On the tautological ring of Mg,n
On the tautological ring of Mg,n
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关于 Mg,n 的同义反复环
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发表时间：
2001
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通讯作者：
R. Vakil
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作者：
R. Vakil
在本节中，我们将简要描述为非专家考虑的对象。[PV]对这些众所周知的观点进行了更详细的非正式阐述。当研究给定g属的黎曼曲面时，人们自然会研究这些物体的模空间。该空间的维数为3g−3，并且由于Deligne和Mumford， [DM]，稳定格g曲线的模空间Mg而具有自然紧化。更一般地说，我们可以定义一个稳定的n点g曲线的模空间，记作Mg，n，在任何给定的代数闭域上（或实际上在SpecZ上）。我们将研究复数。稳定的n点格g复曲线是一个只有节点作为奇异点的紧化曲线，有n个不同的标记光滑点。存在一个稳定性条件：每个有理分量至少有3个特殊点，且属1的每个分量至少有1个特殊点。（特殊点是组件归一化上的一个点，它要么是一个标记点，要么是一个节点的分支。）这个稳定性条件等价于要求点曲线的自同构群是有限的。如果曲线是稳定的，那么一个简短的组合练习表明2g−2 + n > 0。光滑曲线对应的mg，n的开子集记为mg，n。紧型曲线（具有紧雅可比矩阵的曲线，或等价地，具有树状对偶图的曲线）形成一个部分紧化，记为Mg，n。
In this section, we briefly describe the objects under consideration for the sake of non-experts. A more detailed informal exposition of these well-known ideas is given in [PV]. When studying Riemann surfaces of some given genus g, one is naturally led to study the moduli spaceMg of such objects. This space has dimension 3g−3, and has a natural compactification due to Deligne and Mumford, [DM], the moduli space Mg of stable genus g curves. More generally, one can define a moduli space of stable n-pointed genus g curves, denoted Mg,n, over any given algebraically closed field (or indeed over SpecZ). We shall work over the complex numbers. A stable n-pointed genus g complex curve is a compact curve with only nodes as singularities, with n distinct labeled smooth points. There is a stability condition: each rational component has at least 3 special points, and each component of genus 1 has at least 1 special point. (A special point is a point on the normalization of the component that is either a marked point, or a branch of a node.) This stability condition is equivalent to requiring that the automorphism group of the pointed curve be finite. If a curve is stable, then a short combinatorial exercise shows that 2g − 2 + n > 0. The open subset ofMg,n corresponding to smooth curves is denotedMg,n. The curves of compact type (those with compact Jacobian, or equivalently, with a tree as dual graph) form a partial compactification, denoted Mg,n.