Singular intersections in split semiabelian varieties

Abstract

Let $\mathcal G$ be a split semiabelian variety of dimension $n \ge 2$ and let $\mathcal C \subset \mathcal G$ an irreducible curve, where everything is defined over $\overline{\mathbb{Q}}$. We prove that, under the natural hypothesis that $\mathcal C$ is not contained in any proper algebraic subgroup of $\mathcal G$, the set of points $P$ of $\mathcal C$ lying in an algebraic subgroup $H$ which intersects the curve tangentially at $P$ is finite. This generalizes a result for curves in 2-dimensional tori by March\'e and Maurin. We apply these results to square-freeness of certain geometric divisibility sequences.