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Genus two enumerative invariants in del-Pezzo surfaces with a fixed complex structure

Indranil Biswas, Ritwik Mukherjee, Varun Thakre

TL;DR

This work extends Zinger’s symplectic approach to count genus two curves with fixed complex structure on del-Pezzo surfaces. It derives a concrete formula for the genus two enumerative invariant n_{2,β}^j by relating the symplectic RT_{2,β} invariant to the desired count through a boundary correction CR_{2,β}, which is computed via a detailed analysis of boundary strata and affine-bundle zeros following Zinger’s framework. The authors provide explicit expressions for the correction terms in terms of genus-zero data n_{0,β_i} and intersection numbers, and prove genus two regularity and the existence of nice metrics to justify the gluing arguments. They also perform intersection computations in tautological classes and validate the approach with consistency checks, including a programmable method for cases X=ℙ^2 blown up at k≤8 points. Altogether, the paper delivers a practical, explicit formula for n_{2,β}^j on del-Pezzo surfaces and advances the understanding of higher-genus enumerative invariants in this setting.

Abstract

We obtain a formula for the number of genus two 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 is done by extending the symplectic approach of Aleksey Zinger. This enumerative problem is expressed as the difference between the symplectic invariant and an intersection number on the moduli space of rational curves on the surface.

Genus two enumerative invariants in del-Pezzo surfaces with a fixed complex structure

TL;DR

This work extends Zinger’s symplectic approach to count genus two curves with fixed complex structure on del-Pezzo surfaces. It derives a concrete formula for the genus two enumerative invariant n_{2,β}^j by relating the symplectic RT_{2,β} invariant to the desired count through a boundary correction CR_{2,β}, which is computed via a detailed analysis of boundary strata and affine-bundle zeros following Zinger’s framework. The authors provide explicit expressions for the correction terms in terms of genus-zero data n_{0,β_i} and intersection numbers, and prove genus two regularity and the existence of nice metrics to justify the gluing arguments. They also perform intersection computations in tautological classes and validate the approach with consistency checks, including a programmable method for cases X=ℙ^2 blown up at k≤8 points. Altogether, the paper delivers a practical, explicit formula for n_{2,β}^j on del-Pezzo surfaces and advances the understanding of higher-genus enumerative invariants in this setting.

Abstract

We obtain a formula for the number of genus two 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 is done by extending the symplectic approach of Aleksey Zinger. This enumerative problem is expressed as the difference between the symplectic invariant and an intersection number on the moduli space of rational curves on the surface.

Paper Structure

This paper contains 10 sections, 16 theorems, 90 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Enumerative versus symplectic invariant
  3. Computation of the symplectic invariant
  4. Computation of the correction term: boundary contribution
  5. Description of the boundary strata
  6. Calculation of the excess boundary contribution to the Symplectic Invariant
  7. Proof of the Main Theorem
  8. Intersection of Tautological Classes
  9. Existence of a nice metric
  10. Regularity of the complex structure

Key Result

Theorem 1.1

There is an explicit formula for the number of degree $d$ genus one curves with a fixed complex structure in $\mathbb{P}^2_{\mathbb{C}}$ that pass through $3d-1$ generic points.

Figures (4)

  • Figure 1: $\mathcal{B}^1_1$, $\mathcal{B}^2_1$ and $\mathcal{B}^3_1$.
  • Figure 2: Not Possible
  • Figure 3: The stratum $\mathcal{B}^1_2$.
  • Figure 4: Not Possible

Theorems & Definitions (35)

  • Theorem 1.1: R.Pandharipande and E.Ionel
  • Theorem 1.2: S.Katz, Z.Qin, Y.Ruan and A. Zinger
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.7
  • Remark 1.8
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • ...and 25 more