On K-stability of one-nodal prime Fano threefolds of genus $12$

TL;DR

This work proves that a general one-nodal prime Fano threefold of genus $12$ is $K$-polystable by combining explicit Sarkisov-link descriptions of the four nodal families with the Abban–Zhuang framework for K-stability. It constructs concrete symmetric members in Families I–III and performs a meticulous analysis of all $G$-invariant divisors and their centers, showing positivity of the beta-invariant $\beta(\Xi)=A_X(\Xi)-S_X(\Xi)$ in each case; in Families I and II this yields actual $K$-stability, while Family III yields $K$-polystability due to nontrivial automorphisms. The results also exhibit $K$-stable degenerations within Family I and II, and a $K$-polystable (non-$K$-stable) member in Family III, contributing to the understanding of the K-moduli for genus $12$ Fano threefolds and supporting the existence of a K-polystable locus $\,\overline M$ in the moduli. Overall, the paper strengthens the link between explicit birational geometry of nodal Fano threefolds and algebro-geometric stability, with implications for the structure of the K-moduli space $\,\mathrm{M}^{\mathrm{Kps}}_{3,22}$ and its boundary components.

Abstract

We show that general one-nodal prime Fano threefolds of genus $12$ are K-polystable.

Figures (3)

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Theorems & Definitions (64)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Lemma 2.2
• Theorem 2.3
• Remark 2.4
• Lemma 2.5: Fano21