{3}, τ_{21}, τ_{21}, τ_{432}, τ_{432}, τ_{4321}) = 45,
I_{3}(τ_{3}, τ_{21}, τ_{4}, τ_{431}, τ_{432}, τ_{4321}) = 41,  I_{3}(τ_{3}, τ_{21}, τ_{31}, τ_{431}, τ_{432}, τ_{4321}) = 101,
I_{3}(τ_{3}, τ_{41}, τ_{432}, τ_{432}, τ_{4321}) = 13,  I_{3}(τ_{3}, τ_{32}, τ_{432}, τ_{432}, τ_{4321}) = 18,
I_{3}(τ_{3}, τ_{42}, τ_{431}, τ_{432}, τ_{4321}) = 25,  I_{3}(τ_{3}, τ_{321}, τ_{431}, τ_{432}, τ_{4321}) = 11,
I_{3}(τ_{21}, τ_{21}, τ_{21}, τ_{432}, τ_{432}, τ_{4321}) = 31,  I_{3}(τ_{21}, τ_{21}, τ_{4}, τ_{431}, τ_{432}, τ_{4321}) = 28,
I_{3}(τ_{21}, τ_{21}, τ_{31}, τ_{431}, τ_{432}, τ_{4321}) = 73,  I_{3}(τ_{21}, τ_{41}, τ_{432}, τ_{432}, τ_{4321}) = 9,
I_{3}(τ_{21}, τ_{32}, τ_{432}, τ_{432}, τ_{4321}) = 13,  I_{3}(τ_{21}, τ_{42}, τ_{431}, τ_{432}, τ_{4321}) = 17,
I_{3}(τ_{21}, τ_{321}, τ_{431}, τ_{432}, τ_{4321}) = 10,  I_{3}(τ_{4}, τ_{4}, τ_{432}, τ_{432}, τ_{4321}) = 4,
I_{3}(τ_{4}, τ_{31}, τ_{432}, τ_{432}, τ_{4321}) = 9,  I_{3}(τ_{4}, τ_{41}, τ_{431}, τ_{432}, τ_{4321}) = 8,
I_{3}(τ_{4}, τ_{32}, τ_{431}, τ_{432}, τ_{4321}) = 9,  I_{3}(τ_{31}, τ_{31}, τ_{432}, τ_{432}, τ_{4321}) = 22,
I_{3}(τ_{31}, τ_{41}, τ_{431}, τ_{432}, τ_{4321}) = 17,  I_{3}(τ_{31}, τ_{32}, τ_{431}, τ_{432}, τ_{4321}) = 27,
I_{3}(τ_{43}, τ_{432}, τ_{432}, τ_{4321}) = 3,  I_{3}(τ_{421}, τ_{432}, τ_{432}, τ_{4321}) = 3,
I_{3}(τ_{431}, τ_{431}, τ_{432}, τ_{4321}) = 5.

Table 3. Degree 3, Case 2 numbers

I_{3}(τ_{3}, τ_{3}, τ_{3}, τ_{3}, τ_{431}, τ_{431}, τ_{4321}) = 1062, I_{3}(τ_{3}, τ_{3}, τ_{3}, τ_{21}, τ_{431}, τ_{431}, τ_{4321}) = 750,
I_{3}(τ_{3}, τ_{3}, τ_{21}, τ_{21}, τ_{431}, τ_{431}, τ_{4321}) = 534,  I_{3}(τ_{3}, τ_{3}, τ_{41}, τ_{431}, τ_{431}, τ_{4321}) = 120,
I_{3}(τ_{3}, τ_{3}, τ_{32}, τ_{431}, τ_{431}, τ_{4321}) = 174, I_{3}(τ_{3}, τ_{21}, τ_{21}, τ_{21}, τ_{431}, τ_{431}, τ_{4321}) = 385,
I_{3}(τ_{3}, τ_{21}, τ_{41}, τ_{431}, τ_{431}, τ_{4321}) = 83,  I_{3}(τ_{3}, τ_{21}, τ_{32}, τ_{431}, τ_{431}, τ_{4321}) = 128,
I_{3}(τ_{3}, τ_{43}, τ_{431}, τ_{431}, τ_{4321}) = 20,  I_{3}(τ_{3}, τ_{421}, τ_{431}, τ_{431}, τ_{4321}) = 21,
I_{3}(τ_{21}, τ_{21}, τ_{21}, τ_{21}, τ_{431}, τ_{431}, τ_{4321}) = 282,  I_{3}(τ_{21}, τ_{21}, τ_{41}, τ_{431}, τ_{431}, τ_{4321}) = 58,
I_{3}(τ_{21}, τ_{21}, τ_{32}, τ_{431}, τ_{431}, τ_{4321}) = 95,  I_{3}(τ_{21}, τ_{43}, τ_{431}, τ_{431}, τ_{4321}) = 13,
I_{3}(τ_{21}, τ_{421}, τ_{431}, τ_{431}, τ_{4321}) = 16,  I_{3}(τ_{41}, τ_{41}, τ_{431}, τ_{431}, τ_{4321}) = 12,
I_{3}(τ_{41}, τ_{32}, τ_{431}, τ_{431}, τ_{4321}) = 17,  I_{3}(τ_{32}, τ_{32}, τ_{431}, τ_{431}, τ_{4321}) = 28.

