Gökova Geometry Topology Institute

The session will bring together algebraic geometers from the UK, Europe and Asia to work together on higher dimensional Fano varieties. The focus will be made on their birational geometry, derived categories, K-stability and applications to Cremona groups. In particular, the following explicit problems will be considered during the research session: K-stability of smooth Fano threefolds of Picard rank 2 and degree 26, geometric version of Kollar's conjecture about birational rigidity and field extension, classification of maximally non-factorial nodal Fano threefolds after Pavic and Shinder, boundedness of the image of the Lin-Shinder invariant.

This event will be held in person at Hotel Yucelen.

For further questions about this event please contact : Ivan Cheltsov

During our very productive research session in Gokova, we worked on two problems about Fano varieties. The first one is about maximally non-factorial nodal Fano threefolds. The second problem is about K-stability of Casagrande-Druel manifolds. As a result we (Ivan Cheltsov, Igor Krylov, Jesus Martinez Garcia, Evgeny Shinder) wrote the paper “On maximally non-factorial nodal Fano threefolds” which is already available on Arxiv (see https://arxiv.org/abs/2305.09081). Moreover, we (Ivan Cheltsov, Tiago Duarte Guerreiro, Kento Fujita, Igor Krylov, Jesus Martinez Garcia) are preparing the paper “On K-stability of Casagrande-Druel varieties”, which will be ready soon.

In the first paper, we completely classified all non-factorial nodal Fano threefolds with 1 node and class group of rank 2. Namely, we explicitly described 17 3-dimensional Sarkisov links such that their midpoints are maximally non-factorial nodal Fano threefolds with 1 node and Picard group of rank 1 (so their class group is of rank 2). In this paper, we posed the following problem: classify all maximally non-factorial nodal Fano threefolds (of any Picard rank and any number of nodes). We expect that this problem is quite difficult.

In the second paper, we introduced a new class of Fano varieties, which we called Casagrande-Druel varieties. Fano varieties in this class are (n+1)-dimensional Fano varieties whose construction use corresponding n-dimensional Fano double covers. This construction is inspired by a 2012 paper by Casagrande and Druel, so that we decided to name these varieties Casagrande-Druel varieties. Historically, our construction goes back to construction of cyclic central algebras, and de Jonquieres involutions (using hyperelliptic curves instead of Fano double covers). In dimension two, the only smooth Casagrande-Druel variety is the del Pezzo surface of degree 6. In dimension three, there are 3 deformation families of smooth three-dimensional Casagrande-Druel varieties: the family 3-19 (consisting of a single smooth Fano threefold), the family 3.9, and the family 4.2. The corresponding 2-dimensional Fano double covers are the quadric surface (considered as a double cover of the projective plane branched over a conic), del Pezzo surfaces of degree 2 (considered as a double cover of the projective plane branched over a quartic curve), and del Pezzo surfaces of degree 4 (considered as a double cover of a quadric surface branched over a quartic elliptic curve). See https://www.fanography.info/ for the description of all deformation families of smooth Fano threefolds.

Casagrande-Druel varieties are never K-stable, because they automorphism groups are always infinite. Nevertheless, all low dimensional Casagrande-Druel manifolds are known to be K-polystable, and the corresponding double covers are also K-polystable. Inspired by this, we conjectured that an (n+1)-dimensional Casagrande-Druel variety is K-polystable if the corresponding n-dimensional Fano double cover is K-polystable. We verified this conjecture for singular smoothable 3-dimensional Casagrande-Druel varieties, and for infinitely many families of Casagrande-Druel manifolds in higher dimensions whose corresponding Fano covers are K-stable by Dervan's criterion. As a surprising by-product of this, we described all K-polystable singular limits of smooth Fano 3-folds in the families 3.9 and 4.2 - we described their K-moduli.

Our visit to Gokova Geometry Topology Institute was so mathematically enjoyable that we did not want to leave. We hope to return to Gokova in the future for a longer and even more productive visit.