This article studies the relation between the geometry of a smooth projective variety and that of its hyperplane sections from the viewpoint of Mori theory. Let X be a smooth projective variety of dimension n≥4 and Y a smooth hyperplane section of X. Thus H2(Y,R)=H2(X,R) by the Lefschetz hyperplane theorem. Let p:Y→Z be a fibration of Y by Fano varieties. The authors prove several results asserting the existence of an extension of p to X under various conditions. In the main case p is the Mori contraction defined by an extremal ray R of the cone of curves NE−−−(Y) of Y in the region KY<0, and p extends iff R is also an extremal ray of NE−−−(X).
A view on extending morphisms from ample divisors
IONESCU, Paltin;
2009
Abstract
This article studies the relation between the geometry of a smooth projective variety and that of its hyperplane sections from the viewpoint of Mori theory. Let X be a smooth projective variety of dimension n≥4 and Y a smooth hyperplane section of X. Thus H2(Y,R)=H2(X,R) by the Lefschetz hyperplane theorem. Let p:Y→Z be a fibration of Y by Fano varieties. The authors prove several results asserting the existence of an extension of p to X under various conditions. In the main case p is the Mori contraction defined by an extremal ray R of the cone of curves NE−−−(Y) of Y in the region KY<0, and p extends iff R is also an extremal ray of NE−−−(X).I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
