Algebraic Curves and its Applications

设W^r_d（C）是由定义的线丛概型，可定义为P_c（d（C））的一个子概型. Kempfand Kleiman-Laksov证明了簇W^r_d（C）的维数至少为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）是维数p = g-（r+1）（g-d+r）的光滑曲线.“一般曲线”是指在M_g中存在一个开子集U ∈ M_g，对任意曲线C ∈ U，W^r_d（C）在维数p = g-（r+1）（g-d+r）上是光滑的.对于亏格为h的曲线E的二重覆盖π：C→E的曲线C，即不能视为一般曲线的曲线，我们可以期待W^r_d（C）的许多有趣的结构。本文给出了W^r_d（C），特别是W ^1_d（C）的许多好的性质.

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).