Some examples of non-smoothable Gorenstein Fano toric threefolds

TL;DR

This paper introduces a combinatorial criterion for smoothability of Gorenstein Fano toric threefolds, revealing examples with smoothable singularities that are nonetheless not smoothable, highlighting complex obstructions.

Contribution

It provides a new combinatorial obstruction criterion for the smoothability of Gorenstein Fano toric threefolds, with explicit examples.

Findings

Identifies a local-to-global obstruction for smoothability

Constructs examples of non-smoothable Gorenstein Fano toric threefolds with smoothable singularities

Shows that smoothability cannot be inferred solely from singularity types

Abstract

We present a combinatorial criterion on reflexive polytopes of dimension 3 which gives a local-to-global obstruction for the smoothability of the corresponding Fano toric threefolds. As a result, we show examples of singular Gorenstein Fano toric threefolds which have compound Du Val, hence smoothable, singularities but are not smoothable.