## Universal formulas for counting nodal curves on surfaces.(English)Zbl 1271.14084

Jerison, David (ed.) et al., Current developments in mathematics, 2010. Somerville, MA: International Press (ISBN 978-1-57146-228-2/hbk). 95-115 (2011).
Let $$S$$ be any complex projective smooth surface with canonical bundle $$K$$, and let $$r$$ be any nonnegative integer. For every line bundle $$L$$ on $$S$$, denote by $$t_r(S,L)$$ the number of $$r$$-nodal curves in a generic $$r$$-dimensional sublinear system of $$|L|$$. (Here an $$r$$-nodal curve is a nodal curve with exactly $$r$$ nodes.) In [Commun. Math. Phys. 196, No. 3, 523–533 (1998; Zbl 0934.14038)], L. Göttsche conjectured that the number $$t_r(S,L)$$ is a universal polynomial of degree $$r$$ in the numbers $$L^2$$, $$LK$$, $$c_1(S)^2$$, and $$c_2(S)$$ if $$L$$ is sufficiently ample with respect to $$r$$. Note that the polynomial is universal in the sense that its coefficients are independent of $$S$$ and $$L$$.
Suppose $$S^{[3r]}$$ is the Hilbert scheme of $$3r$$ points on $$S$$, and $$S^r_{2,0}$$ is the locally closed subset of $$S^{[3r]}$$ which parametrizes subschemes of the form $$\coprod_{i=1}^r \mathrm{Spec} (\mathcal{O}_{S, x_i}/m^2_{S, x_i})$$, where $$x_1, \ldots, x_r$$ are distinct closed points on $$S$$. Let $$S^r_2 \subset S^{[3r]}$$ be the closure (with the reduced induced structure) of $$S^r_{2,0}$$, and let $$L^{[3r]}$$ be the rank $$3r$$ tautological bundle on $$S^{[3r]}$$ associated to $$L$$, which is constructed from the universal subscheme of $$S \times S^{[3r]}$$ and the obvious projections. L. Göttsche introduced the intersection number $d_r(S,L):=\int_{S^r_2} c_{2r}(L^{[3r]})$ and showed that it coincides with the number $$t_r(S,L)$$ if $$L$$ is $$(5r-1)$$-very ample ($$5$$-very ample if $$r=1$$), i.e., if the natural map $$H^0(S,L) \to H^0(\xi, L\otimes \mathcal{O}_\xi)$$ is surjective for all zero-dimensional subschemes $$\xi \subset S$$ of length $$5r$$ ($$6$$ if $$r=1$$).
In the paper under review, the author studies the generating function $$\phi(S,L):=\sum_{r=0}^\infty d_r(S,L) \, x^r$$ for every line bundle $$L$$ on $$S$$. Using algebraic cobordism theory and deriving a degeneration formula for $$d_r(S,L)$$, she proves that $$\phi$$ gives a homomorphism from the algebraic cobordism group $$\omega_{2,1}$$, which is a four-dimensional vector space over $$\mathbb{Q}$$ spanned by all pairs of surfaces and line bundles modulo extended double point relations, to the group $$\mathbb{Q}[[x]]^{\times}$$ of units of $$\mathbb{Q}[[x]]$$. (More details are provided in her paper [J. Differ. Geom. 90, No. 3, 439–472 (2012; Zbl 1253.14054)]). As a consequence, $$\phi(S,L)$$ can be expressed as the product $A_1^{L^2}A_2^{LK}A_3^{c_1(S)^2}A_4^{c_2(S)}$ for some power series $$A_1, A_2, A_3, A_4 \in \mathbb{Q}[[x]]^{\times}$$ whose coefficients are independent of $$S$$ and $$L$$. (So far, there are no closed-form expressions for the coefficients of $$A_i$$’s, even though these coefficients are determined by the recursive formulas for $$\mathbb{P}^2$$ and $$\mathbb{P}^1 \times \mathbb{P}^1$$ due to L. Caporaso and J. Harris [Invent. Math. 131, No. 2, 345–392 (1998; Zbl 0934.14040)] and R. Vakil [Manuscr. Math. 102, No. 1, 53–84 (2000; Zbl 0967.14036)] respectively.)
By extracting the coefficient of $$x^r$$ in $$A_1^{L^2}A_2^{LK}A_3^{c_1(S)^2}A_4^{c_2(S)}$$, the author deduces that $$d_r(S,L)$$ is a universal polynomial $$T_r(L^2, LK, c_1(S)^2, c_2(S))$$ of degree $$r$$ in $$L^2$$, $$LK$$, $$c_1(S)^2$$, and $$c_2(S)$$. This proves Göttsche’s conjecture if $$L$$ is $$(5r-1)$$-very ample ($$5$$-very ample for $$r=1$$). Note that A.-K. Liu proved the conjecture several years ago using symplectic approach (see [J. Differ. Geom. 56, No. 3, 381–579 (2000; Zbl 1036.14014); “The algebraic proof of the Universality Theorem”, arXiv:math/0402045]). It is also worth mentioning that a different algebro-geometric proof of the conjecture was obtained by M. Kool, V. Shende, and R. Thomas under the weaker condition that $$L$$ is $$r$$-very ample [Geom. Topol. 15, No. 1, 397–406 (2011; Zbl 1210.14011)].
Let $$G_2=-\frac{1}{24}+\sum_{n=1}^\infty (\sum_{d | n} d) \, q^n$$ (the second Eisenstein series), $$\Delta=q\prod_{k =1}^\infty (1-q^k)^{24}$$, and $${D= q \frac{d}{d q}}$$. The author also shows that there exist power series $$B_1$$ and $$B_2$$ in $$q$$ whose coefficients are independent of $$S$$ and $$L$$ such that $\sum_{r=0}^\infty T_r(L^2, LK, c_1(S)^2, c_2(S))(DG_2)^r=\frac{(DG_2/q)^{\chi(L)}B_1^{K^2}B_2^{LK}}{(\Delta D^2G_2/q^2)^{\chi(\mathcal{O}_S)/2}},$ which is referred to as the Göttsche–Yau–Zaslow formula. In fact, the formula follows from the aforementioned results and the corresponding formula for generic $$K3$$ surfaces due to J. Bryan and N. C. Leung [J. Am. Math. Soc. 13, No. 2, 371–410 (2000; Zbl 0963.14031)].
For the entire collection see [Zbl 1245.00031].

### MSC:

 14N10 Enumerative problems (combinatorial problems) in algebraic geometry 14C05 Parametrization (Chow and Hilbert schemes) 14C20 Divisors, linear systems, invertible sheaves