Embedding pointed curves in K3 surfaces

Abstract

We analyze morphisms from pointed curves to K3 surfaces with a distinguished rational curve, such that the marked points are taken to the rational curve, perhaps with specified cross ratios. This builds on work of Mukai and others characterizing embeddings of curves into K3 surfaces via non-abelian Brill–Noether theory. Our study leads naturally to enumerative problems, which we solve in several specific cases. These have applications to the existence of sections of del Pezzo fibrations with prescribed invariants.

Appendix: 71 rational septics through 7 general points on \({\mathrm {OG}}(5,10)\)

1.1 Overview and generalities

The orthogonal Grassmannian ${\mathrm {OG}}(n,2n)$ is, by definition, one component of the space of $n$-dimensional subspaces of $V={\mathbb {C}}^{2n}$, isotropic for a given nondegenerate symmetric bilinear form on $V$. By convention we fix the standard bilinear form, for which $\langle v,w\rangle =\sum _{i=1}^{2n} v_i w_{2n+1-i}$, and we take ${\mathrm {OG}}={\mathrm {OG}}(n,2n)$ to be the component containing $\langle e_1,\ldots ,e_n\rangle$. Then ${\mathrm {OG}}$ is a homogeneous projective variety of dimension $n(n-1)/2$, with $\deg c_1({\mathrm {OG}})=2(n-1)$. 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 $63=7\cdot 9$ and we expect a finite number of rational septics to pass through 7 general points.

It is known (cf. [11, 21]) that the number of rational curves on ${\mathrm {OG}}$ satisfying incidence conditions imposed by Schubert varieties of codimension $\ge 2$ in general position is equal to the corresponding Gromov–Witten invariant:

$$\begin{aligned} \#\left\{ \begin{array}{l} \text {degree}\;d\;\text {rational curves through} \\ \text {general translates of}\;X_{\lambda ^1}, \dots , X_{\lambda ^m} \end{array}\right\} = I_d([X_{\lambda ^1}],\ldots ,[X_{\lambda ^m}]) \end{aligned}$$

for $|\lambda ^i|\ge 2, \sum |\lambda ^i|=n(n-1)/2-3+2(n-1)d+m$, where $I_d([X_{\lambda ^1}],\ldots ,[X_{\lambda ^m}])$ denotes the Gromov–Witten invariant

$$\begin{aligned} \int \limits _{\overline{M}_{0,m}({\mathrm {OG}},d)} \mathrm {ev}_1^*[X_{\lambda ^1}] \cdots \mathrm {ev}_m^*[X_{\lambda ^m}]. \end{aligned}$$

The space $\overline{M}_{0,m}({\mathrm {OG}},d)$ is Kontsevich’s moduli space of stable maps of genus zero $m$-marked curves to ${\mathrm {OG}}$ in degree $d$ (see [18]), and it comes with $m$ evaluation maps (at the marked points) $\mathrm {ev}_1$, $\dots$, $\mathrm {ev}_m$ to ${\mathrm {OG}}$. The Schubert varieties in ${\mathrm {OG}}$ are denoted $X_\lambda$, indexed by strict partitions $\lambda$ whose parts are ${<}n$. (A partition is called strict if has no repeated parts.) The codimension of $X_\lambda$ is equal to $|\lambda |$, the sum of the parts of $\lambda$. The article [19], which includes a determination of the $m=3$ invariants, gives a geometric description of ${\mathrm {OG}}$ including the Schubert varieties.

1.2 Line numbers

The space of lines on ${\mathrm {OG}}$ is known; see [20, Example 4.12]. It is itself a projective homogeneous variety, the space ${\mathrm {OG}}(n-2,2n)$ of isotropic $(n-2)$-dimensional subspaces of $V={\mathbb {C}}^{2n}$. Therefore the computation of the $I_1([X_{\lambda ^1}],\ldots ,[X_{\lambda ^m}])$ 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 Bernšteĭn et al. [4] and Demazure [10]. In the setting of the orthogonal flag variety $\mathrm {OF}(2n)$, parametrizing a space in ${\mathrm {OG}}$ together with a complete flag of subspaces, this has been worked out explicitly by Billey and Haiman [5]. It leads to an explicit formula for the Gromov–Witten invariants counting lines on ${\mathrm {OG}}$ satisfying incidence conditions with respect to Schubert varieties in general position. The formula uses the Schur $P$-polynomials $P_\lambda =P_\lambda (X)$ indexed by strict partitions $\lambda$, which form a ${\mathbb {Q}}$-basis for the ring ${\mathbb {Q}}[p_1,p_3,\ldots ]$ generated by the odd power sums $p_k=p_k(X)=x_1^k+x_2^k+\ldots$ (cf. Proposition 3.1 of op. cit.). Following op. cit., to these we associate polynomials in $z_1$, $\dots$, $z_n$, which we will denote by $P_\lambda (z_1,\ldots ,z_n)$, by sending $p_k(X)$ to $-(1/2)(z_1^k+\cdots +z_n^k)$.

