On K-moduli of quartic threefolds

Abban, Hamid; Cheltsov, Ivan; Kasprzyk, Alexander; Liu, Yuchen; Petracci, Andrea

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 P(1,1,1,1,1,2), denoted by X_{2,4}\subset 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 X_{2,2,4}\subset P(1^5,2^2) 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 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).

Citation

Abban, H., Cheltsov, I., Kasprzyk, A., Liu, Y., & Petracci, A. (in press). On K-moduli of quartic threefolds. Algebraic Geometry,

• Journal Article Type: Article
• Acceptance Date: Mar 29, 2024
• Deposit Date: Mar 29, 2024
• Journal: Algebraic Geometry
• Print ISSN: 2313-1691
• Electronic ISSN: 2214-2584
• Publisher: European Mathematical Society
• Peer Reviewed: Peer Reviewed
• Public URL: https://nottingham-repository.worktribe.com/output/33026937
• Publisher URL: https://ems.press/journals/ag