Kangjin Han, On the Singular Loci of Higher Secants of Veronese Varieties

(This is a part of Algebraic Geometry Day at CCG in IBS.)

For a projective variety X in P^N, the k-secant variety σ_k(X) is defined to be the closure of the union of k-planes in P^N spanned by k-points of X. In this talk, we consider singular loci of higher secant varieties of the image of the d-uple Veronese embedding of projective n-space, ν_d(Pⁿ). For the singular loci of k-secant of ν_d(Pⁿ), it has been known only for k≤3. First, I will review some basic notions and results and then explain projective techniques with respect to an explicit calculation of the Gauss map of X and computation for conormal space via a certain type of Young flattening. As investigating geometry of moving tangents along subvarieties, we determine the (non-)singularity of so-called ‘subsecant loci’ of k-secant of ν_d(Pⁿ) for arbitrary k. This is a joint work with Katsuhisa Furukawa.