Witt groups of smooth toric surfaces

Abstract

In this paper, we calculate the Witt groups of smooth projective toric surfaces over a field of characteristic different from 2. Such a surface is described combinatorially by a fan in the plan. The result is a direct sum of several copies of the Witt group of the basic field, their number depends on the line bundle used in the definition of Witt groups. The technique of the proofs is to filter the derived category of the surface by subcategories with support in unions of orbits closures whose dimension is 1. Then, by excision, we obtain long exact sequences including copies of Witt groups of the basic field.