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


Authors: Giuseppe Favacchio, Elena Guardo and Juan Migliore
Journal: Proc. Amer. Math. Soc. 146 (2018), 2811-2825
MSC (2010): Primary 13C40, 13C14, 13A15, 14M05
DOI: https://doi.org/10.1090/proc/13981
Published electronically: February 21, 2018
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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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Additional Information

Giuseppe Favacchio
Affiliation: Dipartimento di Matematica e Informatica, Viale A. Doria, 6 - 95100 - Catania, Italy
Email: favacchio@dmi.unict.it

Elena Guardo
Affiliation: Dipartimento di Matematica e Informatica, Viale A. Doria, 6 - 95100 - Catania, Italy
Email: guardo@dmi.unict.it

Juan Migliore
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: migliore.1@nd.edu

DOI: https://doi.org/10.1090/proc/13981
Keywords: Points in multiprojective spaces, arithmetically Cohen-Macaulay, linkage
Received by editor(s): February 6, 2017
Received by editor(s) in revised form: August 15, 2017, and October 8, 2017
Published electronically: February 21, 2018
Communicated by: Irena Peeva
Article copyright: © Copyright 2018 American Mathematical Society

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