On the Resonance Varieties of Vector Bundles on Curves

We use the resonance (Papadima, Sociu, Aprodu and others) to study vector bundles E on a smooth curve X of genus g. Our key observation is that if we know it for a specific integer c, then we get strong information on X and E. Fix an integer c≥g+1. Take genus g curves X and Y and vector bundles E on X and F on Y with the same ranks and degrees. Our main result is that if E and F are sufficiently positive and they have birational resonance in degree c, then X and Y have isomorphic Jacobians. We also study the resonance for a singular curve with arithmetic genus 1.