On varieties of almost minimal degree I: Secant loci of rational normal scrolls
Abstract
Description
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 $\tilde X \subset {\mathbb P}^{r+1}_K$ be a variety of minimal degree and of codimension at least 2, and consider $X_p = π_p (\tilde X) \subset {\mathbb P}^r_K$ 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 $Σ_p (\tilde X)$ 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 ${\mathbb P}^{r+1}_K$ via the types of secant loci. We show that there are exactly six possibilities for the secant locus $Σ_p (\tilde X)$, 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 $\tilde X \subset {\mathbb P}^{r+1}_K$ is a variety of minimal degree, $p$ is a closed point in $\mathbb P^{r+1}_K \setminus \tilde X$ and $X_p \subset {\mathbb P}^r _K$ is a Del Pezzo variety.
20 pages
20 pages