From orbifolds to logarithms via birational invariance
May 9, 2022
Logarithmic and orbifold structures provide two different 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 toroidal 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.
For more information see https://personal.math.ubc.ca/~jbryan/Zoominar-UBC-ETH/