The symbolic defect of an ideal
| Autor: | Galetto, Federico, Geramita, Anthony V., Shin, Yong-Su, Van Tuyl, Adam |
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| Rok vydání: | 2016 |
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| Druh dokumentu: | Working Paper |
| Popis: | Let $I$ be a homogeneous ideal of $\Bbbk[x_0,\ldots,x_n]$. To compare $I^{(m)}$, the $m$-th symbolic power of $I$, with $I^m$, the regular $m$-th power, we introduce the $m$-th symbolic defect of $I$, denoted $\operatorname{sdefect}(I,m)$. Precisely, $\operatorname{sdefect}(I,m)$ is the minimal number of generators of the $R$-module $I^{(m)}/I^m$, or equivalently, the minimal number of generators one must add to $I^m$ to make $I^{(m)}$. In this paper, we take the first step towards understanding the symbolic defect by considering the case that $I$ is either the defining ideal of a star configuration or the ideal associated to a finite set of points in $\mathbb{P}^2$. We are specifically interested in identifying ideals $I$ with $\operatorname{sdefect}(I,2) = 1$. Comment: To appear in Journal of Pure and Applied Algebra; revised at referees' suggestion. Fixed typos and clarified writing, included additional references, shortened proof of Thm 6.3 |
| Databáze: | arXiv |
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