We discussed joint
work with Rob Lazarsfeld. We'll describe how to use
Hilbert schemes of points to study syzygies. In particular we are able to solve
a conjecture of Green and Lazarsfeld which says
that one detects the gonality of an algebraic curve
from the Betti numbers of a
minimal resolution of the coordinate ring of a sufficiently ample line bundle
on the curve. We also discuss the generalization of the theorem to higher
dimensions. |