Nef and big divisors on toric weak Fano 3-folds

TL;DR

The paper proves that certain line bundles on nonsingular toric weak Fano 3-folds are normally generated, extending known results without relying on full classifications of Fano polytopes.

Contribution

It establishes normal generation of nef and big line bundles with non-zero sections on toric weak Fano 3-folds, and shows all ample line bundles are normally generated on such varieties.

Findings

Nef and big line bundles with non-zero sections are normally generated.

All ample line bundles on nonsingular toric weak Fano 3-folds are normally generated.

Normal generation holds without full classification of Fano polytopes.

Abstract

We show that a nef and big line bundle whose adjoint bundle has non-zero global sections on a nonsingular toric weak Fano 3-fold is normally generated. As a consequence, we see that all ample line bundles on a nonsingular toric weak Fano 3-fold are normally generated. We also show that an ample line bundle whose adjoint bundle has non-zero global sections on a Gorensein toric Fano 3-fold is normally generated. In our proof we do not use full classifications of Fano polytopes.