Towards an Enumerative Geometry of the Moduli Space of Curves
Towards an Enumerative Geometry of the Moduli Space of Curves
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DOI：
10.1007/978-1-4757-9286-7_12
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发表时间：
1983
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影响因子：
--
通讯作者：
D. Mumford
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文献类型：
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作者：
D. Mumford
本文的目的是形成并开始探索任意属的所有曲线集合的枚举几何。我们的意思是为广义曲线的模空间及其紧化建立一个周氏环，定义这个环中最重要的类，并计算这些类中一些几何上重要的轨迹的类。我们把格拉斯曼人的数列几何作为这个模型。这里的基本类是重言或普遍束的陈类，它们生活在格拉斯曼上，最基本的环是满足各种舒伯特条件的线性空间的轨迹，即所谓的舒伯特环。然而，由于哈里斯和我已经证明了这一点，所以我不是一个人。国家[H-M]不可能期望mg1分解成基本细胞，也不可能期望mg1的周氏环像格拉斯曼环那样简单。但在另一个方向上，J. Harer [Ha]和P. Miller [Mi]有强有力的结果表明，至少mg的低维同调群表现良好。此外，它还证明了许多几何上的自然循环都可以用少量的基本类来表示。
The goal of this paper is to formulate and to begin an exploration of the enumerative geometry of the set of all curves of arbitrary genusg. By this we mean setting up a Chow ring for the moduli spaceMgof curves of genusgand its compactificationMg, defining what seem to be the most important classes in this ring and calculating the class of some geometrically important loci inMgin terns of these classes. We take as a model for this the enumerative geometry of the Grassmannians. Here the basic classes are the Chern classes of the tautological or universal bundle that lives over the Grassmannian, and the most basic cycles are the loci of linear spaces satisfying various Schubert conditions: the so-called Schubert cycles. However, since Harris and I have shown that forglarge,Mgis not unir.ational [H-M] it is not possible to expect thatMghas a decomposition into elementary cells or that the Chow ring ofMgis as simple as that of the Grassmannian. But in the other direction, J. Harer [Ha] and P. Miller [Mi] have strong results indicating that at least the low dimensional homology groups ofMgbehave nicely. Moreover, it appers that many geometrically natural cycles are all expressible in terms of a small number of basic classes.