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Real enumerative invariants relative to the toric boundary and their refinement

Ilia Itenberg, Eugenii Shustin

TL;DR

The paper develops real refined enumerative invariants for toric surfaces, extending Mikhalkin's genus-zero refinement to genus 1 and 2 by counting halves of real curves with a fixed quantum index and two Welschinger-type signs. It builds a framework using constraints in the toric boundary (Menelaus data) and chambers separated by walls, and proves invariance under wall-crossing through local deformation theory, transversality, and cohomology vanishing arguments; genus-two results specialize to toric surfaces with Tor(P_Δ) ≅ P^2. The approach unifies rational, elliptic, and genus-two refinements, connects to tropical Block-Göttsche invariants, and provides a robust foundation for future tropical-combinatorial comparisons. The invariants W_g^κ(Δ) and tilde{W}_g^κ(Δ) vanish outside specific κ-parities and congruences, with tropical counterparts explored in forthcoming work.

Abstract

We introduce new invariants of a class of toric surfaces (including the projective plane) that arise from appropriate enumeration of real curves of genus one and two. These invariants admit a refinement similar to the one introduced by Grigory Mikhalkin in the rational case.

Real enumerative invariants relative to the toric boundary and their refinement

TL;DR

The paper develops real refined enumerative invariants for toric surfaces, extending Mikhalkin's genus-zero refinement to genus 1 and 2 by counting halves of real curves with a fixed quantum index and two Welschinger-type signs. It builds a framework using constraints in the toric boundary (Menelaus data) and chambers separated by walls, and proves invariance under wall-crossing through local deformation theory, transversality, and cohomology vanishing arguments; genus-two results specialize to toric surfaces with Tor(P_Δ) ≅ P^2. The approach unifies rational, elliptic, and genus-two refinements, connects to tropical Block-Göttsche invariants, and provides a robust foundation for future tropical-combinatorial comparisons. The invariants W_g^κ(Δ) and tilde{W}_g^κ(Δ) vanish outside specific κ-parities and congruences, with tropical counterparts explored in forthcoming work.

Abstract

We introduce new invariants of a class of toric surfaces (including the projective plane) that arise from appropriate enumeration of real curves of genus one and two. These invariants admit a refinement similar to the one introduced by Grigory Mikhalkin in the rational case.
Paper Structure (23 sections, 15 theorems, 194 equations, 7 figures)

This paper contains 23 sections, 15 theorems, 194 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preparation
  3. Convex lattice polygons and real toric surfaces
  4. Toric degree and Menelaus condition
  5. Deformations of plane curve germs
  6. Curves on toric surfaces
  7. Real refined enumerative invariants in genus $0$, $1$, and $2$
  8. Preliminary remarks
  9. Refined rational invariants
  10. Refined elliptic invariants
  11. Refined invariants of genus two
  12. Invariance in genus zero: proof of Theorem \ref{['at1']}
  13. Invariance inside chambers
  14. Walls in the space of constraints
  15. Wall-crossing
  16. ...and 8 more sections

Key Result

Lemma 2.2

The sequences ${\boldsymbol z}$ satisfying the Menelaus $\Delta$-condition form an algebraic hypersurface $\operatorname{Men}(\Delta)\subset \prod_{\sigma\in P^1_\Delta}{\operatorname{Tor}}(\sigma)^{n^\sigma}$.

Figures (7)

  • Figure 1: Proof of Lemma \ref{['al8']}, part (2)
  • Figure 2: Proof of Lemma \ref{['al8']}, part (4)
  • Figure 3: Proof of Lemma \ref{['al8']}, part (5), I
  • Figure 4: Proof of Lemma \ref{['al8']}, part (5), II
  • Figure 5: Proof of Lemma \ref{['al8']}, part (5), III
  • ...and 2 more figures

Theorems & Definitions (23)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Remark 3.1
  • Theorem 3.2
  • Remark 3.3
  • Definition 3.4
  • Example 3.5
  • Theorem 3.6
  • ...and 13 more