Proposition

Introduce the divided difference operators on ${\mathbb {Q}}[z_1,\ldots ,z_n]$:

$$\begin{aligned} \begin{array}{cl} \partial _if&{}=\frac{f(z_1,\ldots ,z_n)-f(z_1,\ldots ,z_{i+1},z_i, \ldots ,z_n)}{z_i-z_{i+1}},\\ \partial _{\hat{1}} f&{}=\frac{f(z_1,\ldots ,z_n)-f(-z_2,-z_1,z_3, \ldots ,z_n)}{-z_1-z_2}, \end{array} \end{aligned}$$

and for $i\le j$ let $\partial _{i\ldots j}$ denote $\partial _i \partial _{i+1}\ldots \partial _j$ and let $\partial _{j\ldots i}$ denote $\partial _j\ldots \partial _i$. Then for any $m$ and $\lambda ^1$, $\dots$, $\lambda ^m$ satisfying $|\lambda ^i|\ge 2$, $\sum |\lambda ^i|=n(n-1)/2-3+2(n-1)+m$, if we set

$$\begin{aligned} F=\prod _{i=1}^m \partial _{\hat{1}} P_{\lambda ^i}(z_1,\ldots ,z_n) \end{aligned}$$

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

$$\begin{aligned} I_1&([X_{\lambda ^1}],\ldots ,[X_{\lambda ^m}])\\ {}&= {\left\{ \begin{array}{ll} \partial _{2\ldots n-1} \partial _{1\ldots n-2} \partial _{\hat{1}} \cdots \partial _{2\ldots 3} \partial _{1\ldots 2} \partial _{\hat{1}} \partial _{n-2\ldots 1} \partial _{n-1\ldots 2}F&{} \text {if n is even},\\ \partial _{2\ldots n-1} \partial _{\hat{1}} \partial _{2\ldots n-2} \partial _{1\ldots n-3} \partial _{\hat{1}} \cdots \partial _{2\ldots 3} \partial _{1\ldots 2} \partial _{\hat{1}} \partial _{n-2\ldots 1} \partial _{n-1\ldots 2}F&{} \text {if n is odd}, \end{array}\right. } \end{aligned}$$

where the $\cdots$ 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 $P_\lambda =P_\lambda (X)$, then $\partial _{\bar{1}}P_\lambda$ represents the cycle class of the space of lines incident to $X_\lambda$, and the displayed composition of divided operators sends the polynomial representing the class of a point on the space of lines on ${\mathrm {OG}}$ to 1. So the proposition follows from the observation that the computation may be performed in the polynomial ring ${\mathbb {Q}}[z_1,\ldots ,z_n]$.

When $n=5$, there are $1071$ Gromov–Witten numbers $I_1([X_{\lambda ^1}],\ldots ,[X_{\lambda ^m}])$, 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 ${\mathrm {OG}}$ meeting a given isotropic 3-dimensional space nontrivially). We have $P_2(X)=p_1^2(X)$ sent to $(1/4)(z_1+\cdots +z_5)^2$, which upon applying $\partial _{\hat{1}}$ yields $-z_3-z_4-z_5$. We evaluate

$$\begin{aligned} \partial _2\partial _3\partial _4\partial _{\hat{1}}\partial _2\partial _3 \partial _1\partial _2\partial _{\hat{1}}\partial _3\partial _2\partial _1 \partial _4\partial _3\partial _2(-z_3-z_4-z_5)^{15} \end{aligned}$$

and find

$$\begin{aligned} I_1([X_2],\ldots ,[X_2])=240240. \end{aligned}$$

We list a few more such numbers:

