In this paper, we study the geometry of GIT configurations of $n$ ordered points on ${\mathbb{P}}^1$ both from the birational and the biregular viewpoint. In particular, we prove that any extremal ray of the Mori cone of effective curves of the quotient
${\left({\mathbb{P}}^1\right)}^n$// $PGL\left(2\right)$, taken with the symmetric polarization, is generated by a one dimensional boundary stratum of the moduli space. Furthermore, we develop some technical machinery that we use to compute the canonical divisor and the Hilbert polynomial of ${\left({\mathbb{P}}^1\right)}^n$ // $PGL\left(2\right)$ in its natural embedding, and its automorphism group.

中文翻译：

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线上点配置的模空间的双几何

在本文中，我们研究了GIT配置的几何 $ñ$ 上的有序点 ${\mathbb{P}}^{\mathrm{1个}}$从双理性和双正规的角度来看。特别是，我们证明了商数有效曲线的任何Mori锥极线
${（{\mathbb{P}}^{\mathrm{1个}}）}^{ñ}$// $聚乳酸（\mathrm{2个}）$由对称极化获得的，是由模空间的一维边界层生成的。此外，我们开发了一些技术机制来计算正则除数和Hilbert多项式${（{\mathbb{P}}^{\mathrm{1个}}）}^{ñ}$ // $聚乳酸（\mathrm{2个}）$ 它的自然嵌入和它的同构群。