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 $\mathbb{P} E^\vee$ in $\mathbb{P} V^\vee$ in terms of geometric and cohomological properties of the coherent system $(E, V)$, and give some applications. We show that any $\mathbb{P}^{r-1}$-bundle over $C$ has a linearly stable model in $\mathbb{P}^{n-1}$ for any $n \ge r+2$. Furthermore; linear stability of $(E, V)$ is a necessary condition for stability of the kernel bundle $M_{E, V}$ 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 $M_{E, V}$ for certain $(E, V)$ of type $(r, d, r+2)$ where $E$ is not necessarily stable.