$$\begin{aligned} \begin{aligned} I_1([X_2],[X_3],[X_{421}],[X_{421}])&=2,&I_1([X_2],[X_{21}],[X_{421}],[X_{421}])&=2,\\ I_1([X_2],[X_{42}],[X_{4321}])&=1,&I_1([X_2],[X_{321}],[X_{4321}])&=0,\\ I_1([X_2],[X_{421}],[X_{432}])&=1,&I_1([X_3],[X_{421}],[X_{431}])&=1,\\ I_1([X_{21}],[X_{421}],[X_{431}])&=1,&I_1([X_4],[X_{421}],[X_{421}])&=0,\\ I_1([X_{31}],[X_{421}],[X_{421}])&=1,&I_1([X_{43}],[X_{4321}])&=1,\\ I_1([X_{421}],[X_{4321}])&=0,&I_1([X_{431}],[X_{432}])&=1. \end{aligned} \end{aligned}$$

(7)

1.3 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 [18, Eqn. (3.3)], for the case of ${\mathrm {OG}}$. First, the Poincaré duality involution $\lambda \mapsto \lambda ^\vee$ on the set of partitions indexing the Schubert classes of ${\mathrm {OG}}$ (basis of the classical cohomology ring of ${\mathrm {OG}}$), is such that the set of parts of $\lambda ^{\vee }$ is the complement in $\{1,\ldots ,n-1\}$ of the set of parts of $\lambda$. We have focused on Gromov–Witten invariants involving Schubert classes of codimension $\ge 2$ above, because the ones with fundamental or divisor classes reduce to these by the following identities:

$$\begin{aligned}&I_0([X_\lambda ],[X_\mu ],[X_\nu ])=\int \limits _{{\mathrm {OG}}} [X_\lambda ]\cdot [X_\mu ]\cdot [X _\nu ], \\&I_0([X_{\lambda ^1}],\ldots ,[X_{\lambda ^m}])=0\quad \text { for}\;m\ne 3, \end{aligned}$$

and for $d\ge 1$,

$$\begin{aligned} I_d([X_{\lambda ^1}],\ldots ,[X_{\lambda ^m}],[X_1])= dI_d([X_{\lambda ^1}],\ldots ,[X_{\lambda ^m}]). \end{aligned}$$

Since it is needed for the discussion that follows, we record in Table 1 a portion of the multiplication table for the Schubert classes $\tau _\lambda =[X_\lambda ]$ in the classical cohomology ring. (One can produce this, e.g., using the Pieri formula of [15].)

Table 1 Portion of multiplication table for $H^* ({\mathrm {OG}}(5,10))$

Table 2 Degree 3, Case 1 numbers

Given $d\ge 1, m\ge 4$ and $\lambda ^1, \dots , \lambda ^m$ satisfying

$$\begin{aligned} |\lambda ^i|\ge 1, \quad \quad \sum |\lambda ^i|=n(n-1)/2-4+2d(n-1)+m, \end{aligned}$$

the corresponding associativity relation reads

$$\begin{aligned}&\sum _{d^{\prime },\mu ,A}I_{d^{\prime }}(\tau _{\lambda ^{i_1}},\ldots , \tau _{\lambda ^{i_a}},\tau _{\lambda ^{m-3}},\tau _{\lambda ^{m-2}}, \tau _\mu )I_{d-d^{\prime }}(\tau _{\lambda ^{j_1}},\ldots ,\tau _{\lambda ^{j_b}}, \tau _{\lambda ^{m-1}},\tau _{\lambda ^m},\tau _{\mu ^\vee })\nonumber \\&\quad =\sum _{d^{\prime },\mu ,A}I_{d^{\prime }}(\tau _{\lambda ^{i_1}},\ldots , \tau _{\lambda ^{i_a}},\tau _{\lambda ^{m-3}},\tau _{\lambda ^m}, \tau _\mu )I_{d-d^{\prime }}(\tau _{\lambda ^{j_1}},\ldots ,\tau _{\lambda ^{j_b}}, \tau _{\lambda ^{m-2}},\tau _{\lambda ^{m-1}},\tau _{\mu ^\vee }),\qquad \end{aligned}$$

(8)

where the first, respectively second sum is over integers $0\le d^{\prime }\le d$, strict partitions $\mu$ with parts less than $n$, and subsets $A\subset \{1,\ldots ,m-4\}$ such that

$$\begin{aligned} \sum _{i\in A\cup \{m-3,m-2\}} |\lambda ^i|+|\mu |=n(n-1)/2+2d^{\prime }(n-1)+a, \end{aligned}$$

(9)

respectively the same condition with $m-2$ replaced by $m$. In the Eqs. (8)–(9) $a$, respectively $b$ denotes the cardinality of $A$, respectively $B:=\{1,\ldots ,m-4\}\backslash A$, and we write $A=\{i_1,\ldots ,i_a\}$ and $B=\{j_1,\ldots ,j_b\}$.