Table 4. Degree 3, Case 3 numbers

numbers that have been determined. This is checked on a case-by-case basis for each of the 35 numbers listed in Table 2 and each corresponding application of (8).

I_{3}(τ_{2}, τ_{3}, τ_{3}, τ_{3}, τ_{432}, τ_{432}, τ_{432}) = 1416,  I_{3}(τ_{2}, τ_{3}, τ_{3}, τ_{21}, τ_{432}, τ_{432}, τ_{432}) = 996,
I_{3}(τ_{2}, τ_{3}, τ_{21}, τ_{21}, τ_{432}, τ_{432}, τ_{432}) = 708,  I_{3}(τ_{2}, τ_{3}, τ_{41}, τ_{432}, τ_{432}, τ_{432}) = 189,
I_{3}(τ_{2}, τ_{3}, τ_{32}, τ_{432}, τ_{432}, τ_{432}) = 270, I_{3}(τ_{2}, τ_{21}, τ_{21}, τ_{21}, τ_{432}, τ_{432}, τ_{432}) = 510,
I_{3}(τ_{2}, τ_{21}, τ_{41}, τ_{432}, τ_{432}, τ_{432}) = 129,  I_{3}(τ_{2}, τ_{21}, τ_{32}, τ_{432}, τ_{432}, τ_{432}) = 201,
I_{3}(τ_{2}, τ_{43}, τ_{432}, τ_{432}, τ_{432}) = 42,  I_{3}(τ_{2}, τ_{421}, τ_{432}, τ_{432}, τ_{432}) = 42,
I_{3}(τ_{3}, τ_{3}, τ_{3}, τ_{3}, τ_{431}, τ_{432}, τ_{432}) = 1940,  I_{3}(τ_{3}, τ_{3}, τ_{3}, τ_{21}, τ_{431}, τ_{432}, τ_{432}) = 1362,
I_{3}(τ_{3}, τ_{3}, τ_{21}, τ_{21}, τ_{431}, τ_{432}, τ_{432}) = 966,  I_{3}(τ_{3}, τ_{3}, τ_{4}, τ_{432}, τ_{432}, τ_{432}) = 112,
I_{3}(τ_{3}, τ_{3}, τ_{31}, τ_{432}, τ_{432}, τ_{432}) = 268,  I_{3}(τ_{3}, τ_{3}, τ_{41}, τ_{431}, τ_{432}, τ_{432}) = 228,
I_{3}(τ_{3}, τ_{3}, τ_{32}, τ_{431}, τ_{432}, τ_{432}) = 320, I_{3}(τ_{3}, τ_{21}, τ_{21}, τ_{21}, τ_{431}, τ_{432}, τ_{432}) = 696,
I_{3}(τ_{3}, τ_{21}, τ_{4}, τ_{432}, τ_{432}, τ_{432}) = 77,  I_{3}(τ_{3}, τ_{21}, τ_{31}, τ_{432}, τ_{432}, τ_{432}) = 191,
I_{3}(τ_{3}, τ_{21}, τ_{41}, τ_{431}, τ_{432}, τ_{432}) = 156,  I_{3}(τ_{3}, τ_{21}, τ_{32}, τ_{431}, τ_{432}, τ_{432}) = 236,
I_{3}(τ_{3}, τ_{42}, τ_{432}, τ_{432}, τ_{432}) = 50,  I_{3}(τ_{3}, τ_{321}, τ_{432}, τ_{432}, τ_{432}) = 22,
I_{3}(τ_{3}, τ_{43}, τ_{431}, τ_{432}, τ_{432}) = 42,  I_{3}(τ_{3}, τ_{421}, τ_{431}, τ_{432}, τ_{432}) = 39,
I_{3}(τ_{21}, τ_{21}, τ_{21}, τ_{21}, τ_{431}, τ_{432}, τ_{432}) = 512,  I_{3}(τ_{21}, τ_{21}, τ_{4}, τ_{432}, τ_{432}, τ_{432}) = 52,
I_{3}(τ_{21}, τ_{21}, τ_{31}, τ_{432}, τ_{432}, τ_{432}) = 139,  I_{3}(τ_{21}, τ_{21}, τ_{41}, τ_{431}, τ_{432}, τ_{432}) = 108,
I_{3}(τ_{21}, τ_{21}, τ_{32}, τ_{431}, τ_{432}, τ_{432}) = 176,  I_{3}(τ_{21}, τ_{42}, τ_{432}, τ_{432}, τ_{432}) = 34,
I_{3}(τ_{21}, τ_{321}, τ_{432}, τ_{432}, τ_{432}) = 20,  I_{3}(τ_{21}, τ_{43}, τ_{431}, τ_{432}, τ_{432}) = 27,
I_{3}(τ_{21}, τ_{421}, τ_{431}, τ_{432}, τ_{432}) = 30,  I_{3}(τ_{4}, τ_{41}, τ_{432}, τ_{432}, τ_{432}) = 16,
I_{3}(τ_{4}, τ_{32}, τ_{432}, τ_{432}, τ_{432}) = 18,  I_{3}(τ_{31}, τ_{41}, τ_{432}, τ_{432}, τ_{432}) = 34,
I_{3}(τ_{31}, τ_{32}, τ_{432}, τ_{432}, τ_{432}) = 54,  I_{3}(τ_{41}, τ_{41}, τ_{431}, τ_{432}, τ_{432}) = 24,
I_{3}(τ_{41}, τ_{32}, τ_{431}, τ_{432}, τ_{432}) = 33,  I_{3}(τ_{32}, τ_{32}, τ_{431}, τ_{432}, τ_{432}) = 54,
I_{3}(τ_{431}, τ_{432}, τ_{432}, τ_{432}) = 11.

