arXiv 2204.047722022-04-21math.AGmath.DG

模空间$\mathcal M_{0,n+3}$上由遗忘映射诱导的有理曲线构成的$(n+3)$网

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

Luc Pirio

备注：: 121 pages, 4 figures. Second version : a few typos have been corrected

摘要：

讨论了$n+3$遗忘映射$\mathcal M_{0,n+3}\rightarrow \mathcal M_{0,n+2}$定义的$\mathbf P^1$上$n+3$点的射影构形的模空间$\mathcal M_{0,n+3}$上的曲线网$\boldsymbol{\mathcal W}_{0,n+3}$.我们回顾经典的结果表明，该网络是线性化时$n$是奇数，或相当于一个网络的二次曲线时$n$是偶数。然后，我们转向这些网络的阿贝尔关系（AR）。在回顾了著名的$n=2$（与双对数的5项函数恒等式有关）的情况后，我们重点讨论了6-网$\boldsymbol{\mathcal W}_{0,6}$的情况。我们表明，这个网络是同构的网络所形成的线包含在塞格雷的三次原始$\boldsymbol{S}\subset \mathbf P^4$和一种“阿贝尔定理”允许描述AR的$\boldsymbol{\mathcal W}_{0,6}$通过阿贝尔2-形式的Fano表面上的$F_1(\boldsymbol{S})\subset G_1(\mathbf P^4)$的线包含在$\boldsymbol{S}$。我们由此推导出，$\boldsymbol{\mathcal W}_{0,6}$具有最大秩，其所有AR都是有理数，并且这些AR跨越一个不可约$\mathfrak S_6$ -模的空间。然后，我们采取了一种方法，由于达米亚诺，我们纠正的情况下，当$n$是奇数：它导致了一个抽象的描述空间AR的$\boldsymbol{\mathcal W}_{0,n+3}$作为$\mathfrak S_{n+3}$ -表示。特别是，我们得到，这个网站有最大的秩为任何$n\geq 2$。最后，我们考虑了“Euler交换关系$\boldsymbol{\mathcal E}_n$”，它是Damiano利用Gelfand-MacPherson多对数形式理论从$\mathbf R^{n+3}$中2-平面的Grassmannian上的一个特征类构造的$\boldsymbol{\mathcal W}_{0,n+3}$的一个特殊AR。给出了$\boldsymbol{\mathcal E}_n$的组分的显式结构公式，证明了该公式对$n\leq 12$是正确的。

英文摘要：

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 $n\leq 12$.

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