Arithmetic properties of projective varieties of almost minimal degree

Abstract

We study the arithmetic properties of projective varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely $ 2$. We notably show, that such a variety $ X \subset {\mathbb{P}}^r$ is either arithmetically normal (and arithmetically Gorenstein) or a projection of a variety of minimal degree $ \tilde {X} \subset {\mathbb{P}}^{r + 1}$ from an appropriate point $ p \in {\mathbb{P}}^{r + 1} \setminus \tilde {X}$. We focus on the latter situation and study $ X$ by means of the projection $ \tilde {X} \rightarrow X$.

If $ X$ is not arithmetically Cohen-Macaulay, the homogeneous coordinate ring $ B$ of the projecting variety $ \tilde {X}$ is the endomorphism ring of the canonical module $ K(A)$ of the homogeneous coordinate ring $ A$ of $ X.$ If $ X$ is non-normal and is maximally Del Pezzo, that is, arithmetically Cohen-Macaulay but not arithmetically normal, $ B$ is just the graded integral closure of $ A.$ It turns out, that the geometry of the projection $ \tilde {X} \rightarrow X$ is governed by the arithmetic depth of $ X$ in any case.

We study, in particular, the case in which the projecting variety $ \tilde {X} \subset {\mathbb{P}}^{r + 1}$ is a (cone over a) rational normal scroll. In this case $ X$ is contained in a variety of minimal degree $ Y \subset {\mathbb{P}}^r$ such that $ \operatorname{codim}_Y(X) = 1$. We use this to approximate the Betti numbers of $ X$.

In addition, we present several examples to illustrate our results and we draw some of the links to Fujita's classification of polarized varieties of $ \Delta $-genus $ 1$.