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The projective space and other simple varieties.

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 de ned 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.


Source: Andreatta, Marco - Dipartimento di Matematica, UniversitÓ di Trento


Collections: Mathematics