In case $d=2$ in (8) we observe the following: (i) all terms with $d^{\prime }=1$, and hence $d-d^{\prime }=1$, are known by the previous section; (ii) terms with $d^{\prime }=0$ contribute

$$\begin{aligned} \sum _\mu \left( ~\int \limits _{{\mathrm {OG}}}\tau _{\lambda ^{m-3}}\tau _{\lambda ^{m-2}} \tau _{\mu ^\vee }\right) I_2(\tau _{\lambda ^1},\ldots ,\tau _{\lambda ^{m-4}}, \tau _{\lambda ^{m-1}},\tau _{\lambda ^m},\tau _\mu ) \end{aligned}$$

(10)

to the left-hand side and

$$\begin{aligned} \sum _\mu \left( ~\int \limits _{{\mathrm {OG}}}\tau _{\lambda ^{m-3}}\tau _{\lambda ^m} \tau _{\mu ^\vee }\right) I_2(\tau _{\lambda ^1},\ldots ,\tau _{\lambda ^{m-4}}, \tau _{\lambda ^{m-2}},\tau _{\lambda ^{m-1}},\tau _\mu ) \end{aligned}$$

(11)

to the right-hand side; (iii) terms with $d^{\prime }=2$ contribute

$$\begin{aligned} \sum _\mu \left( ~\int \limits _{{\mathrm {OG}}}\tau _{\lambda ^{m-1}}\tau _{\lambda ^m} \tau _{\mu ^\vee }\right) I_2(\tau _{\lambda ^1},\ldots ,\tau _{\lambda ^{m-4}}, \tau _{\lambda ^{m-3}},\tau _{\lambda ^{m-2}},\tau _\mu ) \end{aligned}$$

(12)

to the left-hand side and

$$\begin{aligned} \sum _\mu \left( ~\int \limits _{{\mathrm {OG}}}\tau _{\lambda ^{m-2}}\tau _{\lambda ^{m-1}} \tau _{\mu ^\vee }\right) I_2(\tau _{\lambda ^1},\ldots ,\tau _{\lambda ^{m-4}}, \tau _{\lambda ^{m-3}},\tau _{\lambda ^m},\tau _\mu ) \end{aligned}$$

(13)

to the right-hand side.

Now it is clear that the associativity relations determine many of the Gromov–Witten numbers $I_2(\tau _{\lambda ^1}, \ldots , \tau _{\lambda ^m})$. 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: $I_2(\ldots ,\tau _{4321},\tau _{4321})$. We apply (8) with $\lambda ^{m-3}=1, \lambda ^{m-2}=432, |\lambda ^{m-1}|\ge 2, \lambda ^m=4321$. Then (11)–(13) vanish, while (10) contributes $I_2(\tau _{\lambda ^1}$, $\ldots$, $\tau _{\lambda ^{m-4}}$, $\tau _{\lambda ^{m-1}}$, $\tau _{4321}$, $\tau _{4321})$.

Case 2

Point and line conditions: $I_2(\ldots ,\tau _{432},\tau _{4321})$. We apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=431$, $|\lambda ^{m-1}|\ge 2$, $\lambda ^m=4321$. Then (11)–(12) vanish, (13) either vanishes or is known by Case 1, and (10) contributes

$$\begin{aligned} I_2(\tau _{\lambda ^1},\ldots ,\tau _{\lambda ^{m-4}},\tau _{\lambda ^{m-1}}, \tau _{432},\tau _{4321}). \end{aligned}$$

Case 3

Point and plane: $I_2(\ldots ,\tau _{431},\tau _{4321})$. We apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=421$, $|\lambda ^{m-1}|\ge 2$, $\lambda ^m=4321$. Then (11)–(12) vanish, (13) either vanishes or is known by previous cases, and (10) contributes

$$\begin{aligned} I_2(\tau _{\lambda ^1},\ldots ,\tau _{\lambda ^{m-4}}, \tau _{\lambda ^{m-1}}, \tau _{431},\tau _{4321}). \end{aligned}$$

Case 4

Point and $X_{421}$: $I_2(\ldots ,\tau _{421},\tau _{4321})$. We apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=321$, $|\lambda ^{m-1}|\ge 2, \lambda ^m=4321$, and proceed as in Case 3.

