Some Definability Results in Abstract Kummer Theory

TL;DR

This paper establishes uniform bounds on the number of irreducible components of preimages of subvarieties under multiplication maps in semiabelian varieties, using Galois-theoretic methods and extending to divisible abelian groups of finite Morley rank.

Contribution

It introduces a Galois-theoretic approach to bounding irreducible components and extends definability results to broader classes of divisible abelian groups under the DMP.

Findings

Uniform bound on irreducible components of [n]^{-1}(X)

Applicability of methods to divisible abelian groups of finite Morley rank

Conditions for finite Morley rank groups to have the DMP

Abstract

Let $S$ be a semiabelian variety over an algebraically closed field, and let $X$ be an irreducible subvariety not contained in a coset of a proper algebraic subgroup of $S$. We show that the number of irreducible components of $[n]^{-1}(X)$ is bounded uniformly in $n$, and moreover that the bound is uniform in families $X_t$. We prove this by purely Galois-theoretic methods. This proof applies in the more general context of divisible abelian groups of finite Morley rank. In this latter context, we deduce a definability result under the assumption of the Definable Multiplicity Property (DMP). We give sufficient conditions for finite Morley rank groups to have the DMP, and hence give examples where our definability result holds.