Projective Q-factorial toric varieties covered by lines

TL;DR

This paper provides a structural theorem for projective Q-factorial toric varieties covered by lines, revealing their fibration structure and applications to dual defectiveness and lattice subset discriminants.

Contribution

It introduces a new structural characterization of Q-factorial toric varieties covered by lines, linking geometry with combinatorics and dual defect properties.

Findings

Existence of a toric fibration with fibers as products of projective joins.

Characterization of dual defective toric varieties via extremal contractions.

Classification of lattice subsets with discriminant equal to one.

Abstract

The main result of this paper is a structural theorem for projective Q-factorial toric varieties X in P^N, covered by lines. We prove that there exists a toric fibration f: X -> Z, locally trivial in the Zariski topology, with fiber a product of projective joins. All lines in X intersecting the open subset isomorphic to the torus, are contained in some fiber of f. This characterization has a geometrical application to dual defective toric varieties, and a combinatorial application to discriminants of lattice subsets. We prove that X has positive dual defect if and only if it has an elementary extremal contraction of fiber type, whose general fiber is a projective join with dual defect bigger than its codimension in X. Turning to combinatorics, we characterize lattice subsets A with discriminant D_A equal to one, under suitable assumptions on the polytope Conv(A).