Let $k$ be a number field and $X$ a smooth projective $k$-variety. In this paper, we study the information obtainable from descent via torsors under finite $k$-group schemes on the location of the $k$-rational points on $X$ within the adelic points. Our main result is that if a curve $C∕k$ maps nontrivially into an abelian variety $A∕k$ such that $A\left(k\right)$ is finite and $\text{ Ш}\left(k,A\right)$ has no nontrivial divisible element, then the information coming from finite abelian descent cuts out precisely the rational points of $C$. We conjecture that this is the case for all curves of genus at least 2. We relate finite descent obstructions to the Brauer–Manin obstruction; in particular, we prove that on curves, the Brauer set equals the set cut out by finite abelian descent. Our conjecture therefore implies that the Brauer–Manin obstruction against rational points is the only one on curves.

The connection between del Pezzo surfaces and root systems goes back to Coxeter and Du Val, and was given modern treatment by Manin in his seminal book Cubic forms. Batyrev conjectured that a universal torsor on a del Pezzo surface can be embedded into a certain projective homogeneous space of the semisimple group with the same root system, equivariantly with respect to the maximal torus action. Computational proofs of this conjecture based on the structure of the Cox ring have been given recently by Popov and Derenthal. We give a new proof of Batyrev’s conjecture using an inductive process, interpreting the blowing-up of a point on a del Pezzo surface in terms of representations of Lie algebras corresponding to Hermitian symmetric pairs.

This paper intends to give a mathematical explanation for results on the zeta function of some families of varieties recently obtained in the context of mirror symmetry. In the process we obtain concrete and explicit examples for some results recently used in algorithms to count points on smooth hypersurfaces in ${\mathbb{P}}^n$.

In particular, we extend the monomial-motive correspondence of Kadir and Yui and we give explicit solutions to the $p$-adic Picard–Fuchs equation associated with monomial deformations of Fermat hypersurfaces.

As a byproduct we obtain Poincaré duality for the rigid cohomology of certain singular affine varieties.

Motivated by affine Schubert calculus, we construct a family of dual graded graphs $\left({\Gamma }_s,{\Gamma }_w\right)$ for an arbitrary Kac–Moody algebra $\mathfrak{g}$. The graded graphs have the Weyl group $W$ of $\mathfrak{g}\mathfrak{e}\mathfrak{h}$ as vertex set and are labeled versions of the strong and weak orders of $W$ respectively. Using a construction of Lusztig for quivers with an admissible automorphism, we define folded insertion for a Kac–Moody algebra and obtain Sagan–Worley shifted insertion from Robinson–Schensted insertion as a special case. Drawing on work of Proctor and Stembridge, we analyze the induced subgraphs of $\left({\Gamma }_s,{\Gamma }_w\right)$ which are distributive posets.