Gaps on the intersection numbers of sections on a rational elliptic surface

TL;DR

This paper explores the possible intersection numbers of sections on rational elliptic surfaces, identifying gap numbers and their distribution using Mordell-Weil lattices and quadratic form representations.

Contribution

It introduces a method to determine gap numbers for intersection counts on rational elliptic surfaces using lattice theory and quadratic forms.

Findings

Identification of conditions for the existence of gap numbers

Analysis of the distribution of gap numbers

Connection between intersection numbers and quadratic form representations

Abstract

Given a rational elliptic surface X over an algebraically closed field, we investigate whether a given natural number k can be the intersection number of two sections of X. If not, we say that k a gap number. We try to answer when gap numbers exist, how they are distributed and how to identify them. We use Mordell-Weil lattices as our main tool, which connects the investigation to the classical problem of representing integers by positive-definite quadratic forms.