The Moduli Space of Curves and Its Tautological Ring
The Moduli Space of Curves and Its Tautological Ring
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曲线模空间及其同义反复环
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发表时间：
2003
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通讯作者：
R. Vakil
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作者：
R. Vakil
曲线的模空间已经证明了它自己是几何学的中心对象。过去十年，在理解曲线模空间方面取得了实质性进展，涉及几何学（代数、辛和微分）、物理学、拓扑学和组合学等思想。许多新的想法都与模空间的重言式环有关，它是上同调（或周）环的一个子环，（i）似乎是高度结构化的，（ii）似乎包含了所有的几何自然类。这篇文章的目的是给读者一个介绍曲线的模空间及其行为的一些直觉。我们这样做的重点是它的重言式环，特别是其美丽的组合结构。作为动机，我们考虑模空间的原型例子，格拉斯曼（参数化n维向量空间的k维子空间）。它的上同调环具有优美的结构，芒福德建议用同样的方法研究曲线的模空间。我们介绍了（亏格g，n-点）曲线的模空间，有足够的信息来感受它的基本地理。接下来我们介绍重言上同调（或Chow）类。最后，我们描述了一些已知的重言式环，强调组合方面。我们将尽可能少地假设，尽管会用到一些几何学和拓扑学的概念。当然，材料的选择是个人的，重要的工作必然会有遗漏。然而，我们希望已经给了足够的读者新的领域，以欣赏这个重要的几何对象的深层结构。对于专家来说，我们希望这一观点将是一个令人耳目一新的观点，也许会有一些组合上的惊喜。这一高度跨学科领域的许多重要的最新发展超出了本文的范围。例如，最近最令人兴奋的新闻是马德森和韦斯的证明马德森的推广芒福德猜想，基于早期的工作马德森和蒂尔曼;方法是拓扑和奇异理论。其他相关领域包括泰希米勒理论、几何拓扑学、共形场论和算术几何（格罗滕迪克的儿童问题）。即使在代数几何，这一讨论遗漏了重要的近期工作法卡斯，GibneyKeel-Morrison，和许多其他人。我们将使用代数几何语言的方案，但读者应该感到自由，以解释这些松散的某种类型的（环，拓扑）空间和思考，而不是在他们的选择范畴（如代数簇，解析空间，流形，拓扑空间等）。除非另有说明，否则所有维度都是复数（=代数），而不是真实的。这里关于模空间的观点是格罗滕迪克、芒福德和德利涅的观点。Ravi Vakil是斯坦福大学的数学助理教授。他的电子邮件地址是vakil@math.stanford.edu。
T he moduli space of curves has proven itself a central object in geometry. The past decade has seen substantial progress in understanding the moduli space of curves, involving ideas, for example, from geometry (algebraic, symplectic, and differential), physics, topology, and combinatorics. Many of the new ideas are related to the tautological ring of the moduli space, a subring of the cohomology (or Chow) ring that (i) seems highly structured, and (ii) seems to include all the geometrically natural classes. The goal of this article is to give the reader an introduction to the moduli space of curves and some intuition for its behavior. We do this by focusing on its tautological ring and in particular on its beautiful combinatorial structure. As motivation we consider the prototypical example of a moduli space, that of the Grassmannian (parameterizing k -dimensional subspaces of an n-dimensional vector space). Its cohomology ring has an elegant structure, and Mumford suggested studying the moduli space of curves in the same way. We introduce the moduli space of (genus g , n-pointed) curves, with enough information to give a feel for its basic geography. We next introduce the tautological cohomology (or Chow) classes. Finally, we describe some of what is known about the tautological ring, emphasizing combinatorial aspects. We will assume as little as possible, although some notions from geometry and topology will be used. The selection of material is, of course, a personal one, and there are necessarily omissions of important work. However, we hope that enough has been given for the reader new to the field to appreciate the deep structure of this important geometrical object. For the expert we hope that the view will be a refreshing one with perhaps some combinatorial surprises. Many important recent developments in this highly interdisciplinary field are beyond the scope of this article. For example, the most exciting news in recent times is Madsen and Weiss’s proof of Madsen’s generalization of Mumford’s conjecture, based on earlier work of Madsen and Tillmann; the methods are topological and singularity-theoretic. Other relevant fields include Teichmüller theory, geometric topology, conformal field theory, and arithmetic geometry (Grothendieck’s dessins d’enfants). Even within algebraic geometry, this discussion leaves out important recent work of Farkas, GibneyKeel-Morrison, and many others. We will use the algebro-geometric language of schemes, but readers should feel free to interpret these loosely as a certain type of (ringed, topological) space and to think instead in their category of choice (such as algebraic varieties, analytic spaces, manifolds, topological spaces, etc.). All dimensions will be complex (= algebraic), not real, unless otherwise specified. The perspective here on moduli spaces is that of Grothendieck, Mumford, and Deligne. The highlights Ravi Vakil is assistant professor of mathematics at Stanford University. His email address is vakil@math. stanford.edu.