Contemporary Mathematics

Volume 388, 2005

Rationally connected varieties

Carolina Araujo

ABSTRACT.

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.

1.

Introduction

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

)

form a

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.

If

Sbo is rational for some bo E B, then Sb is

rational for every

b

E

B.

(2) (Numerical characterization.) LetS be a smooth complex projective sur-

face. Then Sis rational if and only if

H 0 (S,w~

2

)

=

H 1 (S,Os)

=

0.

(3) (Geometric criterion.) Del Pezzo surfaces (i.e., smooth complex projective

surfaces S for which - K

s

is ample) are rational.

(4) (Well behavior under fibration.) Let S be a smooth complex projective

surface. Assume that there exists a morphism

S---+

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

birational to

lP'n, n

2

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

http://dx.doi.org/10.1090/conm/388/07254