On the singular loci of higher secant varieties of Veronese embeddings

The k-th secant variety of a projective variety X P^N, denoted by σ_k(X), is defined to be the closure of the union of (k - 1)-planes spanned by k points on X. In this paper, we examine the k-th secant variety (Formula presented) of the image of the d-uple Veronese embedding v_d of Pⁿ to P^N with (Formula presented), and focus on the singular locus of σ_k(v_d(Pⁿ), which is only known for k ≤ 3. To study the singularity for arbitrary k, d, n, we define the m-subsecant locus of σ_k(v_d(Pⁿ) to be the union of σ_k(v_d(P^m) with any m-plane P^m Pⁿ.Byinvestigatingthe projective geometry of moving embedded tangent spaces along subvarieties and using known results on the secant defectivity and the identifiability of sym metric tensors, we determine whether the m-subsecant locus is contained in the singular locus of σ_k(v_d(Pⁿ) 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 v_d(Pⁿ) other than the trivial one, the (k 1)-th secant variety of v_d(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.