Delta-invariant for projective bundles over a curve and K-semistability

TL;DR

This paper calculates the delta-invariant for ample line bundles on projective bundles over a curve when the vector bundle is strictly Mumford semistable, and explores cases with a single-step Harder-Narasimhan filtration.

Contribution

It provides explicit delta-invariant computations for projective bundles over curves under specific stability conditions, advancing understanding of K-semistability in this context.

Findings

Computed delta-invariants for strictly Mumford semistable bundles.

Analyzed the case with a single-step Harder-Narasimhan filtration.

Enhanced criteria for K-semistability of projective bundles.

Abstract

Consider $E$ a vector bundle over a smooth curve $C$. We compute the $\delta$-invariant of all ample ($\mathbb{Q}$-) line bundles on $\mathbb{P}(E)$ when $E$ is strictly Mumford semistable. We also investigate the case when one assumes that the Harder-Narasimhan filtration of $E$ has only one step.