Secant varieties of the varieties of reducible hypersurfaces in P n

M. V. Catalisano, A. V. Geramita, A. Gimigliano, B. Harbourne, J. Migliore, U. Nagel, Y. S. Shin

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

Abstract

Given the space V=P (d+n−1n−1)−1 of forms of degree d in n variables, and given an integer ℓ>1 and a partition λ of d=d 1 +⋯+d r , it is in general an open problem to obtain the dimensions of the (ℓ−1)-secant varieties σ (X n−1,λ ) for the subvariety X n−1,λ ⊂V of hypersurfaces whose defining forms have a factorization into forms of degrees d 1 ,…,d r . Modifying a method from intersection theory, we relate this problem to the study of the Weak Lefschetz Property for a class of graded algebras, based on which we give a conjectural formula for the dimension of σ (X n−1,λ ) for any choice of parameters n,ℓ and λ. This conjecture gives a unifying framework subsuming all known results. Moreover, we unconditionally prove the formula in many cases, considerably extending previous results, as a consequence of which we verify many special cases of previously posed conjectures for dimensions of secant varieties of Segre varieties. In the special case of a partition with two parts (i.e., r=2), we also relate this problem to a conjecture by Fröberg on the Hilbert function of an ideal generated by general forms.

Original languageEnglish
Pages (from-to)381-438
Number of pages58
JournalJournal of Algebra
Volume528
DOIs
StatePublished - Jun 15 2019

Bibliographical note

Publisher Copyright:
© 2019 Elsevier Inc.

Keywords

  • Fröberg's Conjecture
  • Intersection theory
  • Secant variety
  • Variety of reducible forms
  • Variety of reducible hypersurfaces
  • Weak Lefschetz Property

ASJC Scopus subject areas

  • Algebra and Number Theory

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