Wall crossing morphisms for moduli of stable pairs

Abstract: 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 $0 \lt b \lt a\lt 1$ such that, for every pair (X,D) corresponding to a point in M, the pair (X,D) is smooth and normal crossings, 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. I will explain how 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 $M_a \to M_b$, recovering Hassett's result (for the case of curves) in all dimensions. If time permits, I will explain a slight variation of the moduli functor of varieties with pairs, which has a particularly accessible moduli functor, leads to a simple proof of the projectivity of the moduli of stable pairs, and conjecturally leads to better wall-crossing phenomena. The talk will be based on my work with Kenny Ascher, Dori Bejleri, Zsolt Patakfalvi; and my work with Stefano Filipazzi.

algebraic geometrynumber theory

Audience: researchers in the discipline

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

We normally meet in-person in the indicated room. For online editions, we use Zoom and distribute the link through the mailing list. If you wish to be put on the mailing list, please subscribe to ntag-external using lists.sfu.ca