Abstract:
One of the main problems of birational geometry is the problem of
classifying varieties up to birational equivalence. Generalizing it,
we can introduce an additional structure on the manifold and consider
the classification problem up to birational equivalence that preserves
this structure. An example of such an additional structure would be a
logarithmic volume form, that is, a meromorphic form on a variety with
poles of order no higher than one. In another language, the same
problem can be reformulated as the problem of classifying pairs of the
form variety + boundary divisor up to crepant equivalence. Following
the paper by T. Ducat, we will consider the solution to this problem
for Calabi-Yau pairs of the form projective space + (singular) quartic
in it. If there is time, we will talk about generalizing this
technique to the case of smooth Fano threefolds.
