Genus two enumerative invariants in del-Pezzo surfaces with a fixed complex structure
The number of irreducible, reduced, nodal, degree-\(d\) rational curves passing through the right number of general points in \(\mathbb C \mathbb P^2\) satisfies a nice recursion relation found by \textit{M. Kontsevich} and \textit{Yu. Manin} [AMS/IP Stud. Adv. Math. 1, 607--653 (1997; Zbl 0931.14030)] and also by \textit{Y. Ruan} and \textit{G. Tian} [Math. Res. Lett. 1, No. 2, 269--278 (1994; Zbl 0860.58006)]. This formula was generalized independently by \textit{E. Ionel} [Duke Math. J. 94, No. 2, 279--324 (1998; Zbl 0974.14038)] and \textit{R. Pandharipande} [Proc. Am. Math. Soc. 125, No. 12, 3471--3479 (1997; Zbl 0883.14031)] to the case of elliptic curves with fixed \(j\)-invariant, and by \textit{S. Katz} et al. [J. Algebr. Geom. 7, No. 3, 569--587 (1998; Zbl 0946.14031)] and \textit{A. Zinger} [J. Algebr. Geom. 13, No. 3, 547--561 (2004; Zbl 1063.14038)] to the case of genus-two curves with the fixed complex structure. The authors of the paper under review extended the result for the elliptic curves to del-Pezzo surfaces [\textit{I. Biswas} et al., C. R., Math., Acad. Sci. Paris 354, No. 5, 517--521 (2016; Zbl 1387.14099)]. In this paper they are extending the result for genus-two to del-Pezzo surfaces. To state their result, let \(X\) be a del-Pezzo surface with the second Betti number \(b_2\), \(\beta \in H_2(X,\mathbb Z)\), \(n_{0,\beta}\) the number of rational curves on \(X\) passing through the right number of points, \(\delta_\beta=-K_X\cdot \beta-1\). Let \(A\) be half the number of holomorphic automorphisms of a genus-two Riemann surfaces with a fixed complex structure. Then, under certain conditions to ensure genus-two regularity of the complex structures, \(A\) times the number of genus-two curves with this complex structure passing through a right number of points is \[ n_{0,\beta}\Big((2+b_2)\beta^2-10c_2(X)-K_X^2-12K_X^2/(K_X\cdot \beta) \] \[ +\sum_{\beta_1,\beta_2} \binom{\delta_\beta-1}{\delta_{\beta_1}}n_{0,\beta_1}n_{0,\beta_2}(\beta_1\cdot \beta_2) \] \[ +6(K_X\cdot \beta_1)(K_X\cdot \beta_2)/(K_X\cdot \beta)+\beta_1^2\beta_2^2/2+10\Big), \] where the sum is over all \(0\neq \beta_1, 0=\neq \beta_2 \in H_2(X,\mathbb Z)\) such that \(\beta=\beta_1+\beta_2\).
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