Severi varieties

Pierre-Emmanuel Chaput (1)

1 Géométrie

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.

Domaines: Géométrie algébrique [math.AG]; Théorie des représentations [math.RT]

Pierre-Emmanuel Chaput. Severi varieties. Mathematische Zeitschrift, 2002, 240 (2), pp.451-459. ⟨hal-01256851⟩