论文标题
在$（n+3）$ - 网络上，由Moduli空间上的健忘地图引起的有理曲线$ \ MATHCAL M_ {0，N+3} $
On the $(n+3)$-webs by rational curves induced by the forgetful maps on the moduli spaces $\mathcal M_{0,n+3}$
论文作者
论文摘要
我们讨论曲线网络$ \ boldsymbol {\ Mathcal w} _ {0，n+3} $在Moduli space上m_ {0，n+3} \ rightarrow \ Mathcal M_ {0，n+2} $。我们回想起经典的结果，这些结果表明，当$ n $奇怪时，该网络可线化，或者当$ n $均匀时，圆锥形等同于网络。然后，我们转向这些网的Abelian关系（ARS）。在回忆起$ n = 2 $的众所周知的情况之后（与Diologarithm的5-TERMS功能身份有关），我们重点关注6-Web $ \ boldsymbol {\ Mathcal W} _ {0,6} $的情况。我们表明，这个网络是由Segre的Cutic Primal $ \ boldsymbol {s} \ subset \ Mathbf P^4 $组成的线形成的网络的同构，并且这种“ Abel theorem”允许在$ \ boldsymbol {\ Mathcal W} _ = 0,6,6 _ {0,6,6 _ = 0,6,6 _ {0,6,6,6,6,6,6 _ {0,6,6,6,6 _ = 0,6,6,6 _ {0,6,6,6,6,6,6 _^0,6,6,6 _ = 0,6,6 _ = 0,6,6,6 _ = 0,6,6,6,6,6 _^0,6,6,6 _ = 0,6,6,6}表面$ f_1（\ boldsymbol {s}）\ subset g_1（\ mathbf p^4）$ $ \ boldsymbol {s} $中包含的行。我们从中得出$ \ boldsymbol {\ Mathcal W} _ {0,6} $具有最大排名，其所有ARS合理，并且这些空间跨越了一个不可减至的$ \ Mathfrak S_6 $ -MODULE的空间。然后，由于达米亚诺（Damiano），我们采用了一种方法，即当$ n $奇怪的情况下，我们会纠正：它导致对$ \ boldsymbol {\ mathcal w} _ {0，n+3} $作为$ \ Mathfrak s_ {n+3} $的ARS空间的抽象描述。特别是，我们获得该网络对任何$ n \ geq 2 $的最大排名。最后，我们考虑“ Euler的Abelian Relation $ \ BoldSymbol {\ Mathcal E} _n $'，这是$ \ boldsymbol {\ Mathcal W} _ {\ Mathcal w} _ {0，n+3} $由Damiano构建的Damiano由Damiano构建的，是由2- plane of 2-planes of 2-Planian of 2-planian of 2-planian of 2-planian in $ \ mathbf r^r^r^r^r^r^r. Gelfand-Macpherson的聚类形式理论。我们为$ \ boldsymbol {\ Mathcal e} _n $的组件提供了明确的猜想公式，我们证明是正确的，对于$ n \ leq 12 $。
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 $n\leq 12$.
