Volume 388, 2005
Rationally connected varieties
The aim of these notes is to provide an introduction to the theory
of rationally connected varieties, as well as to discuss a recent result by T.
Graber, J. Harris and J. Starr.
Consider the problem of classifying the "simplest" projective varieties. In di-
mension 1 there is not much to say. Smooth projective curves are classified by their
genus. It is not hard to argue that lP'1 is the simplest projective curve. (See [KolOl]
for a detailed discussion.)
In dimension 2 classification was completed by Enriques at the beginning of
the twentieth century. Rational surfaces (i.e., surfaces birational to 1P'2
distinguished class of surfaces. This class is very well behaved from the point of
view of classification. To illustrate this, we state some nice properties of the class
of rational surfaces.
PROPERTIES 1.1 (Nice properties of rational surfaces).
(1) (Deformation invariance.) Let S
B be a connected family of smooth
complex projective surfaces.
Sbo is rational for some bo E B, then Sb is
rational for every
(2) (Numerical characterization.) LetS be a smooth complex projective sur-
face. Then Sis rational if and only if
H 0 (S,w~
H 1 (S,Os)
(3) (Geometric criterion.) Del Pezzo surfaces (i.e., smooth complex projective
surfaces S for which - K
is ample) are rational.
(4) (Well behavior under fibration.) Let S be a smooth complex projective
surface. Assume that there exists a morphism
lP'1 whose general fiber
is rational. Then S is rational.
Now let us move to higher dimensions. After looking at the 2-dimensional
case, one is naturally led to considering the class of rational varieties (i.e., varieties
1). The drawback is that this class of varieties behaves very
2000 Mathematics Subject Classification. Primary 14M20; Secondary 14005.
This work was partially completed during the period the author was employed by the Clay
Mathematics Institute as a Liftoff Fellow.
@2005 American Mathematical Society