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].


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