On K-moduli of quartic threefolds

The family of smooth Fano 3-folds with Picard rank 1 and anticanonical volume 4 consists of quartic 3-folds and of double covers of the 3-dimensional quadric branched along an octic surface. They can all be parametrised as complete intersections of a quadric and a quartic in the weighted projective space P(1,1,1,1,1,2), denoted by X_{2,4} \subset P(1,1,1,1,1,2); all such smooth complete intersections are K-stable. With the aim of investigating the compactification of the moduli space of quartic 3-folds given by K-stability, we exhibit three phenomena: (i) there exist K-polystable complete intersection Fano 3-folds X_{2,2,4} \subset P(1^5,2^2) which deform to quartic 3-folds and are neither quartic 3-folds nor double covers of quadric 3-folds – in other words, the closure of the locus parametrising complete intersections X_{2,4} \subset P(1^5,2) in the K-moduli contains elements that are not of this type; (ii) any quasi-smooth X_{2,2,4} \subset P(1^5,2^2) is K-polystable; (iii) the closure in the K-moduli space of the locus parametrising complete intersections X_{2,2,4} \subset P(1^5,2^2) which are not complete intersections X_{2,4} \subset P(1^5,2) contains only points which correspond to complete intersections X_{2,2,4} \subset P(1^5,2^2).