Irrationality problems
-- Alena Pirutka, December 2, 2016
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Let X be a projective algebraic variety, the set of solutions of a
system of homogeneous polynomial equations. Several classical notions
describe how "unconstrained" the solutions are, i.e., how close X is
to projective space: there are notions of rational, unirational and
stably rational varieties. Over the field of complex numbers, these
notions coincide in dimensions one and two, but diverge in higher
dimensions. In the last years, many new classes of non-stably rational
varieties were found, using a specialization technique introduced by
C. Voisin. This method was also used to prove that rationality is not
a deformation invariant in smooth projective families of complex
varieties.  This is joint work with B. Hassett and Y. Tschinkel.  In
the first part of my talk I will describe some classical examples as
well as the recent examples obtained by the specialization method.  I
will give more details on this method in the second part of my talk.