On the irreducible components of moduli schemes for affine spherical varieties

TL;DR

This paper provides a combinatorial framework to classify affine spherical varieties with a given weight monoid and analyzes the structure of their moduli schemes, including conditions for irreducibility and examples of reducibility and non-reduced schemes.

Contribution

It introduces a combinatorial description of affine spherical varieties with prescribed weight monoids and characterizes the irreducible components of their moduli schemes.

Findings

Characterization of irreducible components of moduli schemes

Conditions for irreducibility of the moduli scheme

Examples of reducible and non-reduced moduli schemes

Abstract

We give a combinatorial description of all affine spherical varieties with prescribed weight monoid $\Gamma$. As an application, we obtain a characterization of the irreducible components of Alexeev and Brion's moduli scheme $\mathrm M_\Gamma$ for such varieties. Moreover, we find several sufficient conditions for $\mathrm M_\Gamma$ to be irreducible and exhibit several examples where $\mathrm M_\Gamma$ is reducible. Finally, we provide examples of non-reduced $\mathrm M_\Gamma$.