坐标下降、单值性和带标记根映射的有限正规化
Coordinate descents, monodromy, and finite normalization of marked-root maps
- Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究带标记根多项式映射的坐标下降与单值性，确定其通用单值群及阶，构造光滑有限正规化并刻画其几何性质，并证明这些映射在稳定等价下不可视为单一加权提升。
AI中文摘要：
我们研究了Harish引入的维数索引的带标记根多项式映射。对于每个$n\geq 4$，它们的低系数块扩展为一个多项式坐标系，对于固定系数的每个选择，给出几何度为$n(n-2)$的三维Keller映射。我们确定了原始映射及所有连续坐标下降的通用单值性：对于奇数$n$，它是循环圈积；对于偶数$n$，它是一个显式的指数为二的子群。其阶为$(n-2)^n n!/\gcd(2,n-2)$。我们构造了光滑有限正规化，并将其仿射源的补集识别为$n-2$个不相交的仿射因子。其中一个有分歧；其余记录逆叶片的非分歧损失。该正规化的Picard群为$\mathbb{Z}^{n-2}$，同伦型为$n-2$个二维球面的楔积。其有理覆盖群是循环的，而源映射的多项式覆盖群是平凡的。通过基变换，正规化描述了目标中任何子簇的原像；当目标多项式可分解时，一个仅含常数单位的分量是一个因子的简单根的Kummer覆盖的开子集。最后，对于$n\geq 4$，单值性将这些映射与在稳定多项式左右等价下的单一加权提升区分开来。
英文摘要：
We study the dimension-indexed marked-root polynomial maps introduced by Harish. For every $n\geq 4$, their lower coefficient block extends to a polynomial coordinate system, giving three-dimensional Keller maps of geometric degree $n(n-2)$ for every choice of the fixed coefficients. We determine the generic monodromy of the original maps and all successive coordinate descents: it is the cyclic wreath product for odd $n$, and an explicit subgroup of index two for even $n$. Its order is $(n-2)^n n!/\gcd(2,n-2)$. We construct the smooth finite normalization and identify its complement of the affine source as $n-2$ disjoint affine divisors. One is ramified; the others record unramified loss of inverse sheets. The normalization has Picard group $\mathbb{Z}^{n-2}$ and the homotopy type of a wedge of $n-2$ two-spheres. Its rational deck group is cyclic, while the polynomial deck group of the source map is trivial. By base change, the normalization describes the preimage of any subvariety of the target; when the target polynomial factors, a component with only constant units is an open subset of a Kummer cover of the simple roots of one factor. Finally, the monodromy separates these maps, for $n\geq 4$, from single weighted lifts under stable polynomial left-right equivalence.
