On the singular loci of higher secant varieties of Veronese embeddings

The k-th secant variety of a projective variety X ⊂ PN, denoted by σk(X), is defined to be the closure of the union of (k - 1) (k-1) -planes spanned by k points on X. In this paper, we examine the k-th secant variety (Equation presented) of the image of the d-uple Veronese embedding v d of Pnwith (Equation presented), and focus on the singular locus of (Equation presented), which is only known for k ≤ 3. To study the singularity for arbitrary k, d, n k,d,n, we define the m-subsecant locus of (Equation presented) to be the union of (Equation presented) with any m-plane Pm ⊂ Pn. By investigating the projective geometry of moving embedded tangent spaces along subvarieties and using known results on the secant defectivity and the identifiability of symmetric tensors, we determine whether the m-subsecant locus is contained in the singular locus of (Equation presnted) or not. Depending on the value of k, these subsecant loci show an interesting trichotomy between generic smoothness, non-trivial singularity, and trivial singularity. In many cases, they can be used as a new source for the singularity of the k-th secant variety of vd(Pn) other than the trivial one, the (k-1) -th secant variety of vd(Pn). We also consider the case of the fourth secant variety of v d(Pn) by applying main results and computing conormal space via a certain type of Young flattening. Finally, we present some generalizations and discussions for further developments.