A fibered power theorem for pairs of log general type

Let $f: (X,D) \to B$ be a stably family with log canonical general fiber. We prove that, after a birational modification of the base $\tilde{B} \to B$, there is a morphism from a high fibered power of the family to a pair of log general type. If in addition the general fiber is openly canonical, then there is a morphism from a high fibered power of the original family to a pair openly of log general type.Comment: Exposition has been greatly improved. Version to appear in Algebra & Number Theor

Rational points on twisted K3 surfaces and derived equivalences

Using a construction of Hassett--V\'arilly-Alvarado, we produce derived equivalent twisted K3 surfaces over $\mathbb{Q}$, $\mathbb{Q}_2$, and $\mathbb{R}$, where one has a rational point and the other does not. This answers negatively a question recently raised by Hassett and Tschinkel.Comment: To appear in the proceedings volume for the AIM conference "Brauer groups and obstruction problems: moduli spaces and arithmetic

Wall crossing for moduli of stable log pairs

We prove, under suitable conditions, that there exist wall-crossing and reduction morphisms for moduli spaces of stable log pairs in all dimensions as one varies the coefficients of the divisor.Comment: Improved exposition and minor corrections throughout. Final version to appear in Annals of Mathematic