ncts

Smooth Fano threefolds of Picard rank 4 and degree 24 are divisors in $(\mathbb{P}^1)^4$ of degree $(1,1,1,1)$ . All of them are K-stable by a theorem of Grisha Belousov and Costya Loginov from Moscow. It is natural to expect that all singular K-polystable limits of these Fano threefolds are also divisors in $(\mathbb{P}^1)^4$ of degree $(1,1,1,1)$ , and the corresponding K-moduli space can be obtained as a natural GIT quotient, which is classically known (but not well known) to be isomorphic to the weighted projective space $\mathbb{P}(1,3,4,6)$ . Surprisingly, this is not the case - some singular K-polystable limits of smooth Fano threefolds of Picard rank 4 and degree 24 are not divisors in $(\mathbb{P}^1)^4$ of degree $(1,1,1,1)$ . In this talk, I will find all singular K-polystable limits of smooth Fano threefolds of Picard rank 4 and degree 24, and show that the corresponding K-moduli space is a weighted blow up of $\mathbb{P}(1,3,4,6)$ at a smooth point with weights $(1,2,3)$ . This is a joint work with Maksym Fedorchuk (Boston), Kento Fujita (Osaka) and Anne-Sophie Kaloghiros (London).