Numerical obstructions to abelian surfaces in toric Fano 4-folds

TL;DR

This paper investigates which toric Fano 4-folds contain abelian surfaces, using elementary methods and computational tools to exclude certain cases and clarify the presence of such surfaces.

Contribution

It provides new exclusions of abelian surfaces in specific toric Fano 4-folds, advancing understanding of their geometric structure.

Findings

Identified toric Fano 4-folds that do not contain abelian surfaces.

Excluded additional cases where abelian surfaces cannot exist.

Clarified the distribution of abelian surfaces among toric Fano 4-folds.

Abstract

Some of the 124 toric Fano 4-folds contain abelian surfaces but most do not: in a few cases it is not known whether they do or not. By elementary methods, with a little computer help, we exclude some more possibilities.