The notion of linear stability of a variety in a projective space was introduced by Mumford in the context of GIT. It has subsequently been applied by E. Mistretta, L. Stoppino and others to study stability of the dual span bundle (DSB) of a generated linear series $(E,V)$ over a smooth curve. In this talk we explain the first steps we give toward a generalization of linear stability in higher dimension. First, we recall some results on linear stability for generated linear series on curves. We then extend the definition of linear stability to generated coherent systems $(E,V)$, where $E$ is a vector bundle over a smooth curve and $V$ is a subspace of global sections of $E$ generating $E$. We give examples of linearly unstable coherent systems with unstable DSB. We show that linearly stable coherent systems of rank 2 with $\text{dim}(V)=4$ and $\text{deg}(E)=d$ for low enough $d$ have stable DSB, and use this to prove a particular case of Butler’s conjecture. We give an example of a linearly stable generated coherent system with unstable DSB, confirming that in higher rank linear stability of $(E,V)$ in general remains weaker than semistability of the DSB, $M_{V,E}$.

This is part of a joint work with George H. Hitching (Oslo Metropolitan University) and Erick Luna (UNAM).