On non-defectivity of certain Segre-Veronese varieties
type: Item
instance of: scholarly article
title: On non-defectivity of certain Segre-Veronese varieties
zbMATH Open document ID: 1207.14052
author: Hirotachi Abo
published in: Journal of Symbolic Computation
publication date: 2010-11-19T00:00:00Z
review text: The literature on the dimensions of the secant varieties of the two factors Segre-Veronese varieties begin with [\textit{F. London}, Math. Ann. 36, 535-584, Habil.-Schrift Breslau (1890; JFM 22.0735.03)] that studied the case of \(\mathbb{P}^1\times \mathbb{P}^2\) embedded via \(\mathcal{O}(1,3)\). Then many authors developed different techniques to approach this kind of problem. Nevertheless nowadays the classification of defective secant varieties of two factors Segre-Veronese varieties is a problem that in its complete generality is still open. The author of this paper focus his attention on the secant varieties of \(\mathbb{P}^m \times \mathbb{P}^n\), with \(m=n,n+1\), embedded by the section of the sheaf \(\mathcal{O}(1,2)\). For these varieties he uses the notation \(\sigma_s(X_{m,n})\), i.e. a generic element in \(\sigma_s(X_{m,n})\) can be written as a linear combination of \(s\) points of the Segre-Veronse \(X_{m,n}\) obtained embedding \(\mathbb{P}^m \times \mathbb{P}^n\) with \(\mathcal{O}(1,2)\). In a previous paper [\textit{H. Abo, M. C. Brambilla}, Exp. Math. 18, No. 3, 369--384 (2009; Zbl 1198.14051)] the two authors come up with partial results and a conjecture in which they assert that the only defective \(\sigma_s(X_{m,n})\) are those already found in [\textit{E. Carlini, J. Chiapalkatti}, Comment. Math. Helv. 78, No. 3, 494--517 (2003; Zbl 1052.14064)] and in [\textit{G. Ottaviani} Symplectic bundles on the plane, secant varieties and Lüroth quartics revisited. Vector bundles and low codimensional subvarieties: state of the art and recent developments, 315--352, Quad. Mat. 21, Dep. Math., Seconda Univ. di Napoli, Caserta, (2007)]. The main goal of the paper under review is to provide further evidence in support of this conjecture. With a successful merging of computational \texttt{Macaulay2} tools and the classical Horace technique applied to Terracini's Lemma and the Apolarity theory, the author of this paper can prove that \(\sigma_s(X_{n,n})\) …
reviewed by: Alessandra Bernardi
Mathematics Subject Classification ID: 14N05
zbMATH DE Number: 5817621
zbMATH Keywords: Secant varieties
zbMATH Keywords: Segre-Veronese varieties
zbMATH Keywords: Non-defectivity
describes a project that uses: Macaulay2
describes a project that uses: SeDiMO
MaRDI profile type: Publication
full work available at URL: j.jsc.2010.06.008
OpenAlex ID: W2042825883
cites work: Secant Varieties of Segre–Veronese Varieties ℙm× ℙnEmbedded byO(1, 2)
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DOI: 10.1016/J.JSC.2010.06.008
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description: scientific article; zbMATH DE number 5817621