Automorphism loci for the moduli space of rational maps
Automorphism loci for the moduli space of rational maps
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DOI：
10.4064/aa8548-6-2017
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发表时间：
2014-08
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通讯作者：
N. Miasnikov;Brian Stout;P. Williams
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文献类型：
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作者：
N. Miasnikov;Brian Stout;P. Williams
设$k$是特征为$0$的代数闭域，$\数学{M}_d$是$\mathbb{P}^1$上$d$次有理映射的模空间.本文描述了自同构轨迹$A\子集\数学{鼠}_d$和$\数学{A}\子集\数学{M}_d$和奇异轨迹$\数学{S}\子集\数学{M}_d$。特别地，我们确定了对于给定的$d$，哪些群出现在数学{M}_d$中的某个$[\Phi]\的自同构群的子群中，并计算了轨迹的维度。其次，我们证明了一个类似于曲线模格式上奇点的Rauch-Popp-Oort刻画的定理。利用这些区别点的结果计算了$数学{M}_d，数学{M}^S_d，$和$数学{M}^{ss}_d$的Picard和类群。
Let $k$ be an algebraically closed field of characteristic $0$ and $\mathcal{M}_d$ the moduli space of rational maps on $\mathbb{P}^1$ of degree $d$ over $k$. This paper describes the automorphism loci $A\subset \mathrm{Rat}_d$ and $\mathcal{A}\subset \mathcal{M}_d$ and the singular locus $\mathcal{S}\subset\mathcal{M}_d$. In particular, we determine which groups occur as subgroups of the automorphism group of some $[\phi]\in\mathcal{M}_d$ for a given $d$ and calculate the dimension of the locus. Next, we prove an analogous theorem to the Rauch-Popp-Oort characterization of singular points on the moduli scheme for curves. The results concerning these distinguished loci are used to compute the Picard and class groups of $\mathcal{M}_d, \mathcal{M}^s_d,$ and $\mathcal{M}^{ss}_d$.