标题： 固定生成元的单分支曲线奇点的动机类和伊加泽函数 显示英文标题

标题： Motivic classes of fixed-generators Hilbert schemes of unibranch curve singularities and Igusa zeta functions

摘要： 本文深入研究了具有固定最小生成元数的单分支平面曲线的Hilbert概形。 在Oblomkov、Rasmussen和Shende工作的基础上，我们提供了它们的动机类的公式，并探讨了与同一给定单分支曲线的主Hilbert概形之间的关系。 此外，本文将这一研究专门应用于$(p,q)$-曲线的情况，在这种情况下，我们获得了固定生成元Hilbert概形的动机类的更结构化结果：它们的非负性和拓扑不变性，以及与单生成元概形即$\widehat{\mathcal{O}}_{C,0}$中的主理想的显式关系。 最后，我们关注单生成元局部中的一个特殊开分量，其动机类自然与Denef和Loeser引入的平面弧概形$\mathbb A^2_\infty$上的动机测度以及Igusazeta函数相关。 我们还以奇点的嵌入解析为术语，给出了这些动机类的显式表述，证明了它们的多项式性，并使其成为给定曲线的一个有趣的拓扑不变量。 显示英文摘要

摘要： This paper delves into the study of Hilbert schemes of unibranch plane curves whose points have a fixed number of minimal generators. Building on the work of Oblomkov, Rasmussen and Shende we provide a formula for their motivic classes and investigate the relationship with principal Hilbert schemes of the same given unibranch curve. In addition, the paper specializes this study to the case of $(p,q)$-curves, where we obtain more structured results for the motivic classes of fixed-generators Hilbert schemes: their positivity and topological invariance, and an explicit relationship to one-generator schemes i.e. principal ideals in $\widehat{\mathcal{O}}_{C,0}$. Finally, we focus on a special open component in the one-generator locus, whose motivic class is naturally related to the motivic measure on the arc scheme $\mathbb A^2_\infty$ of the plane introduced by Denef and Loeser as well as to the Igusa zeta function. We also provide an explicit formulation of these motivic classes in terms of an embedded resolution of the singularity, proving their polynomiality as well as making them an interesting topological invariant of the given curve.