Table 5. Degree 3, Case 10 numbers

I_{4}(τ_{2}, τ_{2}, τ_{2}, τ_{4321}, τ_{4321}, τ_{4321}, τ_{4321}) = 64, I_{4}(τ_{2}, τ_{2}, τ_{3}, τ_{432}, τ_{4321}, τ_{4321}, τ_{4321}) = 208,
I_{4}(τ_{2}, τ_{2}, τ_{21}, τ_{432}, τ_{4321}, τ_{4321}, τ_{4321}) = 160, I_{4}(τ_{2}, τ_{3}, τ_{3}, τ_{432}, τ_{432}, τ_{4321}, τ_{4321}) = 576,
I_{4}(τ_{2}, τ_{3}, τ_{21}, τ_{432}, τ_{432}, τ_{4321}, τ_{4321}) = 420,  I_{4}(τ_{2}, τ_{3}, τ_{4321}, τ_{4321}, τ_{4321}, τ_{4321}) = 8,
I_{4}(τ_{2}, τ_{21}, τ_{21}, τ_{432}, τ_{432}, τ_{4321}, τ_{4321}) = 304, I_{4}(τ_{2}, τ_{21}, τ_{4321}, τ_{4321}, τ_{4321}, τ_{4321}) = 8,
I_{4}(τ_{2}, τ_{4}, τ_{432}, τ_{4321}, τ_{4321}, τ_{4321}) = 12,  I_{4}(τ_{2}, τ_{31}, τ_{432}, τ_{4321}, τ_{4321}, τ_{4321}) = 38,
I_{4}(τ_{2}, τ_{41}, τ_{432}, τ_{432}, τ_{4321}, τ_{4321}) = 62,  I_{4}(τ_{2}, τ_{32}, τ_{432}, τ_{432}, τ_{4321}, τ_{4321}) = 100,
I_{4}(τ_{3}, τ_{3}, τ_{432}, τ_{4321}, τ_{4321}, τ_{4321}) = 28,  I_{4}(τ_{3}, τ_{21}, τ_{432}, τ_{4321}, τ_{4321}, τ_{4321}) = 22,
I_{4}(τ_{3}, τ_{4}, τ_{432}, τ_{432}, τ_{4321}, τ_{4321}) = 36,  I_{4}(τ_{3}, τ_{31}, τ_{432}, τ_{432}, τ_{4321}, τ_{4321}) = 94,
I_{4}(τ_{21}, τ_{21}, τ_{432}, τ_{4321}, τ_{4321}, τ_{4321}) = 16,  I_{4}(τ_{21}, τ_{4}, τ_{432}, τ_{432}, τ_{4321}, τ_{4321}) = 26,
I_{4}(τ_{21}, τ_{31}, τ_{432}, τ_{432}, τ_{4321}, τ_{4321}) = 68,  I_{4}(τ_{4}, τ_{4321}, τ_{4321}, τ_{4321}, τ_{4321}) = 0,
I_{4}(τ_{31}, τ_{4321}, τ_{4321}, τ_{4321}, τ_{4321}) = 2,  I_{4}(τ_{41}, τ_{432}, τ_{4321}, τ_{4321}, τ_{4321}) = 3,
I_{4}(τ_{32}, τ_{432}, τ_{4321}, τ_{4321}, τ_{4321}) = 6,  I_{4}(τ_{42}, τ_{432}, τ_{432}, τ_{4321}, τ_{4321}) = 14,
I_{4}(τ_{321}, τ_{432}, τ_{432}, τ_{4321}, τ_{4321}) = 8.

