On varieties of almost minimal degree II: A rank-depth formula

Abstract

Let $ X \subset \mathbb{P}^r_K$ denote a variety of almost minimal degree other than a normal del Pezzo variety. Then $ X$ is the projection of a rational normal scroll $ \tilde X \subset {\mathbb{P}}^{r+1}_K$ from a point $ p \in {\mathbb{P}}^{r+1}_K \setminus \tilde X.$ We show that the arithmetic depth of $ X$ can be expressed in terms of the rank of the matrix $ M'(p),$ where $ M'$ is the matrix of linear forms whose $ 3\times 3$ minors define the secant variety of $ \tilde X.$