CIME (Centro
Internazionale
Matematico Estivo) and
CIRM (Centro
Internazionale per la Ricerca Matematica)
organize a summer course
in Levico Terme - Italy
June 22 - June 27, 2015
Rationality Problems in Algebraic Geometry
Historically, rationality problems have been of fundamental
importance in the successful development of Algebraic Geometry.
Motivated by the theory of integration, the parametrization
problem of algebraic curves led to the theory of abelian
integrals, Riemann surfaces and the Abel-Jacobi map;
Castelnuovo's solution of the L
üroth problem in dimension 2 showed the power of the
geometric methods and opened the way to the Enriques
classification of surfaces; the counterexamples to Lüroth
problem in dimension 3, by Clemens- Griffiths, Iskovskih-Manin
and Artin-Mumford, at the end of the 1960's, established the
importance of Griffiths theory of periods, of the birational
geometry of the Cremona and of the Brauer group. Finally, as it
is well-illustrated by the Harris-Mumford work on the Severi
conjecture, the rationality problems had also a prominent
position in the development of moduli theory.
In higher dimension, the rationality problems are
completely open. While it is not easy to make any forecast about
definitive progress, the past year has witnessed a lot of
ingenious new ideas being inserted into the classical picture,
and this makes one hope that some real advance is boiling in the
pot, getting ready to come to the forefront. Many efforts have
been made in the study of special Hodge structures, on the
Cremona Group, new methods (e.g. derived category) have been
introduced, new conjectures have been formulated.
The main aim of the school is to underline the emerging trends
and stimulate young researchers to get involved in this
fascinating area.
Applications
through the CIME
web page are now closed. It is still possible to apply for
participation, but not for financial support, by writing
to michelet@science.unitn.it.