On the singular loci of higher secant varieties of Veronese embeddings

In this paper we study singular loci of higher secant varieties of the image of the $d$-uple Veronese embedding of projective $n$-space, $\nu_d(\mathbb{P}^n) \subset \mathbb{P}^{N={n+d\choose d}-1}$. We call the closure of the union of all the $(k-1)$-planes spanned by $k$ points on $X$ the \textit{$k$-th secant of $X$} for a given embedded variety $X\subset\mathbb{P}^N$ and denote it by $\sigma_k(X)$. For the singular loci of $\sigma_k(\nu_d(\mathbb{P}^n))$, it has been known only for $k\le3$. In the present paper, as investigating geometry of moving tangents along subvarieties, we determine the (non-)singularity of so-called \textit{subsecant loci} of $\sigma_k(\nu_d(\mathbb{P}^n))$ for arbitrary $k$. We also consider the case of the $4$-th secant of $\nu_d(\mathbb{P}^n)$ 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.