Quartics in projective space up to crepant equivalence (based on the work of T. Ducat)

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.