On the singular loci of higher secant varieties of Veronese embeddings

Abstract: For a projective variety X in P^N, the k-secant variety \sigma_k(X) is defined to be the closure of the union of k-planes in P^N spanned by k-points of X. It is well known that \sigma_{k-1}(X) is contained in the singular locus of \sigma_k(X). Let us consider the case when X is the image of the Veronese embedding P^n to P^N of degree d, where N = \binom{d+N}{d}-1. In the case of k=3, K. Han showed that \Sing(\sigma_3(X)) = \sigma_2(X), except when d=4 and n > 2. In the exceptional case, \Sing(\sigma_3(X)) is the union of \sigma_2(X) and D, where D is an irreducible subset. In this talk, we first give a geometric description of this D for k = 3, and next study the case of k > 3. In particular, I will explain projective techniques with respect to an explicit calculation of the Gauss map of X and the projection from the incidence correspondence of \sigma_k(X). This is a joint work with Kangjin Han.

algebraic geometry

Audience: researchers in the topic

ZAG (Zoom Algebraic Geometry) seminar

Series comments: Description: ZAG seminar

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