Table 6. Degree 4, Case 1 numbers

Example. To determine I_{3}(τ_{421}, τ_{431}, τ_{4321}, τ_{4321}) we read off from (8) with d = 3, m = 5, and (λ^{1}, …, λ^{5}) = (431, 1, 432, 421, 4321), the identity (cf. (7), (14)):

I_{3}(τ_{421}, τ_{431}, τ_{4321}, τ_{4321})

= I_{1}(τ_{1}, τ_{43}, τ_{4321})I_{2}(τ_{21}, τ_{421}, τ_{431}, τ_{432}) + I_{1}(τ_{1}, τ_{421}, τ_{4321})I_{2}(τ_{3}, τ_{421}, τ_{431}, τ_{432}) + I_{2}(τ_{1}, τ_{431}, τ_{431}, τ_{4321})I_{1}(τ_{2}, τ_{421}, τ_{432}) − I_{1}(τ_{1}, τ_{431}, τ_{432})I_{2}(τ_{2}, τ_{421}, τ_{431}, τ_{4321}) − I_{1}(τ_{1}, τ_{1}, τ_{431}, τ_{432})I_{2}(τ_{421}, τ_{432}, τ_{4321}) − I_{2}(τ_{1}, τ_{431}, τ_{432}, τ_{432})I_{1}(τ_{1}, τ_{421}, τ_{4321}) = 1 · 4 + 0 · 5 + 2 · 1 − 1 · 3 − 1 · 1 − 4 · 0 = 2.

I_{4}(τ_{3}, τ_{3}, τ_{3}, τ_{432}, τ_{432}, τ_{432}, τ_{4321}) = 1488, I_{4}(τ_{3}, τ_{3}, τ_{21}, τ_{432}, τ_{432}, τ_{432}, τ_{4321}) = 1062,
I_{4}(τ_{3}, τ_{21}, τ_{21}, τ_{432}, τ_{432}, τ_{432}, τ_{4321}) = 764,  I_{4}(τ_{3}, τ_{41}, τ_{432}, τ_{432}, τ_{432}, τ_{4321}) = 154,
I_{4}(τ_{3}, τ_{32}, τ_{432}, τ_{432}, τ_{432}, τ_{4321}) = 232, I_{4}(τ_{21}, τ_{21}, τ_{21}, τ_{432}, τ_{432}, τ_{432}, τ_{4321}) = 552,
I_{4}(τ_{21}, τ_{41}, τ_{432}, τ_{432}, τ_{432}, τ_{4321}) = 108,  I_{4}(τ_{21}, τ_{32}, τ_{432}, τ_{432}, τ_{432}, τ_{4321}) = 170,
I_{4}(τ_{43}, τ_{432}, τ_{432}, τ_{432}, τ_{4321}) = 26,  I_{4}(τ_{421}, τ_{432}, τ_{432}, τ_{432}, τ_{4321}) = 29.

