Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T08:40:19.305Z Has data issue: false hasContentIssue false
  • English
  • Français

Multiprojective spaces and the arithmetically Cohen–Macaulay property

Published online by Cambridge University Press:  03 April 2018

GIUSEPPE FAVACCHIO  and
JUAN MIGLIORE
GIUSEPPE FAVACCHIO
Affiliation:
Dipartimento di Matematica e Informatica, Università di Catania Viale A. Doria, 6 – 95100 – Catania, Italy. e-mail: [email protected]
JUAN MIGLIORE
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, U.S.A. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we study the arithmetically Cohen-Macaulay (ACM) property for sets of points in multiprojective spaces. Most of what is known is for ℙ1 × ℙ1 and, more recently, in (ℙ1)r. In ℙ1 × ℙ1 the so called inclusion property characterises the ACM property. We extend the definition in any multiprojective space and we prove that the inclusion property implies the ACM property in ℙm × ℙn. In such an ambient space it is equivalent to the so-called (⋆)-property. Moreover, we start an investigation of the ACM property in ℙ1 × ℙn. We give a new construction that highlights how different the behavior of the ACM property is in this setting.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 3 , May 2019 , pp. 583 - 597
Copyright
Copyright © Cambridge Philosophical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Abbott, J., Bigatti, A.M. and Robbiano, L. CoCoA: a system for doing computations in commutative algebra. Available at http://cocoa.dima.unige.itGoogle Scholar
[2] Favacchio, G., Guardo, E. and Migliore, J. On the arithmetically Cohen–Macaulay property for sets of points in multiprojective spaces. Proc. Amer. Math. Soc. To appear (2018). DOI: https://doi.org/10.1090/proc/13981Google Scholar
[3] Guardo, E. and Van Tuyl, A. ACM sets of points in multiprojective space. Collect. Math. 59 (2008), no. 2, 191–213.10.1007/BF03191367Google Scholar
[4] Guardo, E. and Van Tuyl, A. Arithmetically Cohen–Macaulay sets of points in ℙ1 × ℙ1. Springer Briefs in Mathematics (2015).Google Scholar
[5] Migliore, J. Introduction to liaison theory and deficiency modules. Progr. Math. 165 (1998).Google Scholar
[6] Migliore, J. and Nagel, U. Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers. Adv. Math. 180 (2003), 1–63.10.1016/S0001-8708(02)00079-8Google Scholar