论文标题
关于方案形态的弧纤维
On arc fibers of morphisms of schemes
论文作者
论文摘要
给定态度$ f \ colon x \ to y $ y $在一个领域的y $，我们证明了关于弧空诱导地图的纤维$ f_ \ f_ \ infty \ infty \ colon x_ \ infty \ infty \ to y__ \ infty $。 Assuming that $f$ is quasi-finite and $X$ is separated and quasi-compact, our theorem states that $f_\infty$ has topologically finite fibers of bounded cardinality and its restriction to $X_\infty \setminus R_\infty$, where $R$ is the ramification locus of $f$, has scheme-theoretically finite reduced fibers.当$ f $是$ f_ \ infty $的纤维的基数时，我们还提供了有效的限制，当$ f $是代数封闭的字段上品种的有限形态时，描述了$ f_ \ infty $的后果座位，并且证明了$ f_ \ suftty $的一般标准是有限类型的形态。我们将这些结果应用于进一步探索弧空的局部结构。一种应用是，在各种弧形空间的稳定点处的局部环有限地产生的最大理想和拓扑结构的频谱，这与这些环通常不是noe夫的事实相反。还获得了这些环的尺寸较低的尺寸。另一个应用程序为嵌入维度和嵌入弧空的嵌入编码赋予了半内态属性，该弧空间延伸到该设置Noetherian Local Rings上的Lech定理，并转化为Mather Log差异的半持续性属性。本文讨论了其他应用程序。
Given a morphism $f \colon X \to Y$ of schemes over a field, we prove several finiteness results about the fibers of the induced map on arc spaces $f_\infty \colon X_\infty \to Y_\infty$. Assuming that $f$ is quasi-finite and $X$ is separated and quasi-compact, our theorem states that $f_\infty$ has topologically finite fibers of bounded cardinality and its restriction to $X_\infty \setminus R_\infty$, where $R$ is the ramification locus of $f$, has scheme-theoretically finite reduced fibers. We also provide an effective bound on the cardinality of the fibers of $f_\infty$ when $f$ is a finite morphism of varieties over an algebraically closed field, describe the ramification locus of $f_\infty$, and prove a general criterion for $f_\infty$ to be a morphism of finite type. We apply these results to further explore the local structure of arc spaces. One application is that the local ring at a stable point of the arc space of a variety has finitely generated maximal ideal and topologically Noetherian spectrum, something that should be contrasted with the fact that these rings are not Noetherian in general; a lower-bound to the dimension of these rings is also obtained. Another application gives a semicontinuity property for the embedding dimension and embedding codimension of arc spaces which extends to this setting a theorem of Lech on Noetherian local rings and translates into a semicontinuity property for Mather log discrepancies. Other applications are discussed in the paper.
