The K-moduli space of a family of conic bundle threefolds

Abstract

We describe the 6-dimensional compact K-moduli space of Fano threefolds in deformation family No 2.18. These Fano threefolds are double covers of $\mathbb P^1\times\mathbb P^2$ branched along smooth $(2,2)$-surfaces, and Cheltsov--Fujita--Kishimoto--Park proved that any smooth Fano threefold in this family is K-stable. A member of family No 2.18 admits the structures of a conic bundle and a quadric surface bundle. We prove that K-polystable limits of these Fano threefolds admit conic bundle structures, but not necessarily del Pezzo fibration structures. We study this K-moduli space via the moduli space of log Fano pairs $(\mathbb P^1\times\mathbb P^2, c R)$ for $c=1/2$ and $R$ a $(2,2)$-divisor, which we construct using wall-crossings. In the case where the divisor is proportional to the anti-canonical divisor, the first author, together with Ascher and Liu, developed a framework for wall crossings in K-moduli and proved that there are only finitely many walls, which occur at rational values of the coefficient $c$. This paper constructs the first example of wall-crossing in K-moduli in the non-proportional setting, and we find a wall at an irrational value of $c$. In particular, we obtain explicit descriptions of the GIT and K-moduli spaces (for $c \leq 1/2$) of these $(2,2)$-divisors. Furthermore, using the conic bundle structure, we study the relationship with the GIT moduli space of plane quartic curves.

Figures (11)

• Figure 1: The wall-crossing morphism in Theorem \ref{['thm:moduli-spaces-2.18']}. In each $\overline{M}^{\mathrm{K}}_{c}$, the strictly polystable (non-stable) locus is a union of three rational curves meeting at the point $[R_0]$, as described in Theorems \ref{['thm:GIT-polystable']} and \ref{['thm:stability-on-X3']}.
• Figure 2: The wall-crossing morphism and the rational map to $\overline{M}^{\mathrm{GIT}}_4$ (Theorem \ref{['thm:wall-crossing-resolves-delta']}). In the diagram, the associated quartic curve $\Delta$ is drawn for each strictly polystable surface.
• Figure 3: Flags used in each of the different cases in Section \ref{['sec:local-delta-A_n:A_1-finite-part-2']}.
• Figure 4: Computing the Nakayama--Zariski decomposition for $L - u Y$ in Section \ref{['sec:local-delta-A_n:A_1-infinite-onesingularity']}.
• Figure 5: Construction of the plt type divisor $Z$ on ${\widetilde{X}}$ in Section \ref{['sec:local-delta-A_n:higher-A_n-part-1']}.

Theorems & Definitions (216)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary 5
• Theorem 6
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Theorem 2.4: Fujita-valuativeLi-valuativeBlumJonssonLXZ-finite-generation