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Combinatorial Göttsche-Schroeter invariants in any genus

Gurvan Mével

Abstract

Göttsche-Schroeter invariants are a genus 0 extension of Block-Göttsche invariants. They interpolate between Welschinger invariants involving pairs of complex conjugated points and genus 0 descendant Gromov-Witten invariants. They can be computed by a floor diagram algorithm. In this paper, we show that this floor diagrams recipe actually leads to some invariants in any genus. This generalizes Göttsche-Schroter invariant in higher genus in a combinatorial way. We then prove some polynomiality result and establish a link with invariants defined by Shustin and Sinichkin. We provide many examples. In particular, we conjecture that these combinatorial invariants satisfy the Abramovich-Bertram formula.

Combinatorial Göttsche-Schroeter invariants in any genus

Abstract

Göttsche-Schroeter invariants are a genus 0 extension of Block-Göttsche invariants. They interpolate between Welschinger invariants involving pairs of complex conjugated points and genus 0 descendant Gromov-Witten invariants. They can be computed by a floor diagram algorithm. In this paper, we show that this floor diagrams recipe actually leads to some invariants in any genus. This generalizes Göttsche-Schroter invariant in higher genus in a combinatorial way. We then prove some polynomiality result and establish a link with invariants defined by Shustin and Sinichkin. We provide many examples. In particular, we conjecture that these combinatorial invariants satisfy the Abramovich-Bertram formula.

Paper Structure

This paper contains 18 sections, 9 theorems, 65 equations, 18 figures, 16 tables.

Table of Contents

  1. Introduction
  2. Enumerative geometry
  3. The tropical approach
  4. Refined invariants
  5. Results of this paper
  6. Floor diagrams and refined invariants in genus $0$
  7. $h$-transverse polygons and floor diagrams
  8. Refined invariants
  9. Operations on floor diagrams
  10. Refined invariants in the non-rational case
  11. Definition of $G_g(\Delta,s)$
  12. Properties of the invariants
  13. Link with other invariants
  14. Examples and conjectures
  15. Some calculations
  16. ...and 3 more sections

Key Result

Theorem 2.8

Let $\Delta$ be a $h$-transverse polygon and $s\in \{0,\dots,s_{\max}(\Delta,0)\}$. For any pairing $S$ of order $s$ of $\{1,\dots, y(\Delta)-1 \}$ one has where the sum runs over the isomorphism classes of marked floor diagrams with Newton polygon $\Delta$ and genus $0$.

Figures (18)

  • Figure 1: Some polygons.
  • Figure 2: The floor diagrams with Newton polygon the polygon of figure \ref{['fig-ex-poly-d3']}.
  • Figure 3: Some marked floor diagrams with Newton polygon the polygon of figure \ref{['fig-ex-poly-d3']}.
  • Figure 4: Operations $A^+$ and $A^-$.
  • Figure 5: Possible configurations with 2 vertices.
  • ...and 13 more figures

Theorems & Definitions (44)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3: Floor diagram
  • Example 2.4
  • Definition 2.5: Marking
  • Example 2.6
  • Definition 2.7: Refined $S$-multiplicity
  • Theorem 2.8: brugalle_polynomiality_2022
  • Remark 2.9
  • Example 2.10
  • ...and 34 more