Hilbert Schemes and Toric Degenerations for Low Degree Fano Threefolds

TL;DR

This paper investigates the Hilbert schemes of low-degree Fano threefolds, classifies their degenerations to toric Fano varieties, and enhances understanding of their geometric structures and singularities.

Contribution

It provides a classification of degenerations of low-degree Fano threefolds to toric varieties with canonical Gorenstein singularities.

Findings

Classification of all degenerations to toric Fano varieties for degrees up to 12

Identification of possible singularities in degenerations

Insights into the structure of Hilbert schemes for these threefolds

Abstract

For fixed degree $d\leq 12$, we study the Hilbert scheme of degree $d$ smooth Fano threefolds in their anticanonical embeddings. We use this to classify all possible degenerations of these varieties to toric Fano varieties with at most canonical Gorenstein singularities.