Dimension of linear systems: a combinatorial and differential approach

L. Evain

Home/Archive/arXiv

arXiv · alg-geom/9709032

Dimension of linear systems: a combinatorial and differential approach

L. Evain

View original source ↗Read source document ↗Export metadata ↓

Abstract

We give upper-bounds for the dimension of some linear systems. The theorem improves the differential Horace method introduced by Alexander-Hirschowitz, and was conjectured by Simpson. Possible applications are the calculus of the dimension of linear systems of hypersurfaces in a projective space $\PP^n$ with generically prescribed singularities, and the calculus of collisions of fat points in $\PP^2$. These applications will be treated independently but a simple example in the introduction explains how the theorem will be used.

Explore related subjects

alg-geom math.AG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

L. Evain. 1997-09-29. Dimension of linear systems: a combinatorial and differential approach. https://arxiv.org/abs/alg-geom/9709032

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2025

Deforming Calabi-Yau Threefolds

This paper first generalises the Bogomolov-Tian-Todorov unobstructedness theorem to the case of Calabi-Yau threefolds with canonical singularities. The deformation space of such a Calabi-Yau threefold is no longer smooth, but the general principle is that the obstructions to deforming such a threefold are precisely the obstructions to deforming the singularities of the threefold. Secondly, these results are applied to smoothing singular Calabi-Yau threefolds with crepant resolutions. Any such Calabi-Yau threefold with isolated complete intersection singularities which are not ordinary double points is smoothable. A Calabi-Yau threefold with non-complete intersection isolated singularities is proved to be smoothable under much stronger hypotheses.

alg-geom↗

arXiv2025

Derived categories of coherent sheaves on abelian varieties and equivalences between them

We study derived categories of coherent sheaves on abelian varieties. We give a criterion for the equivalence of the derived categories on two abelian varieties. We describe the autoequivalence group for the derived category of coherent sheaves of an abelian variety.

alg-geom↗

arXiv2019

Classification of Varieties with Canonical Curve Section via Gaussian maps on Canonical Curves

Let $C \subset P^{g-1}$ be a smooth canonical curve of genus $g \geq 3$. The purpose of this article is to further develop a method to classify varieties having $C$ as their curve section, using Gaussian map computations. In a previous article a careful analysis of the degeneration to the cone over the hyperplane section was made for _prime_ Fano threefolds, that is Fano threefolds whose Picard group is generated by the hyperplane bundle. In this article we extend this method and classify Fano threefolds of higher index (which still have Picard number one). We are also able to classify Mukai varieties, i.e. varieties of dimension four or more with canonical curve sections.

alg-geom↗