An intersection-theoretic proof of the Harer–Zagier fomula

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
A. Giacchetto, Danilo Lewa'nski, P. Norbury
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引用次数: 5

Abstract

We provide an intersection-theoretic formula for the Euler characteristic of the moduli space of smooth curves. This formula reads purely in terms of Hodge integrals and, as a corollary, the standard calculus of tautological classes gives a new short proof of the Harer-Zagier formula. Our result is based on the Gauss-Bonnet formula, and on the observation that a certain parametrisation of the $\Omega$-class - the Chern class of the universal $r$-th root of the twisted log canonical bundle - provides the Chern class of the log tangent bundle to the moduli space of smooth curves. Being $\Omega$-classes by now employed in many enumerative problems, mostly recently found and at times surprisingly different from each other, we dedicate some work to produce an extensive list of their general properties: extending existing ones, finding new ones, and writing down some only known to the experts.
Harer-Zagier公式的一个交点理论证明
给出了光滑曲线模空间的欧拉特性的一个交点理论公式。这个公式纯粹是用Hodge积分来表示的,作为一个推论,重言类的标准演算给出了Harer-Zagier公式的一个新的简短证明。我们的结果是基于高斯-博内公式,并观察到$\Omega$-类的某种参数化-扭曲对数正则束的泛$r$根的Chern类-为光滑曲线的模空间提供了log正切束的Chern类。由于$\Omega$-类现在被用于许多列举性问题,大多数是最近发现的,有时彼此之间的差异令人惊讶,我们花了一些工作来产生它们的一般性质的广泛列表:扩展现有的属性,寻找新的属性,并写下一些只有专家知道的属性。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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