The strict transform in logarithmic geometry
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 exhibitsome 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.