分次Keller映射的定义轨道与域
Orbits and Fields of Definition for Graded Keller Maps
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中文总结 AI 辅助
该研究对分次Keller映射按权向量符号分类，构造两类分次Keller映射族，证明其几何单值群性质，在一般次数12时确立轨道空间含非常数代数族，得到真正模空间。
中文摘要 AI 辅助
我们通过权向量的符号对分次Keller映射进行分类，探讨哪些双曲权支持反例，以及非空轨迹如何分层。我们构造了两类：循环锥在每个复合一般纤维次数N≥6时均分布于K(3,(1,-1,-1))中，包括在Q(√-15)上的6叶成员和在Q上的33叶成员；对角族对每个p≥2给出K(3,(1,-p,-p))≠∅，其雅可比行列式det JF_p=-p，一般纤维次数为2p-1。两类族中每个成员的几何单值群均为全次数的交错群或对称群，因此可解情况仅出现在一般次数3时。在一般次数12时，正规化四次循环种子在Q上形成一条光滑平面三次曲线，其点产生两两不同的分次等价轨道。因此相关轨道空间包含一个非常数代数族，确立了真正的模空间而非孤立例子。
英文摘要
We classify graded Keller maps by the signature of their weight vectors and address which hyperbolic weights support counterexamples and how the nonempty loci are stratified. We construct two families. Cyclic cones populate \(K(3,(1,-1,-1))\) at every composite generic fibre degree \(N\geq 6\), including a six-sheeted member over \(\mathbb{Q}(\sqrt{-15})\) and a 33-sheeted member over \(\mathbb{Q}\). A diagonal family gives \(K(3,(1,-p,-p))\neq\varnothing\) for every \(p\geq2\), with \(\det JF_p=-p\) and generic fibre degree \(2p-1\). The geometric monodromy of every member of both families is alternating or symmetric of full degree; hence the solvable case occurs only in generic degree three. At generic degree twelve, normalized quartic cyclic seeds form a smooth plane cubic over \(\mathbb{Q}\), whose points yield pairwise distinct graded-equivalence orbits. Thus the relevant orbit space contains a nonconstant algebraic family, establishing genuine moduli rather than isolated examples."
