Algebraic Curves and its Applications

设W^r_d (C)是线束的格式，通常可以定义为Pic^d (C)的一个子格式。Kempfand Kleiman-Laksov证明了W^r_d (C)的维数至少为p = g-(r+1)（g-d+r）。Griffith - harris和Fulton-Lazarsfeld证明当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）维数的光滑。“一般曲线”是指在M_g中存在一个开放子集U∧M_g，对于任意曲线C∈U， W^r_d(C)是维数p = g-(r+1)（g-d+r）的光滑曲线。对于曲线C允许π:C→E对h属曲线E的双重覆盖，即我们不能将其视为一般曲线的曲线，我们可以期待W^r_d(C)的许多有趣结构。本文给出了W^r_d(C)的许多优良性质，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).