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Smooth Fano $3$-folds are classified in $105$ families (Iskovskikh, Mori, Mukai). For the description of these families, see https://www.fanography.info  . We know which deformation families have $K$-polystable (Kahler-Einstein) members and which do not (Araujo, Castravet, Cheltsov, Fujita, Kaloghiros, Martinez Garcia, Shramov, Suess, Viswanathan).  Since $K$-polystable Fano threefolds form good moduli spaces, it would be interesting to describe $K$-moduli of smoothable Fano $3$-folds (moduli that parametrize $K$-polystable smooth members of a given deformation family and their $K$-polystable limits). This is a very  active area of research, but the problem has only been solved for the following $52$   deformation families: zero-dimensional families ($47$ families), two one-dimensional   families (families $2-24$ and $2-25$), cubic $3$-folds (Liu, Xu), complete   intersection of two quadrics (Spotti, Sun), quartic double solids (Ascher, DeVleming, Liu).

In this talk, I will speak about K-moduli of Fano $3$-folds in the family $3-10$, see https://www.fanography.info/3-10 . This is a two-dimensional family whose smooth members can be obtained by blowing up a smooth quadric $3$-fold along   two disjoint conics. We know that a general member of this family is $K$-stable (Kahler Einstein and finite automorphism groups), but some smooth members are not $K$-polystable (not Kahler-Einstein), and some members have infinite automorphism group (Cheltsov, Przyjalkowski, Shramov). 

In the talk, I will give explicit classification of all smooth members of the family $3-10$ (normal   forms), explain which smooth Fano $3$-folds in this family are $K$-polystable and which are not (Araujo, Castravet, Cheltsov, Fujita, Kaloghiros, Martinez-Garcia, Shramov, Suess, Viswanathan), and describe all singular $K$-polystable members of this family (work in progress with Alan Thompson from Loughborough). If   time permits, I will explain how to prove $K$-polystability of one singular and very symmetric member of this deformation family.