ON THE WEAK LEFSCHETZ PROPERTY FOR HEIGHT FOUR EQUIGENERATED COMPLETE INTERSECTIONS

Mats Boij, Juan Migliore, Rosa M. Miró-Roig, Uwe Nagel

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

We consider the conjecture that all artinian height 4 complete intersections of forms of the same degree d have the Weak Lefschetz Property (WLP). We translate this problem to one of studying the general hyperplane section of a certain smooth curve in P3, and our main tools are the Socle Lemma of Huneke and Ulrich together with a careful liaison argument. Our main results are (i) a proof that the property holds for d = 3, 4 and 5; (ii) a partial result showing maximal rank in a non-trivial but incomplete range, cutting in half the previous unknown range; and (iii) a proof that maximal rank holds in a different range, even without assuming that all the generators have the same degree. We furthermore conjecture that if there were to exist any height 4 complete intersection generated by forms of the same degree and failing WLP then there must exist one (not necessarily the same one) failing by exactly one (in a sense that we make precise). Based on this conjecture we outline an approach to proving WLP for all equigenerated complete intersections in four variables. Finally, we apply our results to the Jacobian ideal of a smooth surface in P3.

Original languageEnglish
Pages (from-to)1254-1286
Number of pages33
JournalTransactions of the American Mathematical Society Series B
Volume10
DOIs
StatePublished - 2023

Bibliographical note

Publisher Copyright:
©2023 by the author(s) under Creative Commons.

ASJC Scopus subject areas

  • Mathematics (miscellaneous)

Fingerprint

Dive into the research topics of 'ON THE WEAK LEFSCHETZ PROPERTY FOR HEIGHT FOUR EQUIGENERATED COMPLETE INTERSECTIONS'. Together they form a unique fingerprint.

Cite this