Weighted Fano varieties and infinitesimal Torelli problem

TL;DR

This paper addresses the infinitesimal Torelli problem for certain 3-dimensional ${f Q}$-Fano hypersurfaces, providing solutions and exploring chains of double coverings that exhibit both examples and counterexamples related to Hodge properties.

Contribution

It solves the infinitesimal Torelli problem for 3-dimensional quasi-smooth ${f Q}$-Fano hypersurfaces with terminal singularities and investigates infinite chains of double coverings with similar Hodge properties.

Findings

Solved the infinitesimal Torelli problem for specific ${f Q}$-Fano hypersurfaces.

Identified infinite chains of double coverings with varying Torelli properties.

Discovered parallels with Gushel-Mukai chains in Hodge diagram properties.

Abstract

We solve the infinitesimal Torelli problem for $3$-dimensional quasi-smooth ${\mathbb{Q}}$-Fano hypersurfaces with at worst terminal singularities. We also find infinite chains of double coverings of increasing dimension which alternatively distribute themselves in examples and counterexamples for the infinitesimal Torelli claim and which share the analogue, and in some cases the same, Hodge-diagram properties as the length $3$ Gushel-Mukai chain of prime smooth Fanos of coindex $3$ and degree $10$.