Case 5

Point and $X_{43}$: $I_2(\ldots ,\tau _{43},\tau _{4321})$. We apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=42$, $|\lambda ^{m-1}|\ge 2$, $\lambda ^m=4321$. Then (11)–(12) vanish, (13) either vanishes or is known by previous cases, and (10) contributes

$$\begin{aligned} I_2(\tau _{\lambda ^1},\ldots ,\tau _{\lambda ^{m-4}},\tau _{\lambda ^{m-1}}, \tau _{43},\tau _{4321})+I_2(\tau _{\lambda ^1},\ldots ,\tau _{\lambda ^{m-4}}, \tau _{\lambda ^{m-1}}, \tau _{421},\tau _{4321}). \end{aligned}$$

Case 6

Point and $X_{42}$: $I_2(\ldots ,\tau _{42},\tau _{4321})$. We apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=41$, $|\lambda ^{m-1}|\ge 2$, $\lambda ^m=4321$, and proceed as in Case 3.

Case 7

Point and $X_{321}$: $I_2(\ldots ,\tau _{321},\tau _{4321})$. We apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=32$, $|\lambda ^{m-1}|\ge 2$, $\lambda ^m=4321$, and proceed as in Case 5.

Case 8

Point and $X_{41}$: $I_2(\ldots ,\tau _{41},\tau _{4321})$. We apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=4$, $|\lambda ^{m-1}|\ge 2$, $\lambda ^m=4321$, and proceed as in Case 3.

Case 9

Point and $X_{32}$: $I_2(\ldots ,\tau _{32},\tau _{4321})$. We apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=31$, $|\lambda ^{m-1}|\ge 2$, $\lambda ^m=4321$, and proceed as in Case 5.

Case 10

Two line conditions: $I_2(\ldots ,\tau _{432},\tau _{432})$. We apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=431$, $|\lambda ^{m-1}|\ge 2$, $\lambda ^m=432$. Then (12) vanishes, (13) vanishes or is known by Case 2, (11) contributes $I_2(\tau _{\lambda ^1}, \ldots , \tau _{\lambda ^{m-4}}, \tau _{\lambda ^{m-1}}, \tau _{431}, \tau _{4321})$, and (10) contributes $I_2(\tau _{\lambda ^1},\ldots ,\tau _{\lambda ^{m-4}}, \tau _{\lambda ^{m-1}}, \tau _{432}, \tau _{432})$.

Case 11

Line and plane: $I_2(\ldots ,\tau _{431},\tau _{432})$. We apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=421$, $|\lambda ^{m-1}|\ge 2$, $\lambda ^m=432$, and proceed as in Case 10.

Case 12

Line and $X_{421}$: $I_2(\ldots ,\tau _{421},\tau _{432})$. We apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=321$, $|\lambda ^{m-1}|\ge 2$, $\lambda ^m=432$, and proceed as in Case 10.

Case 13

Line and $X_{43}$: $I_2(\ldots ,\tau _{43},\tau _{432})$. We apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=42$, $|\lambda ^{m-1}|\ge 2$, $\lambda ^m=432$. Then (12) vanishes, (13) vanishes or is known by previous cases, (11) contributes $I_2(\tau _{\lambda ^1},\ldots ,\tau _{\lambda ^{m-4}},\tau _{\lambda ^{m-1}}, \tau _{42},\tau _{4321})$, and (10) contributes

$$\begin{aligned} I_2(\tau _{\lambda ^1},\ldots ,\tau _{\lambda ^{m-4}},\tau _{\lambda ^{m-1}}, \tau _{43},\tau _{432})+I_2(\tau _{\lambda ^1},\ldots ,\tau _{\lambda ^{m-4}}, \tau _{\lambda ^{m-1}}, \tau _{421},\tau _{432}). \end{aligned}$$

Case 14

Line and $X_{42}$: $I_2(\ldots ,\tau _{42},\tau _{432})$. We apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=41$, $|\lambda ^{m-1}|\ge 2$, $\lambda ^m=432$, and proceed as in Case 10.

