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Refined enumerative invariants and mixed Welschinger invariants

Eugenii Shustin, Uriel Sinichkin

Abstract

For real toric surfaces and conjugation invariant point conditions with all conjugate pairs on the boundary divisors, we prove that the signed count of real curves of arbitrary genus in the linear system through the given points is invariant under variation of the points, provided the reduced tropicalization is in general position. The proof is based on a new relative refined tropical invariant, which is invariant under variation of the point conditions and specializes at $y\to -1$ to this signed count; at $y\to 1$ the same invariant recovers the count of complex curves with prescribed tangency to the boundary. We extend the invariant to allow arbitrary tangency orders along the boundary and identify its $y\to 1$ limit with the corresponding complex count. Finally, we show that in positive genus the signed real count is not invariant when conjugate pairs are allowed in the interior, even under strong genericity assumptions.

Refined enumerative invariants and mixed Welschinger invariants

Abstract

For real toric surfaces and conjugation invariant point conditions with all conjugate pairs on the boundary divisors, we prove that the signed count of real curves of arbitrary genus in the linear system through the given points is invariant under variation of the points, provided the reduced tropicalization is in general position. The proof is based on a new relative refined tropical invariant, which is invariant under variation of the point conditions and specializes at to this signed count; at the same invariant recovers the count of complex curves with prescribed tangency to the boundary. We extend the invariant to allow arbitrary tangency orders along the boundary and identify its limit with the corresponding complex count. Finally, we show that in positive genus the signed real count is not invariant when conjugate pairs are allowed in the interior, even under strong genericity assumptions.
Paper Structure (18 sections, 25 theorems, 54 equations, 2 figures)

This paper contains 18 sections, 25 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main results
  4. Structure of the paper
  5. Preliminaries
  6. Notations
  7. Tropical curves
  8. Moduli space of plane tropical curves and evaluation
  9. Tropicalization of algebraic curves over a non-Archimedean field
  10. Enhanced tropical curves
  11. Parameterized tropical limit
  12. Real tropical curves
  13. Complex correspondence theorem
  14. Real correspondence theorem
  15. Relative refined tropical invariants
  16. ...and 3 more sections

Key Result

Theorem 1.1

Fix a non-negative integer $g\ge 0$ and let $X=\mathop{\mathrm{Tor}}\nolimits_{\mathbb R}(P)$ be a toric surface corresponding to a convex lattice polygon $P\subset{\mathbb R}^2_{\mathbb Z}$. Let $\boldsymbol{w}\subset X$ be a conjugation invariant set of $|\partial P|+g-1$ points such that all conj

Figures (2)

  • Figure 1: The modification of a marked edge in Construction \ref{['con:modified_tropical_limit']} and Definition \ref{['def:modification_data']}.
  • Figure 2: An example of non-invariance of the signed count of real curves with tangency conditions. In the top row we depict the tropical curve and in the bottom row we depict the dual subdivisions. The black dots in the dual subdivision picture denote the integer points.

Theorems & Definitions (64)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Corollary 2.6
  • Definition 2.7
  • Lemma 2.8
  • Lemma 2.9
  • ...and 54 more