Severi varieties

R. Hartshorne conjectured and F. Zak proved (cf [6,p.9]) that any smooth non-degenerate complex algebraic variety \(X^n \subset{\mathbb P}^m\) with \(m<\frac{3}{2}n+2\) satisfies \(Sec(X)={\mathbb P}^m (Sec(X)\) denotes the secant variety of X; when X is smooth it is simply the union of all the secant and tangent lines to X). In this article, I deal with the limiting case of this theorem, namely the Severi varieties, defined by the conditions \(m=\frac{3}{2}n+2\) and \(Sec(X)\neq{\mathbb P}^m\). I want to give a different proof of a theorem of F. Zak classifying all Severi varieties. F. Zak proves that there exists only four Severi varieties and then realises a posteriori that all of them are homogeneous; here I will work in another direction: I prove a priori that any Severi variety is homogeneous and then deduce more quickly their classification, satisfying R. Lazarsfeld et A. Van de Ven's wish [6, p.18]. By the way, I give a very brief proof of the fact that the derivatives of the equation of Sec(X), which is a cubic hypersurface, determine a birational morphism of \({\mathbb P}^m\). I wish to thank Laurent Manivel for helping me in writing this article.

Authors and Affiliations

1. Institut Fourier, Laboratoire de Mathématiques, UMR5582 (UJF-CNRS) BP 74, 38402 St Martin d'Hères Cedex, France (e-mail: chaput@ujf-grenoble.fr) , , , , , , FR

   Pierre-Emmanuel Chaput

Received in final form: 29 March 2001 / Published online: 1 February 2002

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