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

摘要

To complete the classification theory and the structure theory of varietiesof almost minimal degree, that is of non-degenerate irreducible projectivevarieties whose degree exceeds the codimension by precisely 2, a naturalapproach is to investigate simple projections of varieties of minimal degree.Let $\tilde X \subset {\mathbb P}^{r+1}_K$ be a variety of minimal degree andof codimension at least 2, and consider $X_p = \pi_p (\tilde X) \subset{\mathbb P}^r_K$ where $p \in {\mathbb P}^{r+1}_K \backslash \tilde X$. 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 $\tilde X$ withrespect to $p$. Along these lines, the present paper is devoted to give a geometricdescription of the secant stratification of $\tilde X$, that is of thedecomposition of ${\mathbb P}^{r+1}_K$ via the types of secant loci. We showthat there are exactly six possibilities for the secant locus $\Sigma_p (\tildeX)$, 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 Pezzovarieties by providing a complete list of pairs $(\tilde X, p)$ where $\tilde X\subset {\mathbb P}^{r+1}_K$ is a variety of minimal degree, $p$ is a closedpoint in $\mathbb P^{r+1}_K \setminus \tilde X$ and $X_p \subset {\mathbb P}^r_K$ is a Del Pezzo variety.