A formula for symbolic powers

Paolo Mantero, Cleto B. Miranda-Neto, Uwe Nagel

Research output: Contribution to journalArticlepeer-review

Abstract

Let S be a Cohen-Macaulay ring which is local or standard graded over a field, and let I be an unmixed ideal that is generically a complete intersection. Our goal in this paper is multi-fold. First, we give a multiplicity-based characterization of when an unmixed subideal J⊆I(m) equals the m-th symbolic power I(m) of I. Second, we provide a saturation-type formula to compute I(m) and employ it to deduce a theoretical criterion for when I(m)=Im. Third, we establish an explicit linear bound on the exponent that makes the saturation formula effective, and use it to obtain lower bounds for the initial degree of I(m). Along the way, we prove a generalized version of a conjecture raised by Eisenbud and Mazur about annS(I(m)/Im), and we propose a conjecture connecting the symbolic defect of an ideal to Jacobian ideals.

Original languageEnglish
Pages (from-to)1-21
Number of pages21
JournalJournal of Algebra
Volume658
DOIs
StatePublished - Nov 15 2024

Bibliographical note

Publisher Copyright:
© 2024 Elsevier Inc.

Keywords

  • Jacobian ideal
  • Powers of ideals
  • Symbolic defect
  • Symbolic power

ASJC Scopus subject areas

  • Algebra and Number Theory

Fingerprint

Dive into the research topics of 'A formula for symbolic powers'. Together they form a unique fingerprint.

Cite this