Genus one enumerative invariants in del-Pezzo surfaces with a fixed complex structure
Indranil Biswas, Ritwik Mukherjee, Varun Thakre
TL;DR
This work extends genus-one enumerative counts of curves with fixed ${j}$-invariant from ${\mathbb P}^2$ to complex del-Pezzo surfaces by relating the enumerative invariant ${n_{1,\beta}^j}$ to the genus-zero count ${n_{0,\beta}}$ via ${n_{1,\beta}^j = \frac{2 g_{\beta}}{|\mathrm{Aut}(\Sigma_1, j)|} n_{0,\beta}}$ with ${g_{\beta} = \frac{ {\beta}\cdot{\beta} - c_1(TX)\cdot{\beta} + 2}{2}}$. The authors decompose the genus-one symplectic invariant ${\mathrm{RT}}_{1,\beta}$ into an enumerative part and a correction term ${\mathrm{CR}}$, and provide a self-contained computation of ${\mathrm{CR}}$ via ${c_1}(\mathcal{L}^*)$ and an intersection on the rational-curves moduli space, yielding ${\mathrm{CR}} = ( {\beta} \cdot c_1(TX) - 2) n_{0,\beta}$. They then show ${\mathrm{RT}}_{1,\beta} = (\beta\cdot\beta) n_{0,\beta}$, which together with the CR formula gives the closed form for ${n_{1,\beta}^j}$. Finally, the paper proves genus-one regularity for immersions in del-Pezzo surfaces, ensuring the framework applies to the targeted geometric settings. This work thereby connects enumerative counts, symplectic invariants, and tropical perspectives in the setting of del-Pezzo geometry.
Abstract
We obtain a formula for the number of genus one curves with a fixed complex structure of a given degree on a del-Pezzo surface that pass through an appropriate number of generic points of the surface. This enumerative problem is expressed as the difference between the symplectic invariant and an intersection number on the moduli space of rational curves.