Toric surfaces over an arbitrary field

TL;DR

This paper investigates toric surfaces over arbitrary fields, focusing on their decomposition in the motivic category and linking algebraic, geometric, and categorical properties.

Contribution

It classifies all minimal smooth projective toric surfaces and connects their K-motive decompositions with derived category structures.

Findings

Decomposition of toric varieties into products of central simple algebras

Explicit classification of minimal smooth projective toric surfaces

Relationship between K-motive decomposition and semiorthogonal decompositions

Abstract

We study toric varieties over an arbitrary field with an emphasis on toric surfaces in the Merkurjev-Panin motivic category of "K-motives". We explore the decomposition of certain toric varieties as K-motives into products of central simple algebras, the geometric and topological information encoded in these central simple algebras, and the relationship between the decomposition of the K-motives and the semiorthogonal decomposition of the derived categories. We obtain the information mentioned above for toric surfaces by explicitly classifying all minimal smooth projective toric surfaces using toric geometry.