# On kernel bundles over reducible curves with a node

## TL;DR

This paper investigates the stability properties of kernel bundles over reducible curves with a node, revealing unexpected results that contrast with the smooth case, and provides conditions for their semistability.

## Contribution

It extends the study of kernel bundles to reducible curves with nodes, showing new stability behaviors and establishing conditions for semistability in this setting.

## Key findings

- Kernel bundles over nodal curves are often not semistable, contrary to smooth cases.
- Conditions for semistability of kernel bundles are identified when the subspace V is contained in H^0(E).
- Specific criteria for the semistability of kernel bundles associated to line bundles are provided.

## Abstract

Given a vector bundle $E$ on a complex reduced curve $C$ and a subspace $V$ of $H^0(E)$ which generates $E$, one can consider the kernel of the evaluation map $ev_V:V\otimes \mathcal{O}_C\to E$, i.e. the {\it kernel bundle } $M_{E,V}$ associated to the pair $(E,V)$. Motivated by a well known conjecture of Butler about the semistability of $M_{E,V}$ and by the results obtained by several authors when the ambient space is a smooth curve, we investigate the case of a curve with one node. Unexpectedly, we are able to prove results which goes in the opposite direction with respect to what is known in the smooth case. For example, $M_{E,H^0(E)}$ is actually quite never $w$-semistable. Conditions which gives the $w$-semistability of $M_{E,V}$ when $V\subset H^0(E)$ or when $E$ is a line bundle are then given.