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Optimal Degenerations of K-unstable Fano threefolds

Minghao Miao, Linsheng Wang

Abstract

We explicitly determine the optimal degenerations of Fano threefolds $X$ in family No 2.23 of Mori-Mukai's list as predicted by the Hamilton-Tian conjecture. More precisely, we find a special degeneration $(\mathcal{X}, ξ_0)$ of $X$ such that $(\mathcal{X}_0, ξ_0)$ is weighted K-polystable, which is equivalent to $(\mathcal{X}_0, ξ_0)$ admitting a Kähler-Ricci soliton (KRS) by \cite{HL23} and \cite{BLXZ23}. Furthermore, we study the moduli spaces of $(\mathcal{X}_0, ξ_0)$. The $\mathbf{H}$-invariant of $X$ divides the natural parameter space into two strata, which leads to different moduli spaces of KRS Fano varieties. We show that one of them is isomorphic to the GIT-moduli space of biconic curves $C\subseteq \mathbb{P}^1\times \mathbb{P}^1$, and the other one is a single point.

Optimal Degenerations of K-unstable Fano threefolds

Abstract

We explicitly determine the optimal degenerations of Fano threefolds in family No 2.23 of Mori-Mukai's list as predicted by the Hamilton-Tian conjecture. More precisely, we find a special degeneration of such that is weighted K-polystable, which is equivalent to admitting a Kähler-Ricci soliton (KRS) by \cite{HL23} and \cite{BLXZ23}. Furthermore, we study the moduli spaces of . The -invariant of divides the natural parameter space into two strata, which leads to different moduli spaces of KRS Fano varieties. We show that one of them is isomorphic to the GIT-moduli space of biconic curves , and the other one is a single point.
Paper Structure (19 sections, 17 theorems, 120 equations, 1 table)

This paper contains 19 sections, 17 theorems, 120 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. ${\mathbb R}$-test configurations
  4. $\mathbf{H}$-invariants
  5. Optimal degenerations
  6. Weighted Abban-Zhuang Estimate
  7. Optimal degenerations of Fano threefolds in family №2.23(a)
  8. Optimal degenerations of $X$ in №2.23(a)
  9. Singularities and GIT stability of biconic curves in ${\mathbb P}^1\times{\mathbb P}^1$
  10. Computing $S^g$ for the degeneration $X_0$ of №2.23(a)
  11. The weighted K-moduli space of $(X_0,\xi_0)$
  12. The ${\rm ord}_{\tilde{H}}$-degeneration of $X$ in №2.23(b)
  13. Optimal degenerations of Fano threefolds in family №2.23(b)
  14. Okounkov body and Zariski decomposition
  15. Computing $S^g$ for the degeneration $X_0$ of №2.23(b)
  16. ...and 4 more sections

Key Result

Theorem 1.2

For any $X$ in №2.23, we have $\delta(X)=\frac{12}{13}$, which is minimized by ${\rm ord}_{\tilde{H}}$.

Theorems & Definitions (38)

  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • ...and 28 more