On varieties of almost minimal degree I: Secant loci of rational normal scrolls

Abstract

To complete the classification theory and the structure theory of varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2, a natural approach is to investigate simple projections of varieties of minimal degree. Let $\tilde{X}\subset {\mathbb{P}}_K^{r+1}$ be a variety of minimal degree and of codimension at least 2, and consider $X_p={\pi }_p(\tilde{X})\subset {\mathbb{P}}_K^r$ where $p\in {\mathbb{P}}_K^{r+1}\backslash \tilde{X}$. By \cite{B-Sche}, it turns out that the cohomological and local properties of $X_p$ are governed by the secant locus ${\Sigma }_p(\tilde{X})$ of $\tilde{X}$ with respect to $p$. Along these lines, the present paper is devoted to give a geometric description of the secant stratification of $\tilde{X}$, that is of the decomposition of ${\mathbb{P}}_K^{r+1}$ via the types of secant loci. We show that there are exactly six possibilities for the secant locus ${\Sigma }_p(\tilde{X})$, and we precisely describe each stratum of the secant stratification of $\tilde{X}$, each of which turns out to be a quasi-projective variety. As an application, we obtain the classification of all non-normal Del Pezzo varieties by providing a complete list of pairs $(\tilde{X},p)$ where $\tilde{X}\subset {\mathbb{P}}_K^{r+1}$ is a variety of minimal degree, $p$ is a closed point in ${\mathbb{P}}_K^{r+1}\setminus \tilde{X}$ and $X_p\subset {\mathbb{P}}_K^r$ is a Del Pezzo variety.

20 pages