Logarithmic and orbifold structures provide two different paths to the enumeration of algebraic curves with fixed tangencies along a normal crossings divisor. Simple examples demonstrate that the resulting systems of invariants differ, but a more structural explanation of this defect has remained elusive.

I will explain how the two systems of invariants can be identified by passing to an appropriate blowup. This identifies “birational invariance” as the key property distinguishing the two theories. Our proof hinges on a technique – rank reduction – for reducing questions about normal crossings divisors to questions about smooth divisors.

Time permitting, I will discuss extensions of this result to the setting of negative tangencies, where the pathological geometry of the moduli space is controlled using tropical geometry.

This is joint work with Luca Battistella and Dhruv Ranganathan.