Lang-Vojta's conjecture for the moduli of Fano threefolds of Picard rank 1, index 1 and degree 4

Abstract

We verify Lang-Vojta's conjecture for the moduli of Fano threefolds of Picard rank 1, index 1 and degree 4. This leads us to studying the infinitesimal Torelli problem for quasi-smooth weighted complete intersections. We give a proof for the fact that the infinitesimal Torelli map can be described as a multiplication in the associated Jacobi ring. We also study the geometry of the moduli stack and show that it is stratified via the two types of such Fano threefolds given by Iskovskikh's classification. Furthermore, we work on the persistence conjecture. We generalize a criterion that says that geometric hyperbolicity implies the persistence of arithmetic hyperbolicity to the case of algebraic stacks.