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.
Let $p\neq\operatorname{char}(k)$. We extend the Karpenko--Merkurjev theorem from classifying stacks of finite $p$-groups to arbitrary finite gerbes whose geometric inertia groups are $p$-groups, without assuming that the gerbe is neutral or that its band is represented by a group scheme over the base field. We prove t...
Let $k=\mathbb F_q$. We classify, up to two-sided $k$-M\"obius equivalence, all separable indecomposable tame exceptional maps $\mathbf P^1_k\to\mathbf P^1_k$ whose geometric Galois closure has genus one. The resulting arithmetic fixed-field forms are encoded by Frobenius-stable affine elliptic quotient data recovered...
Let $k=\mathbb F_q$. We reconstruct every $k$-indecomposable $g\in k(X)$ of degree greater than one whose normal closure has genus one. Such a map is automatically separable and, after independent degree-one changes of the source and target coordinates over $k$, arises from a separable equivariant isogeny between ellip...
We obtain a motivic and a cohomological McKay correspondence for finite linearly reductive group schemes in arbitrary characteristic. In particular, we prove that if $V$ is a finite dimensional vector space and $G$ is a finite linearly reductive subgroup scheme of $\mathrm{SL}(V)$, then the Euler number of any crepant...
We determine the defining ideal of the simple affine vertex algebra $L_{2-\ell}(\mathfrak{so}_{2\ell})$ for every odd $\ell\ge5$. Per\v{s}e's quadratic singular vector alone generates the maximal ideal of the universal affine vertex algebra at this level. Together with the established even-rank presentation, this gives...
Let $R$ be a real closed field and let $X_\Sigma$ be a smooth projective split toric variety. For an explicitly chosen basis rigidification $\mathfrak r$ of its Picard grading, we prove that the pair $(\Sigma,\mathfrak r)$ determines the resulting total Chow--Witt ring and give a terminating finite algorithm for all gr...
