On K-moduli of quartic threefolds

Hamid Abban, Ivan Cheltsov, Alexander Kasprzyk, Andrea Petracci

Published 2022-10-26Version 1

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; all of them are K-stable. With the aim of investigating the compactification of the moduli space of quartic 3-folds given by K-stability, we exhibit K-polystable Fano 3-folds which deform to quartic 3-folds, but which are neither quartic 3-folds nor double covers of quadric 3-folds. The examples we find are complete intersections of two quadrics and a quartic in the weighted projective space $\mathbb{P}(1,1,1,1,1,2,2)$.