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

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 is a gap number. We try to answer when gap numbers exist, how they are distributed and how to identify them. Our main tool is the Mordell-Weil lattice, which connects the investigation to the classical problem of representing integers by positive-definite quadratic forms.