Normality and quadraticity for special ample line bundles on toric varieties arising from root systems

TL;DR

This paper proves that certain special ample line bundles on toric varieties from root systems are projectively normal and have quadratic semigroup rings, advancing understanding of their algebraic properties.

Contribution

It establishes projective normality and quadraticity of semigroup rings for special ample line bundles on toric varieties from root systems, a novel result in this context.

Findings

Special ample line bundles are projectively normal.

Semigroup rings associated are quadratic.

Results apply to toric varieties from root systems.

Abstract

We prove that special ample line bundles on toric varieties arising from root systems are projectively normal. Here the maximal cones of the fans correspond to the Weyl chambers, and special means that the bundle is torus-equivariant such that the character of the line bundle that corresponds to a maximal Weyl chamber is dominant with respect to that chamber. Moreover, we prove that the associated semigroup rings are quadratic.