On non-defectivity of certain Segre-Veronese varieties
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 \(\sigma_s(X_{n,n})\) is not defective and that \(\sigma_s(X_{n,n-1})\) is defective iff \(n=4\). As a consequence of this theorem, the author will give two functions \(s_1(m,n)\leq s_2(m,n)\) s.t. if \(s\leq s_1(m,n)\) or if \(s\geq s_2(m,n)\), then \(\sigma_s(X_{m,n})\) has the expected dimension for all positive integers \(m\) and \(n\).
scientific article; zbMATH DE number 5817621
scientific article