Computing Galois groups of Fano problems

TL;DR

This paper investigates the Galois groups associated with Fano problems, which involve enumerating linear spaces on varieties, and uses computational methods to determine these groups for several moderate-sized problems, revealing new cases where the Galois group is the full symmetric group.

Contribution

The paper applies computational techniques to explicitly compute Galois groups of certain Fano problems, expanding known cases where the Galois group is the symmetric group.

Findings

Several Fano problems have Galois group equal to the symmetric group

Previous classifications showed most Galois groups contain the alternating group

New computational results identify specific problems with full symmetric Galois groups

Abstract

A Fano problem consists of enumerating linear spaces of a fixed dimension on a variety, generalizing the classical problem of 27 lines on a cubic surface. Those Fano problems with finitely many linear spaces have an associated Galois group that acts on these linear spaces and controls the complexity of computing them in coordinates via radicals. Galois groups of Fano problems were first studied by Jordan, who considered the Galois group of the problem of 27 lines on a cubic surface. Recently, Hashimoto and Kadets nearly classified all Galois groups of Fano problems by determining them in a special case and by showing that all other Fano problems have Galois group containing the alternating group. We use computational tools to prove that several Fano problems of moderate size have Galois group equal to the symmetric group, each of which were previously unknown.