K-Moduli of Fano Threefolds of Family 3.3

TL;DR

The paper delivers an explicit description of the K-moduli for the Fano threefold family №3.3, proving that K-semistable varieties with volume $V>18$ are Gorenstein canonical and admit general elephants, while relating the K-moduli to lattice-polarized K3 surfaces. Using a moduli-continuity approach and Kirwan blow-ups of the GIT quotient for $(1,1,2)$ divisors in $P^1 imesP^1 imesP^2$, it identifies the K-moduli stack with a smooth, connected component of the K-moduli space $rak M^K_{3,18}$ and provides a complete classification of K-(semi/poly)stable objects. The work combines detailed GIT analysis (including a Luna slice and Kirwan blow-up) with deformation theory to realize the K-moduli as a global GIT quotient, and it characterizes stability conditions in terms of explicit singularities for the $(1,1,2)$ and $(2,2)$ divisors. This yields a concrete, computable picture of K-stability for family №3.3 with direct connections to K3-elephants and to modular descriptions via lattice-polarized K3 surfaces.

Abstract

We explicitly fully describe the K-moduli space of Fano threefold family number 3.3. We first show that K-semistable Fano varieties with volume greater than 18 are Gorenstein canonical and admit general elephants, decreasing the bound on a result by Liu and Zhao. Combining this with the moduli-continuity method via lattice-polarized K3 surfaces, we identify the K-moduli stack parametrising K-semistable varieties in family number 3.3 with a Kirwan blow up of the natural GIT quotient of $(1,1,2)$ divisors in $\mathbb{P}^1\times \mathbb{P}^1\times \mathbb{P}^2$.

Theorems & Definitions (83)

• Theorem 1.1: See Theorem \ref{['gor canonical K-ss limits']}
• Theorem 1.2: See Corollary \ref{['cor:main']}
• Theorem 1.3
• Theorem 1.4: See Theorem \ref{['thm: full K-ss description']}
• Definition 2.1
• Definition 2.2
• Theorem 2.3: cf. Fuj19bLi17BX19
• Definition 2.4
• Definition 2.5
• Theorem 2.6: K-moduli Theorem