A short proof of the multiple cover formula for point insertions

Abstract

A few years ago, G. Oberdieck conjectured a multiple cover fomula that determines the number of curves of fixed genus and degree passing through a configuration of points in an abelian surface. This formula was proved by the author using tropical techniques and Nishinou's correspondence theorem. Using the same techniques, we give a much shorter proof of the multiple cover formula for point insertions, relying on the same geometrical idea, but avoiding any kind of tropical enumeration.