## 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 |
---|---|

Pages (from-to) | 381-438 |

Number of pages | 58 |

Journal | Journal of Algebra |

Volume | 528 |

DOIs | |

State | Published - Jun 15 2019 |

### Bibliographical note

Funding Information:The authors wish to thank Queen's University and NSERC (Canada), in the person of the second author, for kind hospitality during the preparation of this work. Catalisano and Gimigliano were partially supported by GNSAGA of INDAM (Italy) under grant No. U2015/000313, and by MIUR (Italy) under grant No. PRIN 2010-11 prot. 2010S47ARA-004 - Geometria delle Varietà Algebriche. Geramita was partially supported by NSERC (Canada) under grant No. 386080, while Harbourne was partially supported by NSA (US) under grant No. H98230-13-1-0213. Both Migliore and Nagel were partially supported by the Simons Foundation (US) under grants No. 309556 (Migliore) and 317096 (Nagel). Shin was supported by the Basic Science Research Program of the NRF (Korea) under grant No. 2013R1A1A2058240/2. The authors are also grateful to the referees for helpful suggestions and comments.

Funding Information:

Catalisano and Gimigliano were partially supported by GNSAGA of INDAM (Italy) under grant No. U2015/000313 , and by MIUR (Italy) under grant No. PRIN 2010-11 prot. 2010S47ARA-004 - Geometria delle Varietà Algebriche. Geramita was partially supported by NSERC (Canada) under grant No. 386080 , while Harbourne was partially supported by NSA (US) under grant No. H98230-13-1-0213 . Both Migliore and Nagel were partially supported by the Simons Foundation (US) under grants No. 309556 (Migliore) and 317096 (Nagel). Shin was supported by the Basic Science Research Program of the NRF (Korea) under grant No. 2013R1A1A2058240/2 .

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

## Fingerprint

Dive into the research topics of 'Secant varieties of the varieties of reducible hypersurfaces in P^{n}'. Together they form a unique fingerprint.