Gökova Geometry Topology Institute

The research session will bring together the following experts on Algebraic Geometry.

Smooth Fano 3-folds have been classified by Iskovskikh, Mori and Mukai into 105 deformation families. The description of these families are available online at the following web page: https://www.fanography.info. In 2019, Cheltsov organized a collaborative research project on K-stability of smooth Fano 3-folds. This project resulted in many publications including 450 pages long book “The Calabi problem for Fano threefolds” by Carolina Araujo, Ana-Maria Castravet, Ivan Cheltsov, Kento Fujita, Anne-Sophie Kaloghiros, Jesus Martinez-Garcia, Constantin Shramov, Hendrik Suess, Nivedita Viswanathan, which has been published in 2023 at Cambridge University Press (London Mathematical Society Lecture Notes Series, volume 485).

Now it is time to study K-stability of singular Fano 3-folds. Despite efforts of many mathematicians, these geometric objects are not yet classified. However, we know classification of singular Fano 3-folds with Gorenstein canonical singularities such that their anticanonical linear systems give double covers (due to Cheltsov, Przyjalkowski, Shramov), which are known as hyperelliptic Fano 3-folds. The list of hyperelliptic Fano 3-folds consists of 47 deformation families (only 6 families contain smooth members). At the moment, we know very few results about K-stability of these 3-folds, e.g. all smooth hyperelliptic Fano 3-folds are known to be K-polystable (due to Abban, Cheltsov, Denisova, Dervan, Fujita, Park, Tian, Zhuang). We plan to fill this gap during our research session at Gökova Geometry Topology Institute.

Namely, during our visit, we plan to obtain comprehensive results on K-stability of singular hyperelliptic. In particular, we aim to classify all K-polystable double covers of three-dimensional projective space branched along singular sextic surfaces. To start with, we plan to find all K-polystable double covers of three-dimensional projective space branched along singular reduced (but possibly reducible) sextic surfaces - these are hyperelliptic Fano 3-folds of anticanonical degree 2 (in the smooth case, their K-stability has been proven by Cheltsov and Park). If time permits, we plan to study K-stability of trigonal Fano 3-folds. These are Fano 3-folds with canonical Gorenstein singularities such that anticanonical divisors are very ample, but their anticanonical images are not intersection of quadrics. These 3-folds have been classified by Cheltsov, Przyjalkowski, Shramov. We know very little about their K-stability at the moment.

We also plan to study simple finite subgroups of the real Cremona group of the three-dimensional space. All simple finite subgroups of the complex Cremona group of the three-dimensional complex space have been classified by Yuri Prokhorov. During our stay in Gökova, we plan to classify simple finite subgroups of the real Cremona group of the three-dimensional real space (group of birational automorphisms of the real three-dimensional projective space).

For further questions about this event please contact : Ivan Cheltsov