Orbits and fields of definition for graded Keller maps

K. Kistner, Tanush Shaska

I-AIMS preprint sh-132 · 2026

In earlier work, the second author proposed classifying equivariant polynomial maps by the signature of the weight vector and asked which three-dimensional hyperbolic weights carry graded Keller counterexamples and how the nonempty loci are stratified. This paper treats the unit-positive sector, namely weights already of the form (1, −r, −s) with r, s ≥ 1; it does not identify this sector with all primitive hyperbolic weights. Cyclic cones populate (1, −1, −1) at every composite generic fibre degree N ≥ 6. An asymmetric beta–Hermite construction populates every (1, −r, −s) with 2 ≤ r ≤ s over a number field, with determinant -r, generic fibre degree 2r − 1, and full alternating or symmetric geometric monodromy; the row r = 2 is rational. Pullback of the flagship cubic populates every (1, −1, −s) with s ≥ 3 over ℚ, with determinant -16, generic fibre degree three, and monodromy S₃. Together with the balanced and flagship cases, this proves nonemptiness for every three-dimensional unit-positive weight (1, −r, −s). At generic degree twelve the normalised quartic cyclic seeds form a smooth plane cubic E over ℚ. The intrinsic branch-normalisation invariant J = 9b²/(8ac) classifies their unpointed quotient maps up to polynomial left–right equivalence, and J: E → ℙ¹ has degree six with explicit ramification. The distinguished lift point separates the six generic markings, so the six points in a generic J-fibre are six distinct graded orbits. An alternating-shear member lies in a separate left–right class, with geometric monodromy A₁₂ and branch semigroup ⟨ 6, 26, 27 ⟩. Its fixed-split Nielsen class is one braid orbit of size 76,545, and an exact discriminant identity gives arithmetic monodromy S₁₂ over ℚ with constant extension ℚ(√(−3)). The alternating-shear Keller locus is, unconditionally, an explicit rational codimension-three local complete intersection in its pole-normalised Hurwitz chart. Assuming the stable-reduction Hypothesis 11.3, its normalised admissible-cover closure is an orbicurve with thirteen boundary points and the stated rational intersection class. Likewise, assuming the leading-form and chart-exhaustiveness Hypothesis 8.4, the flagship bounded scheme is the product of its degree-bound-preserving gauge orbit with a dual point. Finally, the cyclic lift condition is the vanishing of ∫₀^((−1)ᵏ) ϕ(W)ᵏ W⁻² dW; for even k it has no solution over a field with a real embedding, and at k = 2 it diagonalises over ℚ as ⟨ 3, 5, …, 2n − 1 ⟩.

MSC classes: Primary 14R15; Secondary 14R20, 14G05

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