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 $\tilde X.$