Toric manifolds with degenerate dual variety and defect polytopes

TL;DR

This paper classifies certain toric manifolds with non-hypersurface dual varieties by analyzing associated convex polytopes, introducing invariants that characterize these geometric structures and their properties.

Contribution

It provides a classification of defect polytopes via a combinatorial invariant and links the geometry of toric manifolds to convex polytope properties.

Findings

Defect polytopes are characterized by the vanishing of invariant c(P).

Invariant c*(P) is nonnegative for all simple convex integral polytopes.

The classification connects toric geometry with convex combinatorics.

Abstract

We classify projective toric manifolds whose dual variety is not a hypersurface in the dual projective space. Under the standard dictionary between toric geometry and convex geometry, they correspond to certain convex Delzant integer polytopes, P, which we call defect polytopes. Using the geometrical classification we give a detailed description of defect polytopes and prove that they are characterized by the vanishing of a combinatorial invariant, denoted by c(P). We further prove that a related invariant, c*(P), is nonnegative, for any simple convex integral polytope.