Syzygies, gonality and symmetric products of curves
- Rob Lazarsfeld, February 6, 2015

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In the mid 1980s, Mark Green and I conjectured that one could
read off the gonality of an algebraic curve C from the syzygies
among the equations defining any one sufficiently positive
embedding of C. Ein and I recently noticed that a small variant
of the ideas used by Voisin in her work on canonical curves
leads to a quick proof of this gonality conjecture. The proof
involves the geometry of certain vector bundles on the
symmetric product of C. I will describe this circle of
ideas. If time permits, I will also discuss a partial
generalization, with Ein and Yang, to smooth varieties of all
dimensions, as well as a recent effective statement for curves
by Rathmann.