On K-moduli of quartic threefolds
Hamid Abban, Ivan Cheltsov, Alexander Kasprzyk, Yuchen Liu, Andrea, Petracci

TL;DR
This paper investigates the K-moduli space of quartic threefolds, revealing new phenomena about its boundary and stability properties of certain complete intersection Fano 3-folds.
Contribution
It demonstrates the existence of K-polystable Fano 3-folds deforming to quartic threefolds that are not of the original family, and characterizes stability within certain complete intersections.
Findings
K-polystable complete intersections deform to quartic 3-folds
Quasi-smooth $X_{2,2,4}$ are K-polystable
Boundary points of the moduli space correspond to specific complete intersections
Abstract
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 , denoted by ; 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 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 in…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Geometry and complex manifolds · Historical Studies and Socio-cultural Analysis
