Complete intersections: Moduli, Torelli, and good reduction

We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings. For example, we prove an analogue of the Shafarevich conjecture for cubic and quartic threefolds and intersections of two quadrics.Comment: 37 pages. Typo's fixed. Expanded Section 2.

The moduli of smooth hypersurfaces with level structure

We construct the moduli space of smooth hypersurfaces with level $N$ structure over $\mathbb{Z}[1/N]$. As an application we show that, for $N$ large enough, the stack of smooth hypersurfaces over $\mathbb{Z}[1/N]$ is uniformisable by a smooth affine scheme. To prove our results, we use the Lefschetz trace formula to show that automorphisms of smooth hypersurfaces act faithfully on their cohomology. We also prove a global Torelli theorem for smooth cubic threefolds over fields of odd characteristic.Comment: 10 pages. Added new application to Torelli theorems for cubic threefolds. Corrected mistake in previous version - results now only apply to tame automorphism

Horospherical stacks

We prove structure theorems for algebraic stacks with a reductive group action and a dense open substack isomorphic to a horospherical homogeneous space, and thereby obtain new examples of algebraic stacks which are global quotient stacks. Our results partially generalize the work of Iwanari, Fantechi, Mann and Nironi, and Geraschenko and Satriano for abstract toric stacks

On the distribution of rational points on ramified covers of abelian varieties

We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields of characteristic zero. For example, given a ramified cover, where is an abelian variety over with a dense set of -rational points, we prove that there is a finite-index coset such that is disjoint from. Our results do not seem to be in the range of other methods available at present; they confirm predictions coming from Lang's conjectures on rational points, and also go in the direction of an issue raised by Serre regarding possible applications to the inverse Galois problem. Finally, the conclusions of our work may be seen as a sharp version of Hilbert's irreducibility theorem for abelian varieties