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The notion of linear stability of a variety in projective space wasintroduced by Mumford in the context of GIT. It has subsequently been appliedby Mistretta and others to Butler's conjecture on stability of the dual spanbundle (DSB) $M_{V, E}$ of a general generated coherent system $( E, V )$. Wesurvey recent progress in this direction on rank one coherent systems, prove anew result for hyperelliptic curves, and state some open questions. We thenextend the definition of linear stability to generated coherent systems ofhigher rank. We show that various coherent systems with unstable DSB studied byBrambila-Paz, Mata-Gutierrez, Newstead and Ortega are also linearly unstable.We show that linearly stable coherent systems of type $(2, d, 4)$ for lowenough $d$ have stable DSB, and use this to prove a particular case of ageneralized Butler conjecture. We then exhibit a linearly stable generatedcoherent system with unstable DSB, confirming that linear stability of $( E, V)$ in general remains weaker than semistability of $M_{V, E}$ in higher rank.We end with a list of open questions on the higher rank case.

Linear stability of coherent systems and applications to Butler's conjecture