We construct a three-parameter family of nonhyperelliptic and bielliptic plane genus-three curves whose associated Prym variety is two-isogenous to the Jacobian variety of a general hyperelliptic genus-two curve. Our construction is based on the existence of special elliptic fibrations with the section on the associated Kummer surfaces that provide a simple geometric interpretation for the rational double cover induced by the two-isogeny between the Abelian surfaces.

We define a divisorial motivic zeta function for stable curves with marked points, which agrees with Kapranov’s motivic zeta function when the curve is smooth and unmarked. We show that this zeta function is rational and give a formula in terms of the dual graph of the curve.

We study the equivariant real structures on complex horospherical varieties, generalizing classical results known for toric varieties and flag varieties. We obtain a necessary and sufficient condition for the existence of an equivariant real structure on a given horospherical variety, and we determine the number of equivalence classes of equivariant real structures on horospherical homogeneous spaces. We then apply our results to classifying the equivalence classes of equivariant real structures on smooth projective horospherical varieties of Picard rank 1.

Let S be an n-punctured sphere with $\mathit{n}\ge 3$. We prove that $\left(\genfrac{}{}{0.0pt}{}{\mathit{n}}{3}\right)$ is the maximum size of a family of pairwise nonhomotopic simple arcs on S joining a fixed pair of distinct punctures of S and pairwise intersecting at most twice. On the way, we show that a square annular diagram A has a corner on each of its boundary paths if A contains at least one square and the dual curves of A are simple arcs joining the boundary paths of A and pairwise intersecting at most once.

We continue to explore the numerical nature of the Okounkov bodies focusing on the local behaviors near given points. More precisely, we show that the set of Okounkov bodies of a pseudoeffective divisor with respect to admissible flags centered at a fixed point determines the local numerical equivalence class of divisors, which is defined in terms of refined divisorial Zariski decompositions. Our results extend Roé’s work [R] on surfaces to higher-dimensional varieties although our proof is essentially different in nature.

We prove an injectivity and vanishing theorem for Hodge modules and $\mathbb{R}$-divisors over projective varieties, extending the results for rational Hodge modules and integral divisors in [Wu17]. In particular, the injectivity generalizes the fundamental injectivity of Esnault–Viehweg for normal crossing $\mathbb{Q}$-divisors, whereas the vanishing generalizes Kawamata–Viehweg vanishing for $\mathbb{Q}$-divisors. As a main application, we also deduce a Fujita-type freeness result for Hodge modules in the normal crossing case.

We note that our stable homotopy refinements of Khovanov’s arc algebras and tangle invariants induce refinements of Chen–Khovanov and Stroppel’s platform algebras and tangle invariants, and discuss the topological Hochschild homology of these refinements.