Coordinate descents, monodromy, and finite normalization of marked-root maps

Abstract

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.