An algorithm to compute automorphism groups of K3 surfaces and an application to singular K3 surfaces

TL;DR

This paper introduces an algorithm for computing automorphism groups of K3 surfaces, enabling explicit calculations for complex and supersingular cases, with applications to singular K3 surfaces with small discriminants.

Contribution

The paper presents a novel algorithm to determine generators of automorphism groups of K3 surfaces, advancing computational methods in algebraic geometry.

Findings

Successfully computed automorphism groups for specific K3 surfaces

Applied the algorithm to singular K3 surfaces with small discriminants

Demonstrated the effectiveness of the method in complex and supersingular cases

Abstract

Let X be a complex algebraic K3 surface or a supersingular K3 surface in odd characteristic. We present an algorithm by which, under certain assumptions on X, we can calculate a finite set of generators of the image of the natural homomorphism from the automorphism group of X to the orthogonal group of the N\'eron-Severi lattice of X. We then apply this algorithm to certain complex K3 surfaces, among which are singular K3 surfaces whose transcendental lattices are of small discriminants.