Exploring the interplay of semistable vector bundles and their restrictions on reducible curves
Suhas B. N., Praveen Kumar Roy, Amit Kumar Singh
TL;DR
This work analyzes when the restriction of a $w$-semistable vector bundle $E$ on a comb-like curve $C$ remains semistable on each component, deriving sharp rank-$2$ and higher-rank criteria via the simplified inequalities $w_j \chi \leq \chi_j \leq w_j \chi + n$ and examining destabilizing subbundles. It then applies these results to kernel bundles $M_{E,V}$ attached to generated pairs $(E,V)$, establishing strong instability under nonzero $\\ker(\\rho_j|_V)$ in low rank and giving sufficient and, under Butler's conjecture, necessary conditions for $M_{E,V}$ to be $w$-semistable, with a complete line-bundle ($L,V$) classification under mild degree hypotheses. The paper extends and refines existing results on reducible nodal curves to comb-like degenerations, offering a flexible framework for understanding higher-rank Brill–Noether phenomena on degenerations and linking componentwise semistability to global kernel-bundle semistability through explicit polarization construction. These results provide new tools for constructing semistable kernel bundles on degenerations and illuminate the role of componentwise data in higher-rank stability problems.
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.