The strict transform in logarithmic geometry

Abstract: Let $(X,D)$ be a pair of a smooth variety and a normal crossings divisor. The loci of curves that admit a map to X with prescribed tangency along D exhibit some pathological behavior: for instance, the locus of maps to a product $(X \times Y, D \times E)$ does not coincide with the intersection of the loci of maps to $(X,D)$ and $(Y,E)$. In this talk I want to explain how the root of such pathologies arises from the difference between taking the strict and total of a cycle under a very special kind of birational map, called a logarithmic modification. I will discuss how for a logarithmic modification, the strict transform of a cycle has a modular interpretation, and how its difference with the total transform can be explicitly computed, in terms of certain piecewise polynomial functions on a combinatorial shadow of the original spaces, the tropicalization. Time permitting, I will discuss some applications -- for instance, how these calculations imply that loci of curves with a map to a toric variety lie in the tautological ring.

algebraic geometry

Audience: researchers in the topic

( slides )

Stanford algebraic geometry seminar

Series comments: The seminar was online for a significant period of time, but for now is solely in person. More seminar information (including slides and videos, when available): agstanford.com