Generalized Linear Systems on Curves and Their Weierstrass Points
Generalized Linear Systems on Curves and Their Weierstrass Points
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曲线上的广义线性系统及其Weierstrass点
DOI：
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发表时间：
2009
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通讯作者：
Patricia Nogueira
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作者：
E. Esteves;Patricia Nogueira
设C为特征为0的代数闭域上的戈伦斯坦投影曲线。C上的一个广义线性系统是由C上的一个无扭，秩1轴，和一个向量空间ε: V→Γ（C, k）的映射组成的对（k, ε）。如果系统在C的每一个不可约分量上都是非退化的，我们就给它关联一个0环W，即它的Weierstrass环。然后我们证明了对于每一个退化到C的单参数曲线族，以及沿C t的线性系统族（__t, ε t），且__t可逆，退化到（__t, ε），对应的Weierstrass因子退化到一个相关的0环为w的子方案。我们证明了极限子方案总是包含一个“固有”子方案，正则地与（__t, ε）相关，但极限本身依赖于族__t。
Let C be a projective Gorenstein curve over an algebraically closed field of characteristic 0. A generalized linear system on C is a pair (ℐ, ε) consisting of a torsion-free, rank-1 sheaf ℐ on C, and a map of vector spaces ε: V → Γ(C, ℐ). If the system is nondegenerate on every irreducible component of C, we associate to it a 0-cycle W, its Weierstrass cycle. Then we show that for each one-parameter family of curves C t degenerating to C, and each family of linear systems (ℒ t , ε t ) along C t , with ℒ t invertible, degenerating to (ℐ, ε), the corresponding Weierstrass divisors degenerate to a subscheme whose associated 0-cycle is W. We show that the limit subscheme contains always an “intrinsic” subscheme, canonically associated to (ℐ, ε), but the limit itself depends on the family ℒ t .