Consider a quasi-compact moduli space M of pairs (X,D) consisting of a variety X and a divisor D on X. If M is not proper, it is reasonable to find a compactification of it. Assume furthermore that there are two rational numbers 0<b<a<1 such that, for every pair (X,D) corresponding to a point in M, the pairs (X,aD) and (X,bD) are klt, and the Q-divisors K_X+aD and K_X+bD are ample. Using Kollár's formalism of stable pairs, one can construct two different compactifications of M (M_a and M_b), corresponding to a and b. The goal of this project is to relate these two compactifications. The main result is that, up to replacing M_a and M_b with their normalizations, there are birational morphisms M_a \to M_b.

This project is inspired by Hassett's work on weighted stable curves, and is joint with Kenny Ascher, Dori Bejleri and Zsolt Patakfalvi.