Case 15

Line and $X_{321}$: $I_2(\ldots ,\tau _{321},\tau _{432})$. We apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=32$, $|\lambda ^{m-1}|\ge 2$, $\lambda ^m=432$, and proceed as in Case 13.

Case 16

Two plane conditions: $I_2(\ldots ,\tau _{431},\tau _{431})$. We apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=421$, $|\lambda ^{m-1}|\ge 2$, $\lambda ^m=431$. 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 $I_2(\tau _{\lambda ^1},\ldots ,\tau _{\lambda ^{m-4}},\tau _{\lambda ^{m-1}}, \tau _{431},\tau _{431})$.

Case 17

Plane and $X_{421}$: $I_2(\ldots ,\tau _{421},\tau _{431})$. We apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=321$, $|\lambda ^{m-1}|\ge 2$, $\lambda ^m=431$. Then (11) is known by Case 15, (12) and (13) vanish or are known by previous cases, and (10) contributes $I_2(\tau _{\lambda ^1},\ldots ,\tau _{\lambda ^{m-4}},\tau _{\lambda ^{m-1}}, \tau _{421},\tau _{431})$.

We list a few of the conic numbers:

$$\begin{aligned} \begin{aligned} I_2(\tau _2,\tau _{421},\tau _{431},\tau _{4321})&=3,&I_2(\tau _3,\tau _{421},\tau _{431},\tau _{432})&=5,\\ I_2(\tau _{21},\tau _{421},\tau _{431},\tau _{432})&=4,&I_2(\tau _{421},\tau _{432},\tau _{4321})&=1,\\ I_2(\tau _{431},\tau _{431},\tau _{4321})&=1,&I_2(\tau _{431},\tau _{432},\tau _{432})&=2, \end{aligned} \end{aligned}$$

(14)

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

Example

The number $I_2(\tau _2,\tau _{421},\tau _{431},\tau _{4321})$ falls under Case 3. We have $m=5$, $\lambda ^2=1$, $\lambda ^3=421$, $\lambda ^5=4321$, and either $\lambda ^4=2$, hence $\lambda ^1=421$ with (8) giving

$$\begin{aligned}&I_2(\tau _2,\tau _{421},\tau _{431},\tau _{4321}) + I_1(\tau _1,\tau _4, \tau _{421},\tau _{421})I_1(\tau _2,\tau _{321},\tau _{4321})\\&\quad \quad + I_1(\tau _1,\tau _{31},\tau _{421},\tau _{421})I_1(\tau _2,\tau _{42}, \tau _{4321})\\&\quad = I_2(\tau _1,\tau _{421},\tau _{432},\tau _{4321})+ I_1(\tau _1, \tau _{43},\tau _{4321})I_1(\tau _2,\tau _{21},\tau _{421},\tau _{421}) \\&\quad \quad +I_1(\tau _1,\tau _{421},\tau _{4321})I_1(\tau _2,\tau _3,\tau _{421}, \tau _{421})+ I_1(\tau _1,\tau _1,\tau _{421},\tau _{4321})I_1(\tau _2, \tau _{421},\tau _{432}), \end{aligned}$$

or $\lambda ^4=421$, hence $\lambda ^1=2$ and (8) giving

$$\begin{aligned}&I_2(\tau _2,\tau _{421},\tau _{431},\tau _{4321}) + I_1(\tau _1,\tau _2, \tau _{421},\tau _{432})I_1(\tau _1,\tau _{421},\tau _{4321})\\&\quad =I_1(\tau _1,\tau _{43},\tau _{4321})I_1(\tau _2,\tau _{21}, \tau _{421},\tau _{421})+ I_1(\tau _1,\tau _{421},\tau _{4321})I_1(\tau _2, \tau _3,\tau _{421},\tau _{421}) \\&\quad \quad + I_1(\tau _1,\tau _2,\tau _{42},\tau _{4321})I_1(\tau _{31}, \tau _{421},\tau _{421})+ I_1(\tau _1,\tau _2,\tau _{321},\tau _{4321}) I_1(\tau _4,\tau _{421},\tau _{421}). \end{aligned}$$

Either way, we obtain $I_2(\tau _2,\tau _{421}, \tau _{431}, \tau _{4321})=3$. One way requires the Case 2 number $I_2(\tau _{421}, \tau _{432}, \tau _{4321})$. The needed line numbers appear in (7).