Table 7. Degree 4, Case 2 numbers

I_{5}(τ_{2}, τ_{2}, τ_{4321}, …, τ_{4321}) = 125,  I_{5}(τ_{2}, τ_{3}, τ_{432}, τ_{4321}, …, τ_{4321}) = 250,
I_{5}(τ_{2}, τ_{21}, τ_{432}, τ_{4321}, …, τ_{4321}) = 175, I_{5}(τ_{3}, τ_{3}, τ_{432}, τ_{432}, τ_{4321}, …, τ_{4321}) = 566,
I_{5}(τ_{3}, τ_{21}, τ_{432}, τ_{432}, τ_{4321}, …, τ_{4321}) = 403,  I_{5}(τ_{3}, τ_{4321}, …, τ_{4321}) = 15,
I_{5}(τ_{21}, τ_{21}, τ_{432}, τ_{432}, τ_{4321}, …, τ_{4321}) = 288,  I_{5}(τ_{21}, τ_{4321}, …, τ_{4321}) = 10,
I_{5}(τ_{4}, τ_{432}, τ_{4321}, …, τ_{4321}) = 14,  I_{5}(τ_{31}, τ_{432}, τ_{4321}, …, τ_{4321}) = 33,
I_{5}(τ_{41}, τ_{432}, τ_{432}, τ_{4321}, …, τ_{4321}) = 50,  I_{5}(τ_{32}, τ_{432}, τ_{432}, τ_{4321}, …, τ_{4321}) = 75,
I_{6}(τ_{2}, τ_{4321}, …, τ_{4321}) = 60,  I_{6}(τ_{3}, τ_{432}, τ_{4321}, …, τ_{4321}) = 180,
I_{6}(τ_{21}, τ_{432}, τ_{4321}, …, τ_{4321}) = 130,  I_{7}(τ_{4321}, …, τ_{4321}) = 71.

Table 8. Degree 5, 6, and 7 numbers (all Case 1)

Alternatively (λ^{1}, …, λ^{5}) = (421, 1, 432, 431, 4321) yields I_{3}(τ_{421}, τ_{431}, τ_{4321}, τ_{4321}) = 1 · 4 + 0 · 5 + 0 · 2 + 2 · 1 − 1 · 3 − 1 · 1 = 2.

Reasoning as in Case 2 we obtain the Gromov-Witten numbers I_{3}(…, τ_{432}, τ_{4321}) listed in Table 3. Again it must be checked that each application of (8) requires only known conic numbers.

Similarly we reason as in Case 3 above to obtain the I_{3}(…, τ_{431}, τ_{4321}) listed in Table 4. We conclude our determination of d = 3 numbers with the I_{3}(…, τ_{432}, τ_{432}) listed in Table 5, for which the reasoning is as in Case 10.

An application of (8) with d = 4 requires numbers of degrees 1, 2, and 3. It must be verified on a case-by-case basis that the required conic and cubic numbers are among those already determined. Tables 6 and 7 list the numbers I_{4}(…, τ_{4321}, τ_{4321}), respectively I_{4}(…, τ_{432}, τ_{4321}), which are treated by reasoning as in Case 1 and Case 2, respectively. For d = 5, 6, and 7 we require only numbers with at least two point conditions, hence we use the reasoning of Case 1. Again it must be verified on a case-by-case basis that the required numbers of every smaller degree are among those already determined. The numbers are displayed in Table 8. The final number displayed is the desired

I_{7}(τ_{4321}, τ_{4321}, τ_{4321}, τ_{4321}, τ_{4321}, τ_{4321}, τ_{4321}) = 71,

with the following enumerative interpretation.

Proposition. There are 71 rational curves of degree 7 through 7 general points on OG(5, 10).

Remark. Semi-simplicity allows us to reconstruct the full quantum cohomology even without assuming that the ordinary cohomology is generated by H^{2}, see [BM04] and [Mas11]. (The case where the cohomology is generated by H^{2} is addressed in [KM94].) This property was verified for orthogonal Grassmannians in [CMP10].

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Department of Mathematics, Rice University, MS 136, Houston, TX 77251-1892, USA

E-mail address: hassett@rice.edu

Courant Institute, New York University, New York, NY 10012, USA

E-mail address: tschinkel@cims.nyu.edu

Simons Foundation, 160 Fifth Avenue, New York, NY 10010, USA