## 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 _{ℓ} (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 _{ℓ} (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 language | English |
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Pages (from-to) | 381-438 |

Number of pages | 58 |

Journal | Journal of Algebra |

Volume | 528 |

DOIs | |

State | Published - 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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