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arXiv · 1302.0734

Vanishing ideals over complete multipartite graphs

Jorge Neves·Maria Vaz Pinto
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Abstract

We study the vanishing ideal of the parametrized algebraic toric associated to the complete multipartite graph $\G=\mathcal{K}_{α_1,...,α_r}$ over a finite field of order $q$. We give an explicit family of binomial generators for this lattice ideal, consisting of the generators of the ideal of the torus, (referred to as type I generators), a set of quadratic binomials corresponding to the cycles of length 4 in $\G$ and which generate the \emph{toric algebra of $\G$} (type II generators) and a set of binomials of degree $q-1$ obtained combinatorially from $\G$ (type III generators). Using this explicit family of generators of the ideal, we show that its Castelnuovo--Mumford regularity is equal to $\max\set{α_1(q-2),...,α_r(q-2), \lceil (n-1)(q-2)/2\rceil}$, where $n=α_1+... + α_r$.

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Jorge Neves, Maria Vaz Pinto. 2013-09-28. Vanishing ideals over complete multipartite graphs. https://arxiv.org/abs/1302.0734

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