On the singular loci of higher secant varieties of Veronese embeddings

Abstract

The -th secant variety of a projective variety X subset of P N, denoted by sigma k ( X ), is defined to be the closure of the union of (k-1) -planes spanned by points on . In this paper, we examine the -th secant variety sigma k ( v d ( P n ) ) subset of P N of the image of the -uple Veronese embedding v d v_{d} of P n with N = ( n + d d ) - 1, and focus on the singular locus of sigma k ( v d ( P n ) ), which is only known for k <= 3 k\leq 3 . To study the singularity for arbitrary k , d , n k,d,n , we define the -subsecant locus of sigma k ( v d ( P n ) ) to be the union of sigma k ( v d ( P m ) ) with any -plane P m subset of P n. 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 -subsecant locus is contained in the singular locus of sigma k ( v d ( P n ) ) or not. Depending on the value of , 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 -th secant variety of v d ( P n ) other than the trivial one, the (k-1) -th secant variety of v d ( P n ). We also consider the case of the fourth secant variety of v d ( P n ) 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.