Smoothing Gorenstein toric Fano 3-folds

TL;DR

This paper introduces a combinatorial method using Minkowski decompositions to determine smoothings of Gorenstein toric Fano 3-folds and provides a way to compute their Betti numbers.

Contribution

It defines admissible Minkowski decomposition data for reflexive polytopes and shows how this data determines smoothings of associated toric Fano 3-folds, including Betti number calculations.

Findings

Admissible Minkowski decompositions determine smoothings of Gorenstein toric Fano 3-folds.

Provides an effective method to compute Betti numbers of the smoothings.

The approach is purely combinatorial, relying on the polytope's face structure.

Abstract

We introduce admissible Minkowski decomposition data (amd) for a 3-dimensional reflexive polytope P. This notion is defined purely in terms of the combinatorics of P. Denoting by X the Gorenstein toric Fano 3-fold whose fan is the spanning fan (a.k.a. face fan) of P, our first result states that amd for P determine a smoothing of X. Our second result amounts to an effective recipe for computing the Betti numbers of the smoothing.