On coherent systems of type (n,d,n+1) on Petri curves

We study coherent systems of type \((n,d,n+1)\) on a Petri curve \(X\) of genus \(g\ge2\). We describe the geometry of the moduli space of such coherent systems for large values of the parameter \(\alpha\). We determine the top critical value of \(\alpha\) and show that the corresponding ``flip'' has positive codimension. We investigate also the non-emptiness of the moduli space for smaller values of \(\alpha\), proving in many cases that the condition for non-emptiness is the same as for large \(\alpha\). We give some detailed results for \(g\le5\) and applications to higher rank Brill-Noether theory and the stability of kernels of evaluation maps, thus proving Butler's conjecture in some cases in which it was not previously known.