Syzygies, Hurwitz spaces and Ulrich sheaves

Hurwitz空间是亏格g的光滑曲线对射影直线的k次分歧覆盖的参数空间。通过黎曼存在定理，亏格g的每一条曲线都以这种方式出现，只要适当地选择k。我们建议使用syzygy方法来确定渐近的几何性质（科代拉维），这个空间，并研究其奇异性。我们的方法应该完全描述的决议一般k-R曲线的属g和导致一个著名的猜想的Green-Lazarsfeld关于最小分辨率的坐标环相关的曲线上的一个非特殊的线丛的一个完整的解决方案。提出了绿色猜想的几个推广，并将借助计算机代数进行检验.

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.