# Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces

## TL;DR

This paper investigates the transversality of sections on elliptic surfaces, establishing finiteness results for tangencies, and applies these findings to elliptic divisibility sequences and the classification of algebraic surfaces.

## Contribution

It provides new finiteness theorems for tangencies of sections on elliptic surfaces and explores their implications for elliptic divisibility sequences and surface geography.

## Key findings

- Finitely many points of tangency in characteristic zero
- No tangencies for very general surfaces in certain characteristics
- Construction of surfaces with unbounded invariants

## Abstract

We consider elliptic surfaces $\mathcal{E}$ over a field $k$ equipped with zero section $O$ and another section $P$ of infinite order. If $k$ has characteristic zero, we show there are only finitely many points where $O$ is tangent to a multiple of $P$. Equivalently, there is a finite list of integers such that if $n$ is not divisible by any of them, then $nP$ is not tangent to $O$. Such tangencies can be interpreted as unlikely intersections. If $k$ has characteristic zero or $p>3$ and $\mathcal{E}$ is very general, then we show there are no tangencies between $O$ and $nP$. We apply these results to square-freeness of elliptic divisibility sequences and to geography of surfaces. In particular, we construct mildly singular surfaces of arbitrary fixed geometric genus with $K$ ample and $K^2$ unbounded.