The numerical Amitsur group
Alexander Duncan, Shreya Sharma
TL;DR
The paper introduces the numerical Amitsur group $\operatorname{Am}^\chi(X,J)$ as a computable proxy for the ordinary Amitsur group $\operatorname{Am}(X,G)$ in finite-group actions on smooth projective varieties. By restricting to Euler–Poincaré data via $\operatorname{Pic}^\chi(X,J)$, it proves a canonical surjection $\operatorname{Am}^\chi(X,\operatorname{AutP}(X,G))\to\operatorname{Am}(X,G)$ and derives a dimension–genus–dependent uniform bound on the Amitsur exponent $\mathsf{m}(X,G)$, namely $\mathsf{m}(X,G)\mid (1+(-1)^n p_a)\operatorname{lcm}\{1,\dots,n+1\}$ for dimension $n$ and arithmetic genus $p_a$. The framework further yields finite generating sets for $\operatorname{Pic}^\chi(X,G)$ when $\operatorname{Pic}(X)$ is finitely generated and enables explicit computations, especially for toric varieties. The authors develop a toric refinement via $\operatorname{Am}^T(X,J)$ and show that for smooth Fano toric $X$ with reductive automorphism group, $\operatorname{Am}^T(X,J)\cong\operatorname{Am}^\chi(X,J)$, then illustrate the method with detailed calculations for products of projective spaces, Hirzebruch surfaces, and the del Pezzo surface of degree $6$.
Abstract
The Amitsur subgroup of a variety with a group action measures the failure of the action to lift to the total spaces of its line bundles. We introduce the "numerical Amitsur group," which is an approximation of the ordinary Amitsur subgroup that can be computed using only the Euler-Poincaré characteristic on the Picard group. As an application, we find a uniform upper bound on the exponent of the Amitsur subgroup that depends only on the dimension and arithmetic genus of the variety and is independent of the group. Finally, we compute Amitsur subgroups of toric varieties using these ideas.