论文标题
dame奇异性的算术在尺寸$ 2 $
The arithmetic of tame quotient singularities in dimension $2$
论文作者
论文摘要
让$ k $为一个字段，$ x $ a的品种具有驯服的商奇异性，$ \ tilde {x} \ to x $ to n奇异性的分辨率。 x（k）$中的任何平滑理性点$ x \ in the lang-nishimura定理升至$ \ tilde {x} $，但是如果$ x $是单数的，则可能是错误的。 但是，对于某些类型的奇异性，理性点可以提升。这些称为$ \ mathrm {r} $的奇异性。这个概念在对品种模量领域的研究中具有应用，并产生了lang-nishimura定理的增强版本，其中平滑度假设放松。 我们将$ \ mathrm {r} $ in Dimension $ 2 $的类型$ \ mathrm {r} $完全分类；特别是，我们表明，尺寸$ 2 $中的每个非环状驯服商奇异性都是$ \ mathrm {r} $的类型，并且大多数环状奇异性也是$ \ mathrm {r} $的类型。
Let $k$ be a field, $X$ a variety with tame quotient singularities and $\tilde{X}\to X$ a resolution of singularities. Any smooth rational point $x\in X(k)$ lifts to $\tilde{X}$ by the Lang-Nishimura theorem, but if $x$ is singular this might be false. For certain types of singularities the rational point is guaranteed to lift, though; these are called singularities of type $\mathrm{R}$. This concept has applications in the study of the fields of moduli of varieties and yields an enhanced version of the Lang-Nishimura theorem where the smoothness assumption is relaxed. We classify completely the tame quotient singularities of type $\mathrm{R}$ in dimension $2$; in particular, we show that every non-cyclic tame quotient singularity in dimension $2$ is of type $\mathrm{R}$, and most cyclic singularities are of type $\mathrm{R}$ too.
