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arXiv · 2511.11376

A level initial ideal of the 2-minors determinantal ideal

Francesco Bisio
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Abstract

For $\Bbbk$ a field, let $X$ a $m \times n$ matrix of variables and $S=\Bbbk[X].$ We consider the determinantal ideal $I_2 \subseteq S$ generated by the $2$-minors of $X.$ In this paper we find a suitable monomial order over $S$ such that $I$, the initial ideal of $I_2$ with respect to that order, is level, namely, it is Cohen-Macaulay and the socle of an Artinian reduction of the $\mathbb{N}$-graded algebra $S/I$ is concentrated in only one degree. Moreover, we compare the Betti tables of $I_2$ with the tables of its initial ideals. In the last section, we prove the shellability of the simplicial complex naturally associated to $S/I$ in the case $m<n.$

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Francesco Bisio. 2025-11-14. A level initial ideal of the 2-minors determinantal ideal. https://arxiv.org/abs/2511.11376

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