A note on the smooth blow-ups of P(1,1,1,k) in torus-invariant subvarieties

TL;DR

This paper classifies certain toric Fano 3-folds with specific singularities, explores their birational relationships, and extends previous work by analyzing Fano polytopes and smooth blow-ups.

Contribution

It provides a complete classification of toric Fano 3-folds with singular locus {1/k(1,1,1)} and identifies birational links between them for fixed k>4.

Findings

Exactly two Fano 3-folds are linked by a blow-up for each fixed k>4.

The classification is achieved through the language of Fano polytopes.

The work extends previous classifications by Batyrev and Watanabe-Watanabe.

Abstract

This paper classifies toric Fano 3-folds with singular locus { 1/k(1,1,1) } for any positive integer k, building on the work of Batyrev and Watanabe-Watanabe. This is achieved by completing an equivalent problem in the language of Fano polytopes. Furthermore we identify birational relationships between entries of the classification. For a fixed value k>4, there are exactly two such Fano 3-folds linked by a blow-up in a torus-invariant line.