Algebraic Curves and its Applications

设W^r_d(C)是通常定义的线丛方案，可定义为Pic^d(C)的一个子方案。Kempfand Kleiman-Laksov证明了簇W^r_d(C)至少有p=g-(r+1)(g-d+r)。Griffith-Harris和Fulton-Lazarsfeld证明了当C是模空间M_g中的一般曲线时，W^r_d(C)是p=g-(r+1)(g-d+r)维光滑的。“一般曲线”是指M_g中有一个开子集U⊂M_g，W^r_d(C)对任意曲线C∈U都是p=g-(r+1)(g-d+r)维光滑的。对于一条允许h亏格曲线E的二重覆盖π：C→E的曲线C，我们可以期待W^r_d(C)的许多有趣的结构。在本文中，我们给出了W^r_d(C)的许多性质，特别是W^1_d(C)。

Let W^r_d (C)be a scheme of line bundles defined by usually, can be defined as a subscheme of Pic^d (C)). Kempfand Kleiman-Laksov prove that the variety W^r_d (C) hasdimension at least p = g-(r+1)(g-d+r). Griffith -Harris and Fulton-Lazarsfeld prove that W^r_d(C) is smooth of dimension p = g-(r+1)(g-d+r) when C is a general curve in the moduli space M_g. "A general curve" means there is an open subset U⊂M_g in M_g, W^r_d(C) is smooth of dimension p = g-(r+1)(g-d+r) for any curve C ∈ U. For a curve C which admits a double covering π:C→E to a curve E of genus h, i.e. a curve which we can not regard as a general curve, we can expect many interesting structure of W^r_d (C). In the papers, we give many good properties of W^r_d(C), especially W^1_d(C).