 
Summary: The projective space
and other simple varieties.
Marco Andreatta
In this paper I would like to present some recent results on the problem of
nding conditions under which a smooth projective variety X dened over an
algebraically closed eld k is actually the projective space P n .
In the 1979 S. Mori proved the following celebrated theorem, which was previ
ously conjectured by Frenkel and Hartshorne.
Theorem [17]. X = P n if and only if TX is ample.
In his amazing proof Mori introduced the use of rational curves on Fano man
ifolds. Since then the method has been much developed to study (uniruled)
projective manifolds. The book of J. Kollar, see [16], is a wonderful reference
for many fundamental results, I will frequently refer to it (in particular in section
V.3, Characterization of P n , one can nd the proof of the above theorem).
I will focus on P n but I will also present some characterizations of other simple
projective varieties as the hyperquadrics and the projective bundles (scrolls)
over a smooth variety.
The paper is an extended version of the talk I gave at the Fano conference in
Torino, october 2002; I would like to thank and to congratulate the organizers
of this beautiful conference.
