Some homogeneous coordinate rings that are Koszul algebras

TL;DR

This paper demonstrates that homogeneous coordinate rings of certain algebraic varieties, specifically proper smooth toric and Schubert varieties, are Koszul algebras by employing Frobenius splitting techniques in positive characteristic.

Contribution

It introduces a novel application of Frobenius splitting to prove Koszulity of coordinate rings for these classes of varieties.

Findings

Homogeneous coordinate rings of proper smooth toric varieties are Koszul.

Homogeneous coordinate rings of Schubert varieties are Koszul.

Frobenius splitting in positive characteristic is effective for establishing Koszulity.

Abstract

Using reduction to positive characteristic and the method of Frobenius splitting of diagonals, due to Mehta and Ramanathan, it is shown that homogeneous coordinate rings for either proper and smooth toric varieties or Schubert varieties are Koszul algebras.