A vanishing theorem in Siefring’s intersection theory

In 2009, Siefring introduced a homotopy-invariant generalized intersection number and singularity index for punctured pseudoholomorphic curves, by adding contributions from the curve’s asymptotic behavior to the standard intersection number and singularity index. In this paper, we provide a stratification of the moduli space that describes the rate of asymptotic convergence of the pseudoholomorphic curves. Using this stratification, we provide a more intricate characterization of the curves for which these added contributions to the intersection number and singularity index vanish. In doing so, we prove that the asymptotic contribution to intersection number and singularity index vanishes under generic perturbations. This means that in generic situations we only need to consider the usual intersections of the curves.