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Wall crossing for K-moduli space of degree 5 pairs

Long Pan

Abstract

In this paper, we describe the wall-crossing of the two parameter K-moduli space of pairs (P 2,aQ + bL), where Q is a plane quintic curve and L is a line.

Wall crossing for K-moduli space of degree 5 pairs

Abstract

In this paper, we describe the wall-crossing of the two parameter K-moduli space of pairs (P 2,aQ + bL), where Q is a plane quintic curve and L is a line.

Paper Structure

This paper contains 11 sections, 31 theorems, 79 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. K-semistable degeneration
  4. K-stability of complexity one pairs
  5. The critical $\mathbb{P}^2$-pair
  6. The critical $\mathbb{P}(1,1,4)$-pair
  7. The critical $X_{26}$-pair
  8. The critical $\mathbb{P}(1,4,25)$-pair
  9. Explicit wall crossings
  10. The Type(I) vertical wall crossing
  11. The Type(II) vertical wall crossing

Key Result

Theorem 1.1

If K-semistable log Fano pair $(X,aC+bL)$ is parameterized by $\overline{\mathcal{P}}^K_{(a,b)}$, then $X$ can only possibly be $\mathbb{P}^2$, $\mathbb{P}(1,1,4)$, $\mathbb{P}(1,4,25)$ or $X_{26}$.

Figures (1)

  • Figure 1: wall-chamber structure of $\overline{P}^K_{a,b}$

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 54 more