Slope Stability and Exceptional Divisors of High Genus

TL;DR

This paper explores the relationship between slope stability of smooth surfaces and exceptional divisors, establishing conditions under which certain divisors influence stability and providing examples distinguishing slope stability from K-stability.

Contribution

It demonstrates that surfaces with exceptional divisors of genus at least two are slope unstable and characterizes destabilizing divisors, linking geometric properties to stability criteria.

Findings

Surfaces with high-genus exceptional divisors are slope unstable.

Destabilizing divisors must have negative self-intersection and genus at least two.

A surface can be slope stable but not K-stable.

Abstract

We study slope stability of smooth surfaces and its connection with exceptional divisors. We show that a surface containing an exceptional divisor with arithmetic genus at least two is slope unstable for some polarisation. In the converse direction we show that slope stability of surfaces can be tested with divisors, and prove that for surfaces with non-negative Kodaira dimension any destabilising divisor must have negative self-intersection and arithmetic genus at least two. We also prove that a destabilising divisor can never be nef, and as an application give an example of a surface that is slope stable but not K-stable.