Secant varieties of the varieties of reducible hypersurfaces in Pn

JOURNAL OF ALGEBRA

卷:528

Article

Catalisano, M. V.¹  Geramita, A. V.^2,3  Gimigliano, A.^4,5  Harbourne, B.⁶  Migliore, J.⁷  Nagel, U.⁸  Shin, Y. S.^9,10 

[1] Univ Genoa, Dipartimento Ingn Meccan Energet Gest & Trasporti, Genoa, Italy

[2] Queens Univ, Dept Math & Stat, Kingston, ON, Canada

[3] Univ Genoa, Dipartimento Matemat, Genoa, Italy

[4] Univ Bologna, Dipartimento Matemat, Bologna, Italy

[5] Univ Bologna, CIRAM, Bologna, Italy

[6] Univ Nebraska, Dept Math, Lincoln, NE 68588 USA

[7] Univ Notre Dame, Dept Math, Notre Dame, IN 46556 USA

[8] Univ Kentucky, Dept Math, 715 Patterson Off Tower, Lexington, KY 40506 USA

[9] Sungshin Womens Univ, Dept Math, Seoul 02844, South Korea

[10] KIAS, Seoul 02455, South Korea

关键词: Secant variety;    Variety of reducible hypersurfaces;    Variety of reducible forms;    Intersection theory;    Weak Lefschetz Property;    Froberg's Conjecture;   

DOI  :  10.1016/j.jalgebra.2019.03.014

来源: Elsevier

【 摘 要 】

Given the space V = P((d+n-1)(n-)(1))(-1) of forms of degree d in n variables, and given an integer l > 1 and a partition lambda of d = d(1) + ... + d(r), it is in general an open problem to obtain the dimensions of the (l - 1)-secant varieties sigma(l)(X-n-1,X-lambda) for the subvariety X-n-1,X-lambda subset of 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 sigma(l)(X-n-1,X-lambda) for any choice of parameters n, l and lambda. 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 Froberg on the Hilbert function of an ideal generated by general forms. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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