Example

To determine $I_3(\tau _{421},\tau _{431},\tau _{4321},\tau _{4321})$ we read off from (8) with $d=3$, $m=5$, and $(\lambda ^1,\ldots ,\lambda ^5)=(431, 1, 432, 421, 4321)$, the identity [cf. (7), (14)]:

$$\begin{aligned}&I_3(\tau _{421},\tau _{431},\tau _{4321},\tau _{4321}) \\&\quad = I_1(\tau _1,\tau _{43},\tau _{4321})I_2(\tau _{21},\tau _{421}, \tau _{431},\tau _{432})+I_1(\tau _1,\tau _{421},\tau _{4321})I_2(\tau _3, \tau _{421},\tau _{431},\tau _{432})\\&\quad \quad + I_2(\tau _1,\tau _{431},\tau _{431},\tau _{4321})I_1(\tau _2, \tau _{421},\tau _{432}) -I_1(\tau _1,\tau _{431},\tau _{432})I_2(\tau _2, \tau _{421},\tau _{431},\tau _{4321})\\&\quad \quad - I_1(\tau _1,\tau _1,\tau _{431},\tau _{432})I_2(\tau _{421}, \tau _{432},\tau _{4321})- I_2(\tau _1,\tau _{431},\tau _{432},\tau _{432}) I_1(\tau _1,\tau _{421},\tau _{4321}) \\&\quad = 1\cdot 4 + 0\cdot 5 + 2\cdot 1 - 1\cdot 3 - 1\cdot 1 - 4\cdot 0=2. \end{aligned}$$

Alternatively $(\lambda ^1,\ldots ,\lambda ^5)=(421, 1, 432, 431, 4321)$ yields

$$\begin{aligned} I_3(\tau _{421},\tau _{431},\tau _{4321},\tau _{4321})= 1\cdot 4 + 0\cdot 5 + 0\cdot 2 + 2\cdot 1 - 1\cdot 3 - 1\cdot 1=2. \end{aligned}$$

Reasoning as in Case 2 we obtain the numbers $I_3(\ldots ,\tau _{432},\tau _{4321})$ listed in Table 3. Again it must be checked that each application of (8) requires only known conic numbers.

Table 3 Degree 3, Case 2 numbers

Similarly we reason as in Case 3 above to obtain the $I_3(\ldots ,\tau _{431},\tau _{4321})$ listed in Table 4. We conclude our determination of $d=3$ numbers with the $I_3(\ldots ,\tau _{432},\tau _{432})$ listed in Table 5, for which the reasoning is as in Case 10.

Table 4 Degree 3, Case 3 numbers

Table 5 Degree 3, Case 10 numbers

Table 6 Degree 4, Case 1 numbers

Table 7 Degree 4, Case 2 numbers

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

1.4 Higher degree numbers

The associativity relations also determine many higher-degree Gromov–Witten numbers. For instance, taking $d=3$ we may apply (8) with $\lambda ^{m-3}=1$, $\lambda ^{m-2}=432$, $|\lambda ^{m-1}|\ge 2$, $\lambda ^m=4321$ just as in Case 1 above, and obtain many Gromov–Witten numbers $I_3(\ldots ,\tau _{4321}, \tau _{4321})$. 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 $d^{\prime }=2$ or $d-d^{\prime }=2$ involve only degree 2 Gromov–Witten 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).

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 $I_4(\ldots ,\tau _{4321},\tau _{4321})$, respectively $I_4(\ldots ,\tau _{432},\tau _{4321})$, 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

$$\begin{aligned} I_7(\tau _{4321},\tau _{4321},\tau _{4321},\tau _{4321},\tau _{4321}, \tau _{4321},\tau _{4321})=71, \end{aligned}$$

with the following enumerative interpretation.

Proposition

There are 71 rational curves of degree 7 through 7 general points on ${\mathrm {OG}}(5,10)$.

Remark

Semi-simplicity allows us to reconstruct the full quantum cohomology even without assuming that the ordinary cohomology is generated by $H^2$, see [3] and [22]. (The case where the cohomology is generated by $H^2$ is addressed in [18].) This property was verified for orthogonal Grassmannians in [6].