Arithmetic properties of projective varieties of almost minimal degree

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 $1$.