On kernel bundles over reducible curves with a node

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 $e{v}_V:V\otimes {\mathcal{O}}_C\to E$, 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.

Some typos corrected. Last version is accepted for publication in "International Journal of Mathematics"

References in corpus (1)

• Genus 2 curves and generalized theta divisors

Cited by in corpus (4)

• Coherent systems on curves of compact type
• Coherent systems and BGN extensions on nodal reducible curves
• Nodal curves and polarizations with good properties
• On the stability of kernel bundles over chain-like curves