An algorithm for the classification of twisted forms of toric varieties

TL;DR

This paper presents an algorithm to classify twisted forms of toric varieties over a base field by computing a specific Galois cohomology set, enabling systematic classification of various toric varieties.

Contribution

The paper introduces a novel algorithm for computing Galois cohomology sets that classify twisted forms of toric varieties, extending classification capabilities.

Findings

Algorithm effectively computes $H^1(G,Aut_{\sigma}^T)$ for classification.

Successfully classifies forms of toric surfaces and 3D affine toric varieties.

Applicable to cyclic Galois extensions for 3D quasi-projective toric varieties.

Abstract

Let $K/k$ be a finite Galois extension, $G=\text{Gal}(K/k)$, $\Sigma$ be a fan in a lattice $N$ and $X_{\Sigma}$ be an associated toric variety over $k$. It is well known that the set of $K/k$-forms of $X_{\Sigma}$ is in bijection with $H^1(G,\text{Aut}_{\Sigma}^T)$, where $\text{Aut}_{\Sigma}^T$ is an algebraic group of toric automorphisms of $X_{\Sigma}$. In this paper, we suggest an algorithm to compute $H^1(G,\text{Aut}_{\Sigma}^T)$ and find that followings can be classified via this algorithm : $K/k$-forms of all toric surfaces, $K/k$-forms of all 3-dimensional affine toric varieties with no torus factor, $K/k$-forms of all 3-dimensional quasi-projective toric varieties when $K/k$ is cyclic.