Complete intersections: Moduli, Torelli, and good reduction

TL;DR

This paper investigates the arithmetic properties of complete intersections in projective space, establishing Torelli theorems and Shafarevich conjecture analogues for specific classes like cubic and quartic threefolds.

Contribution

It provides new arithmetic Torelli theorems and verifies the Shafarevich conjecture for certain classes of complete intersections, extending known results to higher-dimensional varieties.

Findings

Proved an analogue of the Shafarevich conjecture for cubic and quartic threefolds.

Established arithmetic Torelli theorems for complete intersections.

Extended Faltings' results to new classes of algebraic varieties.

Abstract

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.