366 research outputs found
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 is either arithmetically normal (and arithmetically
Gorenstein) or a projection of a variety of minimal degree from an appropriate point . We focus on the latter situation and study by means
of the projection .
If is not arithmetically Cohen-Macaulay, the homogeneous coordinate ring
of the projecting variety is the endomorphism ring of the
canonical module of the homogeneous coordinate ring of If
is non-normal and is maximally Del Pezzo, that is arithmetically Cohen-Macaulay
but not arithmetically normal is just the graded integral closure of
It turns out, that the geometry of the projection is
governed by the arithmetic depth of in any case.
We study in particular the case in which the projecting variety is a cone (over a) rational normal scroll. In this
case is contained in a variety of minimal degree
such that \codim_Y(X) = 1. We use this to approximate the Betti numbers of
.
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 -genus 1.Comment: corrected, revised version. J. Algebraic Geom., to appea
On an endomorphism ring of local cohomology
Let be an ideal of a local ring with For
the local cohomology module it is a well-known fact that it vanishes
for and is an Artinian -module for In the case that the
Hartshorne-Lichtenbaum Vanishing Theorem fails, that is we
explore its fine structure. In particular, we investigate its endomorphism ring
and related connectedness properties. In the case is complete we prove - as
a technical tool - that for a certain
ideal Thus, properties of and its Matlis dual might
be described in terms of the local cohomology supported in the maximal ideal.Comment: 8 pages, The paper will appear in Journal "Communications in Algebra
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