$\delta$-invariants of log Fano planes

TL;DR

This paper calculates the delta-invariant for certain log Fano pairs involving plane curves of degree up to 4, providing new examples of K-stable and K-semistable pairs, and advancing the understanding of K-stability via surface case analysis.

Contribution

It offers explicit delta-invariant computations for specific log Fano pairs, illustrating the application of the Abban-Zhuang method to surface cases.

Findings

Identified new K-stable and K-semistable log Fano pairs.

Extended the application of the Abban-Zhuang method to plane curves.

Provided explicit delta-invariant values for pairs involving degree ≤ 4 curves.

Abstract

We compute the $\delta$-invariant for pairs $(\mathbb{P}^2, \lambda C_d)$, where $C_d$ is a plane curve of degree $d \leq 4$. These computations provide new examples of $K$-stable and $K$-semistable log Fano pairs, and contribute to the study of $K$-stability of log Fano varieties via the Abban-Zhuang method, which reduces higher-dimensional problems to the surface case.