Abstract
This paper studies the defectivity of secant varieties of Segre varieties. We prove that there exists an asymptotic lower estimate for the greater non-defective secant variety (without filling the ambient space) of any given Segre variety. In particular, we prove that the ratio between the greater non-defective secant variety of a Segre variety and its expected rank is lower bounded by a value depending just on the number of factors of the Segre variety. Moreover, in the final section, we present some results obtained by explicit computation, proving the non-defectivity of all the secant varieties of Segre varieties of the shape \((\mathbb{P }^{n})^4\), with \(2 \le n\le 10\), except at most \(\sigma _{199}((\mathbb{P }^8)^4)\) and \(\sigma _{357}((\mathbb{P }^{10})^4)\).
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Gesmundo, F. An asymptotic bound for secant varieties of Segre varieties. Ann Univ Ferrara 59, 285–302 (2013). https://doi.org/10.1007/s11565-013-0175-y
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DOI: https://doi.org/10.1007/s11565-013-0175-y