# The Rees algebra of a two-Borel ideal is Koszul

## TL;DR

This paper proves that the Rees algebra of a two-Borel ideal is Koszul by constructing a quadratic Gr"obner basis, using graph constructions related to fibers of the toric map, thus establishing Koszulness of the associated toric ring and Rees algebra.

## Contribution

It introduces a method to prove the Koszul property of Rees algebras of two-Borel ideals via quadratic Gr"obner bases and graph constructions.

## Key findings

- The toric ring of a two-Borel ideal is Koszul.
- The Rees algebra of a two-Borel ideal is Koszul.
- A quadratic Gr"obner basis can be constructed for the toric ideal.

## Abstract

Let $M$ and $N$ be two monomials of the same degree, and let $I$ be the smallest Borel ideal containing $M$ and $N$. We show that the toric ring of $I$ is Koszul by constructing a quadratic Gr\"obner basis for the associated toric ideal. Our proofs use the construction of graphs corresponding to fibers of the toric map. As a consequence, we conclude that the Rees algebra is also Koszul.