On the arithmetically Cohen-Macaulay property for sets of points in multiprojective spaces

Favacchio, Giuseppe; Guardo, Elena; Migliore, Juan

doi:10.1090/proc/13981

On the arithmetically Cohen-Macaulay property for sets of points in multiprojective spaces
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by Giuseppe Favacchio, Elena Guardo and Juan Migliore PDF

Proc. Amer. Math. Soc. 146 (2018), 2811-2825 Request permission

Abstract:

We study the arithmetically Cohen-Macaulay (ACM) property for finite sets of points in multiprojective spaces, especially $(\mathbb P^1)^n$. A combinatorial characterization, the $(\star )$-property, is known in $\mathbb P^1 \times \mathbb P^1$. We propose a combinatorial property, $(\star _s)$ with $2\leq s\leq n$, that directly generalizes the $(\star )$-property to $(\mathbb P^1)^n$ for larger $n$. We show that $X$ is ACM if and only if it satisfies the $(\star _n)$-property. The main tool for several of our results is an extension to the multiprojective setting of certain liaison methods in projective space.

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