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: Description: Research/departmental seminar

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