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Perazzo hypersurfaces and the weak Lefschetz property.
- Source :
-
Journal of Algebra . May2024, Vol. 646, p357-375. 19p. - Publication Year :
- 2024
-
Abstract
- We deal with Perazzo hypersurfaces X = V (f) in P n + 2 defined by a homogeneous polynomial f (x 0 , x 1 , ... , x n , u , v) = p 0 (u , v) x 0 + p 1 (u , v) x 1 + ⋯ + p n (u , v) x n + g (u , v) , where p 0 , p 1 , ... , p n are algebraically dependent but linearly independent forms of degree d − 1 in K [ u , v ] and g is a form in K [ u , v ] of degree d. Perazzo hypersurfaces have vanishing hessian and, hence, the associated graded artinian Gorenstein algebra A f fails the strong Lefschetz property. In this paper, we first determine the maximum and minimum Hilbert function of A f , we prove that the Hilbert function of A f is always unimodal and we determine when A f satisfies the weak Lefschetz property. We illustrate our results with many examples and we show that our results do not generalize to Perazzo hypersurfaces X = V (f) in P n + 3 defined by a homogeneous polynomial f (x 0 , x 1 , ... , x n , u , v , w) = p 0 (u , v , w) x 0 + p 1 (u , v , w) x 1 + ⋯ + p n (u , v , w) x n + g (u , v , w) , where p 0 , p 1 , ... , p n are algebraically dependent but linearly independent forms of degree d − 1 in K [ u , v , w ] and g is a form in K [ u , v , w ] of degree d. [ABSTRACT FROM AUTHOR]
- Subjects :
- *HILBERT functions
*HOMOGENEOUS polynomials
*ALGEBRA
*HYPERSURFACES
Subjects
Details
- Language :
- English
- ISSN :
- 00218693
- Volume :
- 646
- Database :
- Academic Search Index
- Journal :
- Journal of Algebra
- Publication Type :
- Academic Journal
- Accession number :
- 175985253
- Full Text :
- https://doi.org/10.1016/j.jalgebra.2024.02.009