Computing automorphisms of Mori dream spaces

TL;DR

This paper introduces an algorithm for computing the automorphism groups of Mori dream spaces, with applications to singular cubic surfaces and potential uses in symmetry-based computational methods.

Contribution

It develops a new algorithm leveraging graded automorphisms of affine algebras and Cox rings to determine automorphism groups of Mori dream spaces.

Findings

Successfully computed automorphism groups of singular cubic surfaces

Demonstrated the algorithm's applicability to Cox rings and affine algebras

Potential for use in symmetry-based computational techniques

Abstract

We present an algorithm to compute the automorphism group of a Mori dream space. As an example calculation, we determine the automorphism groups of singular cubic surfaces with general parameters. The strategy is to study graded automorphisms of affine algebras graded by a finitely generated abelian groups and apply the results to the Cox ring. Besides the application to Mori dream spaces, our results could be used for symmetry based computing, e.g. for Gr\"obner bases or tropical varieties.