In this paper we construct an action of the affine Hecke category (in its ‘Soergel bimodules’ incarnation) on the principal block of representations of a simply connected semisimple algebraic group over an algebraically closed field of characteristic bigger than the Coxeter number. This confirms a conjecture of G. Williamson and the second author, and provides a new proof of the tilting character formula in terms of antispherical $p$-Kazhdan–Lusztig polynomials.

Investigating a conjecture of Zannier, we study irreducible subvarieties of abelian schemes that dominate the base and contain a Zariski dense set of torsion points that lie on pairwise isogenous fibers. If everything is defined over the algebraic numbers and the abelian scheme has maximal variation, we prove that the geometric generic fiber of such a subvariety is a union of torsion cosets. We go on to prove fully or partially explicit versions of this result in fibered powers of the Legendre family of elliptic curves. Finally, we apply a recent result of Galateau and Martínez to obtain uniform bounds on the number of maximal torsion cosets in the Manin–Mumford problem across a given isogeny class. For the proofs, we adapt the strategy, due to Lang, Serre, Tate, and Hindry, of using Galois automorphisms that act on the torsion as homotheties to the family setting.

We describe a method to show that certain elliptic surfaces do not admit purely inseparable multisections (equivalently, that genus 1 curves over function fields admit no points over the perfect closure of the base field) and use it to show that any non-Jacobian elliptic structure on a very general supersingular K3 surface has no purely inseparable multisections. We also describe specific examples of genus 1 fibrations on supersingular K3 surfaces without purely inseparable multisections.

The pentagram map takes a planar polygon $P$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$. This map is known to interact nicely with Poncelet polygons, that is, polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of Schwartz states that if $P$ is a Poncelet polygon, then the image of $P$ under the pentagram map is projectively equivalent to $P$. In the present paper, we show that in the convex case this property characterizes Poncelet polygons: if a convex polygon is projectively equivalent to its pentagram image, then it is Poncelet. The proof is based on the theory of commuting difference operators, as well as on properties of real elliptic curves and theta functions.

We study derived categories of Gorenstein varieties $X$ and $X^+$ connected by a flop. We assume that the flopping contractions $f\colon X\to Y$, $f^+ \colon X^+ \to Y$ have fibers of dimension bounded by one and $Y$ has canonical hypersurface singularities of multiplicity two. We consider the fiber product $W=X\times _YX^+$ with projections $p\colon W\to X$, $p^+\colon W\to X^+$ and prove that the flop functors $F = Rp^+_*Lp^* \colon {\mathcal {D}}^b(X) \to {\mathcal {D}}^b(X^+)$, $F^+= Rp_*L{p^+}^* \colon {\mathcal {D}}^b(X^+) \to {\mathcal {D}}^b(X)$ are equivalences, inverse to those constructed by Van den Bergh. The composite $F^+ \circ F \colon {\mathcal {D}}^b(X) \to {\mathcal {D}}^b(X)$ is a non-trivial auto-equivalence. When variety $Y$ is affine, we present $F^+ \circ F$ as the spherical cotwist of a spherical couple $(\Psi ^*,\Psi )$ which involves a spherical functor $\Psi$ constructed by deriving the inclusion of the null category $\mathscr {A}_f$ of sheaves ${\mathcal {F}} \in \mathop {{\rm Coh}}\nolimits (X)$ with $Rf_*({\mathcal {F}} )=0$ into $\mathop {{\rm Coh}}\nolimits (X)$. We construct a spherical pair (${\mathcal {D}}^b(X)$, ${\mathcal {D}}^b(X^+)$) in the quotient ${\mathcal {D}}^b(W) /{\mathcal {K}}^b$, where ${\mathcal {K}}^b$ is the common kernel of the derived push-forwards for the projections to $X$ and $X^+$, thus implementing in geometric terms a schober for the flop. A technical innovation of the paper is the $L^1f^*f_*$ vanishing for Van den Bergh's projective generator. We construct a projective generator in the null category and prove that its endomorphism algebra is the contraction algebra.