Arithmetic properties of projective varieties of almost minimal degree
| dc.creator | Brodmann, Markus | |
| dc.creator | Schenzel, Peter | |
| dc.date | 2005-06-14 | |
| dc.date | 2006-05-18 | |
| dc.date.accessioned | 2026-07-07T06:42:26Z | |
| dc.date.available | 2026-07-07T06:42:26Z | |
| dc.description | 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 $\tilde {X} \to X$. 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 $\tilde {X} \to X$ 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 $\codim_Y(X) = 1$. 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 $Δ$-genus 1. | |
| dc.description | corrected, revised version. J. Algebraic Geom., to appear | |
| dc.identifier | https://arxiv.org/abs/math/0506277 | |
| dc.identifier | http://arxiv.org/abs/math/0506277 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/102038 | |
| dc.subject | Commutative Algebra | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 14H45; 13D02 | |
| dc.title | Arithmetic properties of projective varieties of almost minimal degree | |
| dc.type | text |