On the singular loci of higher secant varieties of Veronese embeddings

Abstract

In this paper we study singular loci of higher secant varieties of the image of the d-uple Veronese embedding of projective n-space, ν d (P n ) ⊂ P N=( n+d d )−1 . We call the closure of the union of all the (k − 1)-planes spanned by k points on X the k-th secant of X for a given embedded variety X ⊂ P N and denote it by σ k (X). For the singular loci of σ k (ν d (P n )), it has been known only for k ≤ 3. In the present paper, as investigating geometry of moving tangents along subvarieties, we determine the (non-)singularity of so-called subsecant loci of σ k (ν d (P n )) for arbitrary k. We also consider the case of the 4-th secant of ν d (P n ) by applying main results and computing conormal space via a certain type of Young flattening. In the end, some discussion for further development is presented as well.