Vanishing Hessian, wild forms and their border VSP

Huang, Hang; Michałek, Mateusz; Ventura, Emanuele (2020). Vanishing Hessian, wild forms and their border VSP. Mathematische Annalen, 378(3-4), pp. 1505-1532. Springer 10.1007/s00208-020-02080-8

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Wild forms are homogeneous polynomials whose smoothable rank is strictly larger than their border rank. The discrepancy between these two ranks is caused by the difference between the limit of spans of a family of zero-dimensional schemes and the span of their flat limit. For concise forms of minimal border rank, we show that the condition of vanishing Hessian is equivalent to being wild. This is proven by making a detour through structure tensors of smoothable and Gorenstein algebras. The equivalence fails in the non-minimal border rank regime. We exhibit an infinite series of minimal border rank wild forms of every degree d ≥ 3 as well as an infinite series of wild cubics. Inspired by recent work on border apolarity of Buczyńska and Buczyński, we study the border varieties of sums of powers VSP of these forms in the corresponding multigraded Hilbert schemes.

Item Type:

Journal Article (Original Article)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Ventura, Emanuele

Subjects:

500 Science > 510 Mathematics

ISSN:

0025-5831

Publisher:

Springer

Funders:

[UNSPECIFIED] Netherlands Organisation for Scientific Research

Language:

English

Submitter:

Sebastiano Don

Date Deposited:

10 Feb 2021 15:15

Last Modified:

10 Feb 2021 15:17

Publisher DOI:

10.1007/s00208-020-02080-8

ArXiv ID:

1912.13174

Uncontrolled Keywords:

14C05, 15A69

BORIS DOI:

10.48350/151278

URI:

https://boris.unibe.ch/id/eprint/151278

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