Embedding codimension of the space of arcs
Embedding codimension of the space of arcs
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DOI：
10.1017/fmp.2021.19
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发表时间：
2020-01
期刊：
影响因子：
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通讯作者：
C. Chiu;Tommaso de Fernex;Roi Docampo
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文献类型：
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作者：
C. Chiu;Tommaso de Fernex;Roi Docampo
本文引入了任意局部环的嵌入余维的概念，建立了一些一般性质，并详细研究了域上有限型概型弧空间的情形。把嵌入余维看作是奇异性的度量，我们的主要结果可以解释为弧空间的奇异性在完全嵌入基础方案的奇异轨迹中的弧处最大，并且随着我们远离所述轨迹而逐渐改善。作为一个应用程序，我们补充了定理的德林费尔德，格林伯格和Kazhdan正式的邻域在弧空间提供一个匡威定理，最佳界的嵌入余维的正式模型中出现的声明，一个精确的公式嵌入维的模型构建在德林费尔德的证明和几何意义的方式实现分解定理中所述。
Abstract We introduce a notion of embedding codimension of an arbitrary local ring, establish some general properties and study in detail the case of arc spaces of schemes of finite type over a field. Viewing the embedding codimension as a measure of singularities, our main result can be interpreted as saying that the singularities of the arc space are maximal at the arcs that are fully embedded in the singular locus of the underlying scheme, and progressively improve as we move away from said locus. As an application, we complement a theorem of Drinfeld, Grinberg and Kazhdan on formal neighbourhoods in arc spaces by providing a converse to their theorem, an optimal bound for the embedding codimension of the formal model appearing in the statement, a precise formula for the embedding dimension of the model constructed in Drinfeld’s proof and a geometric meaningful way of realising the decomposition stated in the theorem.