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arXiv · 2303.05190

Componentwise Linearity Under Square-Free Gr\"obner Degenerations

Hongmiao Yu
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Abstract

Using the recent results on square-free Gr\"obner degenerations by Conca and Varbaro, we proved that if a homogeneous ideal $I$ of a polynomial ring is such that its initial ideal $\mathrm{in}_<(I)$ is square-free and $\beta_0(I) = \beta_0(\mathrm{in}_<(I))$, then $I$ is a componentwise linear ideal if and only if $\mathrm{in}_<(I)$ is a componentwise linear ideal. In particular, if furthermore one of $I$ and $\mathrm{in}_<(I)$ is componentwise linear, then their graded Betti numbers coincide.

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Hongmiao Yu. 2023-03-09. Componentwise Linearity Under Square-Free Gr\"obner Degenerations. https://arxiv.org/abs/2303.05190

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