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

Brodmann, M; Park, E; Schenzel, P (2011). On varieties of almost minimal degree II: A rank-depth formula. Proceedings of the American Mathematical Society, 139(6):2025-2032.

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 $\tilde X.$

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 $\tilde X.$

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