Binomial edge ideals and rational normal scrolls

TL;DR

This paper investigates binomial ideals generated by 2-minors associated with closed graphs, establishing their Cohen-Macaulay property, describing their minimal primes, and providing bounds on their regularity.

Contribution

It proves that these binomial ideals are Cohen-Macaulay, identifies their minimal primes, and offers a sharp upper bound for their regularity, linking graph theory and algebraic geometry.

Findings

$I_G$ is Cohen-Macaulay.

$I_G$ is a set-theoretic complete intersection.

A sharp upper bound for the regularity of $I_G$ is established.

Abstract

Let $X$ be the Hankel matrix of size $2\times n$ and let $G$ be a closed graph on the vertex set $[n].$ We study the binomial ideal $I_G\subset K[x_1,\ldots,x_{n+1}]$ which is generated by all the $2$-minors of $X$ which correspond to the edges of $G.$ We show that $I_G$ is Cohen-Macaulay. We find the minimal primes of $I_G$ and show that $I_G$ is a set theoretical complete intersection. Moreover, a sharp upper bound for the regularity of $I_G$ is given.