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--- events:2025:researchsession [2025/06/12 13:54]
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+++ events:2025:researchsession [2025/09/23 00:11] (current)
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-===== K-stability of Hyperelliptic Fano 3-folds and Mukai 4-folds =====
+===== K-stability of singular Fano 3-folds and simple subgroups of Cremona groups =====
 =====(Sep 1 - Sep 14, 2025) =====
+{{:events:2025:25rsFano.jpeg|}}
 The research session will bring together the following experts on Algebraic Geometry.
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 ^        List of Participants       ^^^
 |   Ivan Cheltsov (University of Edinburgh, UK)   |   Adrien Dubouloz (University of Poitiers, France)   |  Tiago Duarte Guerreiro (University of Basel, Switzerland)   |
-|Yuri Prokhorov (Steklov Institute of Mathematics, Russia)   |       |  Jihun Park (IBS and POSTECH, Korea)   |
+|Yuri Prokhorov (Steklov Institute of Mathematics, Russia)   |  Antoine Pinardin (University of Edinburgh, UK)  |  Jihun Park (IBS and POSTECH, Korea)   |
-During our visit, we plan to obtain comprehensive results on K-stability of singular hyperelliptic Fano 3-folds and smooth Mukai 4-folds. In particular, we aim to classify all K-polystable double covers of three-dimensional projective space branched along singular sextic surfaces, and we plan to prove K-polystability of all smooth Fano-Mukai 4-folds of genus 10 that have reductive automorphism group.
+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).
-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.
-Now it is time to study K-stability of singular Fano 3-folds and smooth Fano 4-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. Similarly, we know classification of smooth Fano 4-folds whose anticanonical divisors are divisible by 2 in the Picard group (due to Mukai and Wisniewski), which are known as Mukai 4-folds (or Fano-Mukai 4-folds).  The list of hyperelliptic Fano 3-folds consists of 47 deformation families (only 6 families contain smooth members), and the list of smooth Mukai 4-folds consists of 29 deformation families. At the moment, we know very few results about K-stability of these 3-folds and 4-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 Gokova 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.
-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). Then we plan to classify K-polystable smooth Mukai 4-folds of genus 10. These Mukai 4-folds have been studied by Prokhorov and Zaidenberg in a series of papers, and their geometry is well-understood. Up to isomorphisms, they form a one-dimensional family, and one 4-fold in this family has a non-reductive automorphism group, so it is not K-polystable by the Matsushima obstruction. We expect that the remaining smooth Mukai 4-folds of genus 10 are K-polystable and we plan to prove this during our visit to Gokova. If time permits, we plan to study K-stability of other singular hyperelliptic Fano 3-folds and smooth Mukai 4-folds.
+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 : [[I.Cheltsov@ed.ac.uk |Ivan Cheltsov ]]

Gökova Geometry Topology Institute

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