BPS polynomials and Welschinger invariants

TL;DR

This paper extends Block-G"ottsche polynomials to all surfaces via BPS polynomials, and conjectures their evaluation at q=-1 gives Welschinger invariants, which is proven for certain blow-up surfaces.

Contribution

It introduces BPS polynomials for arbitrary surfaces and establishes a link to Welschinger invariants for specific blow-up surfaces, advancing the understanding of real enumerative geometry.

Findings

BPS polynomials generalize Block-G"ottsche polynomials to all surfaces.

Conjecture that evaluation at q=-1 yields Welschinger invariants.

Proven the conjecture for surfaces obtained by up to 6 blow-ups of P^2.

Abstract

We generalize Block-G\"ottsche polynomials, originally defined for toric del Pezzo surfaces, to arbitrary surfaces. To do this, we show that these polynomials arise as special cases of BPS polynomials, defined for any surface $S$ as Laurent polynomials in a formal variable $q$ encoding the BPS invariants of the $3$-fold $S \times \mathbb{P}^1$. We conjecture that for surfaces $S_n$ obtained by blowing up $\mathbb{P}^2$ at $n$ general points, the evaluation of BPS polynomials at $q=-1$ yields Welschinger invariants, given by signed counts of real rational curves. We prove this conjecture for all surfaces $S_n$ with $n \leq 6$.