Computing N\'eron-Severi groups and cycle class groups

Assuming the Tate conjecture and the computability of \'etale cohomology with finite coefficients, we give an algorithm that computes the N\'eron-Severi group of any smooth projective geometrically integral variety, and also the rank of the group of numerical equivalence classes of codimension p cycles for any p.Comment: 22 pages; to appear in Compositio Mat

Connectivity of tropicalizations

We show that the tropicalization of an irreducible variety over a complete or algebraically closed valued field is connected through codimension 1, giving an affirmative answer in all characteristics to a question posed by Einsiedler, Lind, and Thomas in 2003.Comment: 7 page

On Mordell-Weil groups of Jacobians over function fields

We study the arithmetic of abelian varieties over $K=k(t)$ where $k$ is an arbitrary field. The main result relates Mordell-Weil groups of certain Jacobians over $K$ to homomorphisms of other Jacobians over $k$. Our methods also yield completely explicit points on elliptic curves with unbounded rank over \Fpbar(t) and a new construction of elliptic curves with moderately high rank over \C(t).Comment: v1: 25 pages; v2=v1, ignore; v3: Corrects rank formula when the covers C_d or D_d are reducible and includes other minor improvements and simplification

Separating algebras and finite reflection groups

A separating algebra is, roughly speaking, a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this paper, we introduce a geometric notion of separating algebra. This allows us to prove that only groups generated by reflections may have polynomial separating algebras, and only groups generated by bireflections may have complete intersection separating algebras.Comment: 12 pages, corrected yet another typ