A level initial ideal of the 2-minors determinantal ideal
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914158642987008 |
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| author | Bisio, Francesco |
| author_facet | Bisio, Francesco |
| contents | 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 $m<n.$ |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_11376 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A level initial ideal of the 2-minors determinantal ideal Bisio, Francesco Commutative Algebra 13F55 (Primary) 13C40, 13P10 (Secondary) 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 $m<n.$ |
| title | A level initial ideal of the 2-minors determinantal ideal |
| topic | Commutative Algebra 13F55 (Primary) 13C40, 13P10 (Secondary) |
| url | https://arxiv.org/abs/2511.11376 |