A short proof of the multiple cover formula for point insertions

TL;DR

This paper provides a concise proof of Oberdieck's multiple cover formula for point insertions on abelian surfaces, simplifying previous tropical-based proofs by avoiding tropical enumeration techniques.

Contribution

The paper offers a shorter, more direct proof of the multiple cover formula for point insertions, expanding the applicability of geometric methods without tropical enumeration.

Findings

Shorter proof of the multiple cover formula

Avoids tropical enumeration techniques

Confirms the validity of the formula using geometric methods

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.