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 P^r-1-bundle over C has a linearly stable model in P^n-1 for any n r+2. Furthermore; linear stability of (E, V) is a necessary condition for stability of the kernel bundle M₄, ₕ 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 M₄, ₕ for certain (E, V) of type (r, d, r+2) where E is not necessarily stable.
Castorena et al. (Sun,) studied this question.