Exploring the interplay of semistable vector bundles and their restrictions on reducible curves

TL;DR

This paper studies conditions under which vector bundles on comb-like reducible curves remain semistable when restricted to components, extending known results to more complex curve structures using new techniques.

Contribution

It provides new criteria for semistability of vector bundle restrictions on reducible curves and extends existing results to more general curve configurations.

Findings

Criteria for semistability of restrictions on components

Extension of results to higher rank bundles

Analysis of kernel bundles on general reducible curves

Abstract

Let $C$ be a comb-like curve over $\mathbb{C}$, and $E$ be a vector bundle of rank $n$ on $C$. In this paper, we investigate the criteria for the semistability of the restriction of $E$ onto the components of $C$ when $E$ is given to be semistable with respect to a polarization $w$. As an application, assuming each irreducible component of $C$ is general in its moduli space, we investigate the $w$-semistability of kernel bundles on such curves, extending the results (completely for rank two and partially for higher rank) known in the case of a reducible nodal curve with two smooth components, but here, using different techniques.