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A relative orientation for the moduli space of stable maps to a del Pezzo surface

Jesse Leo Kass, Marc Levine, Jake P. Solomon, Kirsten Wickelgren

TL;DR

The paper develops a framework to orient evaluation maps from genus zero stable-map moduli to del Pezzo surfaces, yielding quadratically enriched curve counts valued in the Grothendieck–Witt group ${\rm GW}(k)$. A two-step approach identifies the ramification locus with cusp/tacnode data via a discriminant construction, then orients the discriminant to obtain a relative orientation of $\mathrm{ev}$ and a well-defined degree. The authors extend these orientations to twisted and symmetrized evaluation maps, relate the degrees to genus-zero Gromov–Witten-type invariants, and provide explicit formulas in degree, including the case $D=-K_S$. They further adapt the results to positive characteristic by lifting to characteristic zero over a DVR, establishing good moduli spaces and compatible discriminant orientations, and defining twisted degrees in that setting. The work yields a robust toolkit for quadratically enriched curve counts on del Pezzo surfaces, compatible with both complex and real realizations and with arithmetic refinements over non-algebraically closed fields.

Abstract

We prove orientation results for evaluation maps of moduli spaces of rational stable maps to del Pezzo surfaces over a field, both in characteristic $0$ and in positive characteristic. These results and the theory of degree developed in a sequel produce quadratically enriched counts of rational curves over non-algebraically closed fields of characteristic not $2$ or $3$. Orientations are constructed in two steps. First, the ramification locus of the evaluation map is shown to be the divisor in the moduli space of stable maps where image curves have a cusp. Second, this divisor is related to the discriminant of a branched cover of the moduli space given generically by pairs of points on the universal curve with the same image.

A relative orientation for the moduli space of stable maps to a del Pezzo surface

TL;DR

The paper develops a framework to orient evaluation maps from genus zero stable-map moduli to del Pezzo surfaces, yielding quadratically enriched curve counts valued in the Grothendieck–Witt group . A two-step approach identifies the ramification locus with cusp/tacnode data via a discriminant construction, then orients the discriminant to obtain a relative orientation of and a well-defined degree. The authors extend these orientations to twisted and symmetrized evaluation maps, relate the degrees to genus-zero Gromov–Witten-type invariants, and provide explicit formulas in degree, including the case . They further adapt the results to positive characteristic by lifting to characteristic zero over a DVR, establishing good moduli spaces and compatible discriminant orientations, and defining twisted degrees in that setting. The work yields a robust toolkit for quadratically enriched curve counts on del Pezzo surfaces, compatible with both complex and real realizations and with arithmetic refinements over non-algebraically closed fields.

Abstract

We prove orientation results for evaluation maps of moduli spaces of rational stable maps to del Pezzo surfaces over a field, both in characteristic and in positive characteristic. These results and the theory of degree developed in a sequel produce quadratically enriched counts of rational curves over non-algebraically closed fields of characteristic not or . Orientations are constructed in two steps. First, the ramification locus of the evaluation map is shown to be the divisor in the moduli space of stable maps where image curves have a cusp. Second, this divisor is related to the discriminant of a branched cover of the moduli space given generically by pairs of points on the universal curve with the same image.
Paper Structure (20 sections, 73 theorems, 267 equations)

This paper contains 20 sections, 73 theorems, 267 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Rational curves on a del Pezzo surface
  4. Definitions of moduli spaces of stable maps to del Pezzo surfaces
  5. Over a field
  6. Properties of rational curves
  7. Some geometry of moduli stacks of birational and/or unramified maps
  8. Non-birational and non-free maps
  9. Fine structure of the evaluation map
  10. The double point locus
  11. Orienting the evaluation map
  12. The symmetrized moduli space
  13. Twisting the degree map
  14. Positive characteristic
  15. Lifting to characteristic zero
  16. ...and 5 more sections

Key Result

Theorem 1.1

Suppose $k$ is a field of characteristic zero and that $(S,D)$ satisfies Hypothesis hyp:basic. Let $n=d-1$. Then there is a codimension $\geq 2$ closed subset $A$ of $S^n$ such that admits an orientation.

Theorems & Definitions (173)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 163 more