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 P 1 5 , 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 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 P 1 5 , 2 in the K-moduli contains elements that are not of this type; (ii) any quasi-smooth X 2,2,4 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 P 1 5 , 2 2 which are not complete intersections X 2,4 P 1 5 , 2 contains only points which correspond to complete intersections X 2,2,4 P 1 5 , 2 2 .