$\mathbb{Z}/2\mathbb{Z}$-Equivariant smoothings of cusp singularities

TL;DR

This paper establishes conditions under which cusp and certain elliptic singularities with antisymplectic involutions can be smoothly deformed in a way compatible with the involution, linking geometric structures to smoothability.

Contribution

It provides new criteria for $bZ/2bZ$-equivariant smoothability of cusp and elliptic singularities using Looijenga pairs and involution properties.

Findings

A sufficient condition for equivariant smoothability of cusp singularities with antisymplectic involution.

An analogue necessary and sufficient condition for $bZ/2bZ$-equivariant smoothability of simple elliptic singularities.

Characterization of involution properties on Looijenga pairs related to smoothability.

Abstract

Let $p\in X$ be the germ of a cusp singularity and let $\iota$ be an antisymplectic involution, that is an involution such that there exists a nowhere vanishing holomorphic 2-form $\Omega$ on $X\setminus \{p\}$ for which $\iota^*(\Omega)=-\Omega$. Assume also that the involution is fixed point free on $X\setminus\{p\}$. We prove that a sufficient condition for such a singularity equipped with an antisymplectic involution to be equivariantly smoothable is the existence of a Looijenga (or anticanonical) pair $(Y,D)$ that admits an involution free on $Y\setminus D$ and that reverses the orientation of $D$. This work also contains the proof of an analogue necessary and sufficient condition for the $\mathbb{Z}/2\mathbb{Z}$-equivariant smoothability of simple elliptic singularities $p\in C(E)$ with $E$ an elliptic curve of degree $d\leq 8$ and even equipped with a $\mathbb{Z}/2\mathbb{Z}$-action.