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.
Similar content being viewed by others
References
Araujo, C., Kollár, J.: Rational curves on varieties. In: Higher Dimensional Varieties and Rational Points (Budapest, 2001), Bolyai Society, Mathematical Studies, vol. 12, pp. 13–68. Springer, Berlin (2003).
Arbarello, E., Bruno, A., Sernesi, E.: Mukai’s program for curves on a \(K3\) surface (2013) ArXiv:1309.0496
Bayer, A., Manin, Y.I.: (Semi)simple exercises in quantum cohomology. In: The Fano Conference, pp. 143–173. University of Turin, Turin (2004).
Bernšteĭn, I.N., Gel’fand, I.M., Gel’fand, S.I.: Schubert cells, and the cohomology of the spaces \(G/P\). Uspehi Mat. Nauk, 28(3(171)):3–26 (1973) English transl. Russian Math. Surveys 28(3), 1–26 (1973)
Billey, S., Haiman, M.: Schubert polynomials for the classical groups. J. Am. Math. Soc. 8(2), 443–482 (1995)
Chaput, P.E., Manivel, L., Perrin, N.: Quantum cohomology of minuscule homogeneous spaces III. Semi-simplicity and consequences. Can. J. Math. 62(6), 1246–1263 (2010)
Ciliberto, C., Lopez, A., Miranda, R.: Projective degenerations of \(K3\) surfaces, Gaussian maps, and Fano threefolds. Invent. Math. 114(3), 641–667 (1993)
de Jong, A.J., He, X., Starr, J.M.: Families of rationally simply connected varieties over surfaces and torsors for semisimple groups. Publ. Math. Inst. Hautes Études Sci. 114, 1–85 (2011)
Degtyarëv, A.I.: Classification of quartic surfaces that have a nonsimple singular point. Izv. Akad. Nauk SSSR Ser. Mat. 53(6), 1269–1290, 1337–1338 (1989).
Demazure, M.: Désingularisation des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup. (4), 7, 53–88 (1974) Collection of articles dedicated to Henri Cartan on the occasion of his 70th birthday, I.
Fulton, W., Pandharipande, R.: Notes on stable maps and quantum cohomology. In: Algebraic geometry (Santa Cruz 1995), Proceedings of Symposia in Pure Mathematics, vol. 62, pp. 45–96. American Mathematical Society, Providence, RI (1997).
Green, M., Lazarsfeld, R.: Special divisors on curves on a \(K3\) surface. Invent. Math. 89(2), 357–370 (1987)
Hassett, B., Kresch, A., Tschinkel, Y.: On the moduli of degree 4 Del Pezzo surfaces (2013) ArXiv:1312.6734
Hassett, B., Tschinkel, Y.: Quartic del Pezzo surfaces over function fields of curves. Cent. Eur. J. Math. 12(3), 395–420 (2014)
Hiller, H., Boe, B.: Pieri formula for \({\rm SO}_{2n+1}/{\rm U}_n\) and \({\rm Sp}_n/{\rm U}_n\). Adv. Math. 62(1), 49–67 (1986)
Ishii, Y., Nakayama, N.: Classification of normal quartic surfaces with irrational singularities. J. Math. Soc. Jpn. 56(3), 941–965 (2004)
Kawamata, Y.: On deformations of compactifiable complex manifolds. Math. Ann. 235(3), 247–265 (1978)
Kontsevich, M., Manin, Yu.: Gromov-Witten classes, quantum cohomology, and enumerative geometry. Commun. Math. Phys. 164(3), 525–562 (1994)
Kresch, A., Tamvakis, H.: Quantum cohomology of orthogonal Grassmannians. Compos. Math. 140(2), 482–500 (2004)
Landsberg, J.M., Manivel, L.: On the projective geometry of rational homogeneous varieties. Comment. Math. Helv. 78(1), 65–100 (2003)
Martín, López: A.: Gromov-Witten invariants and rational curves on Grassmannians. J. Geom. Phys. 61(1), 213–216 (2011)
Maszczyk, T.: Computing genus zero Gromov-Witten invariants of Fano varieties. J. Geom. Phys. 61(6), 1079–1092 (2011)
Mukai, S.: On the moduli space of bundles on \(K3\) surfaces. I. In: Vector Bundles on Algebraic Varieties (Bombay, 1984), Tata Institute of Fundamental Research Studies in Mathematics, vol. 11, pp. 341–413. Tata Institute of Fundamental Research, Bombay (1987).
Mukai, S.: Curves and symmetric spaces. I. Am. J. Math. 117(6), 1627–1644 (1995)
Mukai, S.: Curves and \(K3\) surfaces of genus eleven. In: Moduli of Vector Bundles (Sanda, 1994; Kyoto, 1994), Lecture Notes in Pure and Applied Mathematics, vol. 179, pp. 189–197. Dekker, New York (1996).
Mukai, S.: Non-abelian Brill-Noether theory and Fano 3-folds [translation of Sūgaku 49(1), 1997, pp. 1–24;]. Sugaku Expositions 14(2), 125–153 (2001)
Shah, J.: Degenerations of \(K3\) surfaces of degree 4. Trans. Am. Math. Soc. 263(2), 271–308 (1981)
Urabe, T.: Classification of nonnormal quartic surfaces. Tokyo J. Math. 9(2), 265–295 (1986)
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 National Science Foundation Grants 0968349, 0901645, and 1148609; the second author is supported by National Science Foundation Grants 0739380, 0968349, and 1160859.
Author information
Authors and Affiliations
Corresponding author
Appendix: 71 rational septics through 7 general points on \({\mathrm {OG}}(5,10)\)
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:
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
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 \({\mathbb {Q}}[z_1,\ldots ,z_n]\):
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
with the above convention on \(P\)-polynomials, then we have
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
and find
We list a few more such numbers:
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:
and for \(d\ge 1\),
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].)
Given \(d\ge 1, m\ge 4\) and \(\lambda ^1, \dots , \lambda ^m\) satisfying
the corresponding associativity relation reads
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
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
to the left-hand side and
to the right-hand side; (iii) terms with \(d^{\prime }=2\) contribute
to the left-hand side and
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
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
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
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
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:
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
or \(\lambda ^4=421\), hence \(\lambda ^1=2\) and (8) giving
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)]:
Alternatively \((\lambda ^1,\ldots ,\lambda ^5)=(421, 1, 432, 431, 4321)\) yields
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.
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.
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
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].
Rights and permissions
About this article
Cite this article
Hassett, B., Tschinkel, Y. Embedding pointed curves in K3 surfaces. Math. Z. 278, 927–953 (2014). https://doi.org/10.1007/s00209-014-1339-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-014-1339-x