The isomorphism problem of projective schemes and related algorithmic problems

TL;DR

This paper investigates the algorithmic problem of determining isomorphism between projective schemes, providing positive results for specific classes such as one-dimensional schemes, certain varieties, and K3 surfaces with finite automorphism groups.

Contribution

It offers new algorithms and decidability results for the isomorphism problem in particular classes of projective schemes and related positivity and cone approximation problems.

Findings

Decidability for one-dimensional projective schemes.

Decidability for smooth irreducible varieties with big canonical or anti-canonical sheaves.

Decidability for K3 surfaces with finite automorphism groups.

Abstract

We discuss the isomorphism problem of projective schemes; given two projective schemes, can we algorithmically decide whether they are isomorphic? We give affirmative answers in the case of one-dimensional projective schemes, the case of smooth irreducible varieties with a big canonical sheaf or a big anti-canonical sheaf, and the case of K3 surfaces with a finite automorphism group. As related algorithmic problems, we also discuss decidability of positivity properties of invertible sheaves, and approximation of the nef cone and the pseudo-effective cone.