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.