摘要翻译：
我们证明了g\ge3$属4-正则嵌入曲线的Hilbert和Chow格式中合适轨迹的GIT商是Schubert在引用Schubert中用Chow变体和3-正则模型构造的伪稳定曲线的模空间$\bar{M}_g^{ps}}$。在Hilbert格式变体中唯一需要的新成分是更仔细地分析具有椭圆尾的曲线$x$的$m^{\text{th}}$Hilbert点关于某个1-ps$\lambda$的稳定性。我们计算$\lambda$作用的确切权重，而不仅仅是这个权重的$M$中的前导项。对有理尖尾曲线稳定性的类似分析允许我们确定稳定的半可定的4-正则Chow轨迹。虽然这里的商的几何更加复杂，因为存在严格的半稳定轨道，但我们可以再次将其识别为$\bar{M}_g^{\text{ps}}$。作为副产品，我们的计算得到了M$-Hilbert不稳定和M$-Hilbert稳定的例子，它们都是严格半稳定的。
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英文标题：
《Stability of Tails and 4-Canonical Models》
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作者：
Donghoon Hyeon and Ian Morrison
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最新提交年份：
2009
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分类信息：

一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  We show that the GIT quotients of suitable loci in the Hilbert and Chow schemes of 4-canonically embedded curves of genus $g\ge 3$ are the moduli space $\bar{M}_g^{\text{ps}}$ of pseudo-stable curves constructed by Schubert in \cite{Schubert} using Chow varieties and 3-canonical models. The only new ingredient needed in the Hilbert scheme variant is a more careful analysis of the stability with respect to a certain 1-ps $\lambda$ of the $m^{\text{th}}$ Hilbert points of curves $X$ with elliptic tails. We compute the exact weight with which $\lambda$ acts, and not just the leading term in $m$ of this weight. A similar analysis of stability of curves with rational cuspidal tails allows us to determine the stable and semistable 4-canonical Chow loci. Although here the geometry of the quotient is more complicated because there are strictly semi-stable orbits, we are able to again identify it as $\bar{M}_g^{\text{ps}}$. Our computations yield, as byproducts, examples of both $m$-Hilbert unstable and $m$-Hilbert stable $X$ that are Chow strictly semi-stable.
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PDF链接：
https://arxiv.org/pdf/0806.1269