We show that the Kulikov surfaces form a connected component of the moduli space of surfaces of general type with $p_g=0$ and $K^2=6$. We also give a new description for these surfaces, extending ideas of Inoue. Finally, we calculate the bicanonical degree of Kulikov surfaces and prove that they verify the Bloch conjecture.

For a smooth proper variety over a $p$-adic field, its Brauer group and abelian fundamental group are related to higher Chow groups by the Brauer–Manin pairing and class field theory. We generalize this relation to smooth (possibly nonproper) varieties, using motivic homology and a variant of Wiesend’s ideal class group. Several examples are discussed.

We illustrate the theory of log abelian varieties and their moduli in the case of log elliptic curves.

We address the problem of existence and uniqueness of a factorization of a given element of the modular group into a product of two Dehn twists. As a geometric application, we conclude that any maximal real elliptic Lefschetz fibration is algebraic.

Given an ideal $\mathfrak{a}$ on a smooth variety in characteristic zero, we estimate the $F$-jumping numbers of the reductions of $\mathfrak{a}$ to positive characteristic in terms of the jumping numbers of $\mathfrak{a}$ and the characteristic. We apply one of our estimates to bound the Hartshorne–Speiser–Lyubeznik invariant for the reduction to positive characteristic of a hypersurface singularity.