On derived categories of arithmetic toric varieties

TL;DR

This paper systematically studies the derived categories of smooth projective toric varieties over arbitrary fields, demonstrating the existence of full exceptional collections in many cases, including all toric surfaces and certain Fano varieties.

Contribution

It introduces a general Galois descent method for exceptional collections, enabling the analysis of derived categories over non-closed fields and expanding known cases of full exceptional collections.

Findings

All toric surfaces admit full exceptional collections.

Many toric Fano 3-folds and some Fano 4-folds have full exceptional collections.

A Galois descent technique is developed for exceptional collections over non-closed fields.

Abstract

We begin a systematic investigation of derived categories of smooth projective toric varieties defined over an arbitrary base field. We show that, in many cases, toric varieties admit full exceptional collections. Examples include all toric surfaces, all toric Fano 3-folds, some toric Fano 4-folds, the generalized del Pezzo varieties of Voskresenskii and Klyachko, and toric varieties associated to Weyl fans of type $A$. Our main technical tool is a completely general Galois descent result for exceptional collections of objects on (possibly non-toric) varieties over non-closed fields.