Fano 3-folds and double covers by half elephants

TL;DR

This paper constructs deformation families for all 34 Hilbert series of index 2 Fano 3-folds, revealing topological distinctions in some cases, advancing understanding of their geometric classification.

Contribution

It provides explicit deformation families for all known Hilbert series of index 2 Fano 3-folds, including multiple families distinguished by topology.

Findings

Constructed deformation families for all 34 Hilbert series.

Identified two distinct families in 18 cases based on topology.

Enhanced classification of index 2 Fano 3-folds.

Abstract

We construct a deformation family for each of the 34 Hilbert series of index 2 Fano 3-folds. In 18 cases we construct two different families, distinguished by the topology of their general members.