Roots and logs in the enumerative forest

Logarithmic and orbifold structures provide two paths to the enumeration of curves with fixed tangencies to 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 discuss joint work with Luca Battistella and Dhruv Ranganathan, in which we identify birational invariance as the key property distinguishing the two theories. The logarithmic theory is stable under strata blowups of the target, while the orbifold theory is not. By identifying a suitable system of “slope-sensitive” blowups, we define a limit orbifold theory and prove that it coincides with the logarithmic theory. Our proof hinges on a technique – rank reduction – for reducing questions about normal crossings divisors to questions about smooth divisors, where the situation is much-better understood.