On the $(n+3)$-webs by rational curves induced by the forgetful maps on the moduli spaces $\mathcal M_{0,n+3}$

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
Luc Pirio
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引用次数: 2

Abstract

We discuss the curvilinear web $\boldsymbol{\mathcal W}_{0,n+3}$ on the moduli space $\mathcal M_{0,n+3}$ of projective configurations of $n+3$ points on $\mathbf P^1$ defined by the $n+3$ forgetful maps $\mathcal M_{0,n+3}\rightarrow \mathcal M_{0,n+2}$. We recall classical results which show that this web is linearizable when $n$ is odd, or is equivalent to a web by conics when $n$ is even. We then turn to the abelian relations (ARs) of these webs. After recalling the well-known case when $n=2$ (related to the 5-terms functional identity of the dilogarithm), we focus on the case of the 6-web $\boldsymbol{\mathcal W}_{{0,6}}$. We show that this web is isomorphic to the web formed by the lines contained in Segre's cubic primal $\boldsymbol{S}\subset \mathbf P^4$ and that a kind of `Abel's theorem' allows to describe the ARs of $\boldsymbol{\mathcal W}_{{0,6}}$ by means of the abelian 2-forms on the Fano surface $F_1(\boldsymbol{S})\subset G_1(\mathbf P^4)$ of lines contained in $\boldsymbol{S}$. We deduce from this that $\boldsymbol{\mathcal W}_{{0,6}}$ has maximal rank with all its ARs rational, and that these span a space which is an irreducible $\mathfrak S_6$-module. Then we take up an approach due to Damiano that we correct in the case when $n$ is odd: it leads to an abstract description of the space of ARs of $\boldsymbol{\mathcal W}_{0,n+3}$ as a $\mathfrak S_{n+3}$-representation. In particular, we obtain that this web has maximal rank for any $n\geq 2$. Finally, we consider `Euler's abelian relation $\boldsymbol{\mathcal E}_n$', a particular AR for $\boldsymbol{\mathcal W}_{0,n+3}$ constructed by Damiano from a characteristic class on the grassmannian of 2-planes in $\mathbf R^{n+3}$ by means of Gelfand-MacPherson theory of polylogarithmic forms. We give an explicit conjectural formula for the components of $\boldsymbol{\mathcal E}_n$ that we prove to be correct for $n\leq 12$.
模空间$\mathcal M_{0,n+3}$上的遗忘映射诱导有理曲线的$(n+3)$-网
我们讨论了由$n+3$遗忘映射$\mathcal M_{0,n+3}\rightarrow \mathcal M_{0,n+2}$定义的$\mathbf P^1$上的$n+3$点的投影位形的模空间$\mathcal M_{0,n+3}$上的曲线web $\boldsymbol{\mathcal W}_{0,n+3}$。我们回想一下经典的结果,这些结果表明当$n$为奇数时,这个网络是线性的,或者当$n$为偶数时,它等价于一个由圆锥组成的网络。然后我们转向这些网络的阿贝尔关系(ARs)。在回顾了众所周知的情况下$n=2$(涉及到五项功能同一性的二对数),我们关注6 web的情况$\boldsymbol{\mathcal W}_{{0,6}}$。我们证明了这个网与Segre的三次原$\boldsymbol{S}\subset \mathbf P^4$中包含的线构成的网是同构的,并且证明了一种“Abel定理”允许用$\boldsymbol{S}$中包含的线的Fano曲面$F_1(\boldsymbol{S})\subset G_1(\mathbf P^4)$上的abelian 2-form来描述$\boldsymbol{\mathcal W}_{{0,6}}$的ar。由此我们推导出$\boldsymbol{\mathcal W}_{{0,6}}$在其所有有理数下具有极大秩,并且它们张成一个不可约的$\mathfrak S_6$ -模空间。然后我们采用Damiano的一种方法,当$n$是奇数时,我们纠正了这种方法:它导致对$\boldsymbol{\mathcal W}_{0,n+3}$的ar空间的抽象描述为$\mathfrak S_{n+3}$ -表示。特别地,我们得到了这个网站对于任何$n\geq 2$都有最大的排名。最后,我们考虑了由Damiano利用多对数形式的Gelfand-MacPherson理论从$\mathbf R^{n+3}$的2平面的grassmannian上的一个特征类构造的$\boldsymbol{\mathcal W}_{0,n+3}$的特殊AR“Euler’s abelian关系$\boldsymbol{\mathcal E}_n$”。我们给出了一个明确的推测公式$\boldsymbol{\mathcal E}_n$的组成部分,我们证明了$n\leq 12$是正确的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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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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