Geometry of linearly stable coherent systems over curves

Let E be a vector bundle over a smooth curve C, and V a generating space of sections of E. We characterise Mumford linear stability of the associated projective model of P E^ in P V^ in terms of geometric and cohomological properties of the coherent system $(E, V)$, and give some applications. We show that any Pʳ⁻¹-bundle over C has a linearly stable model in Pⁿ⁻¹ for any n ≥ r+2. Furthermore; linear stability of $(E, V)$ is a necessary condition for stability of the kernel bundle ME, V of $(E, V)$, which is predicted by Butler's conjecture for general C and $(E, V)$. We give new examples showing that it is not in general sufficient; in particular, a general bundle E of large degree fits into a linearly stable coherent system $(E, V)$ with nonsemistable kernel bundle. Finally, we use these ideas to show the stability of ME, V for certain $(E, V)$ of type $(r, d, r+2)$ where E is not necessarily stable.