On the singular locus of Grassmann secant varieties

Abstract

Let $X\subset \mathbb{P}^N$ be an irreducible non-degenerate variety. If the $(h,k)$-Grass\-mann secant variety $G_{h,k}(X)$ of $X$ is not the whole Grassmannian $\mathbb{G}(h,N)$, we have that the singular locus of $G_{h,k}(X)$ contains $G_{h,k-1}(X)$. Moreover, if $X$ is a smooth curve without $(2k+2)$-secant $2k$-space divisors, we obtain the equality $\text{Sing}(G_{h,k}(X))=G_{h,k-1}(X)$.