Severi varieties
Algebraic Geometry
2007-05-23 v2
Abstract
R. Hartshorne conjectured and F. Zak proved that any n-dimensional smooth non-degenerate complex algebraic variety X in a m-dimensional projective space P satisfies Sec(X)=P if m<3n/2+2. In this article, I deal with the limiting case of this theorem, namely the Severi varieties, defined by the conditions m=3n/2+2 and Sec(X) different from P. I want to give a different proof of a theorem of F. Zak classifying all Severi varieties: I will prove that any Severi variety is homogeneous and then deduce their classification and the following geometric property : the derivatives of the equation of Sec(X), which is a cubic hypersurface, determine a birational morphism of P.
Cite
@article{arxiv.math/0102042,
title = {Severi varieties},
author = {P. E. Chaput},
journal= {arXiv preprint arXiv:math/0102042},
year = {2007}
}
Comments
8 pages