A Fano compactification of the $\mathrm{SL}_2(\mathbb{C})$ free group character variety

TL;DR

This paper proves that a specific compactification of the $ ext{SL}_2( ext{C})$ free group character variety is Fano, involving the construction of integral reflexive polytopes, advancing understanding of its geometric properties.

Contribution

It establishes the Fano property of a particular compactification of the $ ext{SL}_2( ext{C})$ free group character variety, using novel polytope constructions.

Findings

The compactification $rak{X}_g$ is Fano.

Construction of a large family of integral reflexive polytopes.

Connections to previous studies by other researchers.

Abstract

We show that a certain compactification $\mathfrak{X}_g$ of the $\mathrm{SL}_2(\mathbb{C})$ free group character variety $\mathcal{X}(F_g, \mathrm{SL}_2(\mathbb{C}))$ is Fano. This compactification has been studied previously by the second author, and separately by Biswas, Lawton, and Ramras. Part of the proof of this result involves the construction of a large family of integral reflexive polytopes.