| It is a classical theorem that if X is a Riemann surface of genus g, and if X is embedded in projective space in a sufficiently positive manner, then X is cut out by equations of degree 2. Mark Green realized in the early 1980s that one should see this as the first case of a more general picture involving higher syzygies. Around that time, he and I conjectured that one should be able to read off the "gonality" of X -- i.e. the least degree with which X can be expressed as a branched covering of the sphere -- from the resolution of the homogeneous ideal of X with respect to any one sufficiently positive embedding. A couple of years ago Lawrence Ein and I recently noticed that this gonality conjecture in fact follows very simply from a small variant of ideas introduced by Voisin, involving vector bundles on the symmetric product of X. In this talk aimed at non specialists, I will survey this circle of ideas. |