First Stiefel-Whitney class of real moduli spaces of stable maps to a convex surface

TL;DR

This paper computes the first Stiefel-Whitney class of real moduli spaces of stable maps to convex surfaces, providing a homological description in specific cases, which advances understanding of their real algebraic geometry.

Contribution

It offers a homological description of the first Stiefel-Whitney class for real moduli spaces of stable maps to convex surfaces, specifically when the number of marked points is one less than the first Chern class times the degree.

Findings

Explicit representative of the first Stiefel-Whitney class in certain cases

Homological description involving boundary divisors

Extension of real algebraic geometry understanding for moduli spaces

Abstract

Let $(X,c_X)$ be a convex projective surface equipped with a real structure. The space of stable maps $\bar{\mathcal{M}}_{0,k}(X,d)$ carries different real structures induced by $c_X$ and any order two element $\tau$ of permutation group $S_k$ acting on marked points. Each corresponding real part $\R_{\tau}\bar{\mathcal{M}}_{0,k}(X,d)$ is a real normal projective variety. As the singular locus is of codimension bigger than two, these spaces thus carry a first Stiefel-Whitney class for which we determine a representative in the case $k=c_1(X)d-1$ where $c_1(X)$ is the first Chern class of $X$. Namely, we give a homological description of these classes in term of the real part of boundary divisors of the space of stable maps.