Second flip in the Hassett–Keel program: a local description
Second flip in the Hassett–Keel program: a local description
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哈塞特-基尔计划的第二次翻转：本地描述
DOI：
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发表时间：
2017
影响因子：
1.8
通讯作者：
Frederick van der Wyck
中科院分区：
文献类型：
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作者：
J. Alper;M. Fedorchuk;D. Smyth;Frederick van der Wyck
这是三篇文章中的第一篇，其中我们对$OVERLINE{M}_{g}$的对数最小模型程序中的第二个翻转给出了模解释，将具有亏格$2$Weerstrass尾巴的曲线轨迹替换为具有斜面尖点的曲线轨迹。本文对(2/3-unicode[STIX]{x1D716}，2/3+unicode[STIX]{x1D716})$中的$UNICODE[STIX]{x1D6FC}，引入了新的$UNICODE[STIX]{x1D6FC}-曲线稳定条件，并证明了它们是形变开的。这产生了由开放浸入$overline{{mathcal{M}}}_{g}(unicode[STIX]{x1D6FC})$关联的代数堆栈$overline{{mathcal{M}}}_{g}(2/3+unicode[STIX]{x1D716}){hookrightarrow}overline{{mathcal{M}}}_{g}(2/3){hookleftarrow}overline{{mathcal{M}}}_{g}(2/3-unicode[STIX]{x1D716})$。我们证明了在$OVERLINE{{mathcal{M}}_{g}(2/3)$中对应于闭点的曲线$C$周围，这些开放浸入是由$ext{Aut}(C)$在$C$的一阶变形空间上作用的几何不变量理论的变分局部模拟的.
This is the first of three papers in which we give a moduli interpretation of the second flip in the log minimal model program for $overline{M}_{g}$ , replacing the locus of curves with a genus $2$ Weierstrass tail by a locus of curves with a ramphoid cusp. In this paper, for $unicode[STIX]{x1D6FC}in (2/3-unicode[STIX]{x1D716},2/3+unicode[STIX]{x1D716})$ , we introduce new $unicode[STIX]{x1D6FC}$ -stability conditions for curves and prove that they are deformation open. This yields algebraic stacks $overline{{mathcal{M}}}_{g}(unicode[STIX]{x1D6FC})$ related by open immersions $overline{{mathcal{M}}}_{g}(2/3+unicode[STIX]{x1D716}){hookrightarrow}overline{{mathcal{M}}}_{g}(2/3){hookleftarrow}overline{{mathcal{M}}}_{g}(2/3-unicode[STIX]{x1D716})$ . We prove that around a curve $C$ corresponding to a closed point in $overline{{mathcal{M}}}_{g}(2/3)$ , these open immersions are locally modeled by variation of geometric invariant theory for the action of $ ext{Aut}(C)$ on the first-order deformation space of $C$ .