International Journal of Mathematics ( IF 0.604 ) Pub Date : 2020-06-23 , DOI: 10.1142/s0129167x20500548
Sonia Brivio; Filippo F. Favale

Given a vector bundle $E$ on a complex reduced curve $C$ and a subspace $V$ of $H0(E)$ which generates $E$, one can consider the kernel of the evaluation map $evV:V⊗𝒪C→E$, i.e. the kernel bundle$ME,V$ associated to the pair $(E,V)$. Motivated by a well-known conjecture of Butler about the semistability of $ME,V$ and by the results obtained by several authors when the ambient space is a smooth curve, we investigate the case of a reducible 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, $ME,H0(E)$ is actually quite never $w$-semistable. Conditions which gives the $w$-semistability of $ME,V$ when $V⊂H0(E)$ or when $E$ is a line bundle are then given.

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