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arXiv · 1309.5082

Symbolic Powers of Monomial Ideals

Susan M. Cooper·Robert J. D. Embree·Huy Tài Hà·Andrew H. Hoefel
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Abstract

We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal $I$ in $k[x_0, \ldots, x_n]$ we show $I^{t(m+e-1)-e+r)}$ is a subset of $M^{(t-1)(e-1)+r-1}(I^{(m)})^t$ for all positive integers $m$, $t$ and $r$, where $e$ is the big-height of $I$ and $M = (x_0, \ldots, x_n)$. This captures two conjectures ($r=1$ and $r=e$): one of Harbourne-Huneke and one of Bocci-Cooper-Harbourne. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals.

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Susan M. Cooper, Robert J. D. Embree, Huy Tài Hà, Andrew H. Hoefel. 2013-09-19. Symbolic Powers of Monomial Ideals. https://arxiv.org/abs/1309.5082

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