Binomial edge ideals and rational normal scrolls

Document Type: Research Paper

Authors

1 Abdus Salam School of Mathematical Sciences‎, ‎GC University‎, ‎68-B‎, ‎New Muslim Town‎, ‎Lahore 54600‎, ‎Pakistan

2 Faculty of Mathematics and Computer Science‎, ‎Ovidius University‎ ‎Bd‎. ‎Mamaia 124‎, ‎900527 Constanta, and Lumina-The University of South-East Europe‎ ‎Sos‎. ‎Colentina nr‎. ‎64b‎, ‎Bucharest‎, ‎Romania

3 Faculty of Mathematics and Computer Science‎, ‎Ovidius University‎, ‎Bd. Mamaia 124‎, ‎900527 Constanta‎, ‎Romania‎, ‎and‎ Simion Stoilow Institute of Mathematics of the Romanian Academy‎, ‎Research group of the project ID-PCE-2011-1023‎, ‎P.O.Box 1-764‎, ‎Bucharest 014700‎, ‎Romania

Abstract

‎Let $X=\left(‎
‎\begin{array}{llll}‎
‎ x_1 & \ldots & x_{n-1}& x_n\\‎
‎ x_2& \ldots & x_n & x_{n+1}‎
‎\end{array}\right)$ 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.‎

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