Orbit and Orbit Closure Containments for Cubic Surfaces

TL;DR

This paper develops techniques to analyze orbit and orbit closure containments for cubic surfaces, providing a partial classification in complex cases with many singular points and discussing computational challenges.

Contribution

It introduces optimized algorithms for orbit closure problems and offers a partial classification for cubic surfaces with numerous singularities, highlighting computational obstacles.

Findings

Partial classification of orbit closures for cubic surfaces with many singular points

Development of optimized algorithms for orbit containment problems

Discussion of computational challenges in classifying orbit closures

Abstract

Given two elements of a vector space acted on by a reductive group, we ask whether they lie in the same orbit, and if not, whether one lies in the orbit closure of the other. We develop techniques to optimize the orbit and orbit closure algorithms and apply these to give a partial classification of orbit closure containments in the case of cubic surfaces with infinitely many singular points, which are known to fall into 13 normal forms. We also discuss the computational obstructions to completing this classification, and discuss tools for future work in this direction.