SPP 1489: Syzygies, Hurwitz Spaces and Ulrich Sheaves

Combination of methods in algebra, geometry and number theory. The Hurwitz space is the parameter space of degree k ramified coverings of the projective line by smooth curves of genus g. Via the Riemann existence theorem, every curve of genus g appears in this way, for a suitable choice of k. We propose to use syzygy methods to determine asymptotically the geometric nature (Kodaira dimension) of this space and study its singularities. Our methods should completely describe the resolution of a general k-gonal curve of genus g and lead to a full solution of a well-known conjecture of Green-Lazarsfeld concerning the minimal resolution of the coordinate ring associated to a non-special line bundle on a curve. Several generalizations of Green’s conjecture are proposed and will be tested with the help of computer algebra.

Combination of methods in algebra, geometry and number theory. The Hurwitz space is the parameter space of degree k ramified coverings of the projective line by smooth curves of genus g. Via the Riemann existence theorem, every curve of genus g appears in this way, for a suitable choice of k. We propose to use syzygy methods to determine asymptotically the geometric nature (Kodaira dimension) of this space and study its singularities. Our methods should completely describe the resolution of a general k-gonal curve of genus g and lead to a full solution of a well-known conjecture of Green-Lazarsfeld concerning the minimal resolution of the coordinate ring associated to a non-special line bundle on a curve. Several generalizations of Green’s conjecture are proposed and will be tested with the help of computer algebra.

Duration of project

Start date: 08/2013

End date: 09/2017

Research Areas