发表机构
Morningside Center of Mathematics, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明半阿贝尔簇中支配不可约子簇纤维挠元稠密当且仅当其混合 Betti 秩为 $2(g+r)$，并给出小点满秩转移及环面高度界，结合代数商、二次 nef 补偿与单值计算。
AI 中文摘要
设 $\mathcal G\to S$ 是光滑复代数基上的半阿贝尔簇，其阿贝尔相对维数为 $g$，环面秩为 $r$。我们证明，一个不可约支配子簇若不包含在真相对特殊子簇中，则其纤维挠元在 Zariski 稠密意义下存在当且仅当其混合 Betti 秩为 $2(g+r)$。特别地，该子簇的维数至少为 $g+r$。对每个素数，主挠元甚至在极大秩点中也是稠密的。例外轨迹包括广义 Ribet 轨迹。在代数数上，我们证明当阿贝尔 Betti 秩为极大时，一般小点强制满混合秩。这给出了从阿贝尔相对 Bogomolov 维数定理到任意环面扩张的转移，以及环面高度间隙（高于阿贝尔挠元）和结点 Pell 方程的阶线性粘合高度界。证明结合了带标记的 $1$-动机的代数商、二次 nef 补偿、中心单值计算以及混合 Betti 层的代数性。
英文摘要
Let $\mathcal G\to S$ be a semiabelian scheme over a smooth complex algebraic base, with abelian relative dimension $g$ and toric rank $r$. We prove that a dominating irreducible subvariety not contained in a proper relative special subvariety has Zariski-dense fibrewise torsion if and only if its mixed Betti rank is $2(g+r)$. In particular, its dimension is at least $g+r$. For every prime, primary torsion is then dense even among points of maximal rank. The exceptional loci include generalized Ribet loci. Over algebraic numbers, we prove that generic small points force full mixed rank whenever the abelian Betti rank is maximal. This yields a transfer from abelian relative Bogomolov dimension theorems to arbitrary toric extensions, together with toric height gaps above abelian torsion and order-linear gluing-height bounds for nodal Pell equations. The proof combines algebraic quotients of marked $1$-motives, quadratic nef compensation, a central-monodromy calculation, and algebraicity of mixed Betti strata.
Comments62 pages, preliminary version, comments very welcome!
