On the singular loci of higher secant varieties of Veronese embeddings

Abstract

The $k$-th secant variety of a projective variety $X\subset {\mathbb{P}}^N$, denoted by ${\sigma }_k(X)$, 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 ${\sigma }_k(v_d({\mathbb{P}}^n))\subset {\mathbb{P}}^N$ of the image of the $d$-uple Veronese embedding $v_d$ of ${\mathbb{P}}^n$ to ${\mathbb{P}}^N$ with $N=\left(\genfrac{}{}{0ex}{}{n+d}{d}\right)-1$, and focus on the singular locus of ${\sigma }_k(v_d({\mathbb{P}}^n))$, which is only known for $k\le 3$. To study the singularity for arbitrary $k,d,n$, we define \emph{the $m$-subsecant locus} of ${\sigma }_k(v_d({\mathbb{P}}^n))$ to be the union of ${\sigma }_k(v_d({\mathbb{P}}^m))$ with any $m$-plane ${\mathbb{P}}^m\subset {\mathbb{P}}^n$. 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 $m$-subsecant locus is contained in the singular locus of ${\sigma }_k(v_d({\mathbb{P}}^n))$ 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 $v_d({\mathbb{P}}^n)$ other than the trivial one, the $(k-1)$-th secant variety of $v_d({\mathbb{P}}^n)$. We also consider the case of the $4$-th secant variety of $v_d({\mathbb{P}}^n)$ 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.

The final version, 40 pages, Theorem 2 and 3 are revised as considering few missing non-identifiable cases, some references updated, typos corrected