摘要翻译：
设F是一个平面射影格式族，其几何泛型纤维是约化和不可约的。我们给出一种特殊纤维的条件（家庭的“极限”），以保证它也被减少。这些情况往往也意味着一般纤维是正常的。在“几何顶点分解”[Knutson-Miller-Yong'07]的设置中，这些条件尤其容易检查。所用的主要工具是相应的Limit_BranchVariety_[Alexeev-Knutson'06],它通过构造进行约简，并映射到Limited子格式；我们的技巧是用正规性来表明branchvariety映射一定是同构的。作为证明，我们给出了有限型Schubert变体是正规的和Cohen-Macaulay的一个本质朴素的证明。该证明不涉及任何奇异性的解析或上同调消失技术（例如诉诸特征p）。
---
英文标题：
《Automatically reduced degenerations of automatically normal varieties》
---
作者：
Allen Knutson
---
最新提交年份：
2007
---
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
--

---
英文摘要：
  Let F be a flat family of projective schemes, whose geometric generic fiber is reduced and irreducible. We give conditions on a special fiber (a "limit" of the family) to guarantee that it too is reduced. These conditions often imply also that the generic fiber is normal. The conditions are particularly easy to check in the setup of a "geometric vertex decomposition" [Knutson-Miller-Yong '07].   The primary tool used is the corresponding limit _branchvariety_ [Alexeev-Knutson '06], which is reduced by construction, and maps to the limit subscheme; our technique is to use normality to show that the branchvariety map must be an isomorphism.   As a demonstration, we give an essentially naive proof that Schubert varieties in finite type are normal and Cohen-Macaulay. The proof does not involve any resolution of singularities or cohomology-vanishing techniques (e.g. appeal to characteristic p).
---
PDF链接：
https://arxiv.org/pdf/0709.3999