The minimal formal models of curve singularities
The minimal formal models of curve singularities
复制标题
曲线奇点的最小形式模型
DOI：
10.1142/s0129167x17500811
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发表时间：
2017
影响因子：
0.6
通讯作者：
J. Sebag
中科院分区：
文献类型：
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作者：
David Bourqui;J. Sebag
设k是一个域。我们引入了一个新的几何不变量，即最小形式模型，它与每个曲线奇点(定义在k上)有关。这是一种Notherian仿射形式k-格式，它是利用本原k-参数化的伴随圆弧格式中的形式邻域来定义的。对于平面曲线A2n-奇点，我们证明了这个不变量是SPF(k[[Z]]/<Zn+1>)。我们还得到了所谓的广义尖点的最小形式模型的信息。我们从奇点理论的角度介绍了这些极小形式模型研究中的各种问题。我们的结果提供了答案的第一个积极因素。作为前人结果的直接应用，我们证明了，一般而言，满足Drinfeld-Grinberg-Kazhdan关于非退化弧上弧格式形式邻域结构的定理的同构不是来自喷流水平。从某种意义上说，这表明Drinfeld-Grinberg-Kazhdan定理不是形式上的推论。
Let k be a field. We introduce a new geometric invariant, namely the minimal formal models, associated with every curve singularity (defined over k). This is a noetherian affine adic formal k-scheme, defined by using the formal neighborhood in the associated arc scheme of a primitive k-parametrization. For the plane curve A2n-singularity, we show that this invariant is Spf(k[[Z]]/〈Zn+1〉). We also obtain information on the minimal formal model of the so-called generalized cusp. We introduce various questions in the direction of the study of these minimal formal models with respect to singularity theory. Our results provide the first positive elements of answer. As a direct application of the former results, we prove that, in general, the isomorphisms satisfying the Drinfeld–Grinberg–Kazhdan theorem on the structure of the formal neighborhoods of arc schemes at non-degenerate arcs do not come from the jet levels. In some sense, this shows that the Drinfeld–Grinberg–Kazhdan theorem is not a formal consequenc...