Primitive Collections and Toric Varieties

TL;DR

This paper characterizes the nef cone of certain toric varieties using primitive collections, extending Batyrev's concept to non-simplicial fans and discussing related open problems.

Contribution

It introduces a modified definition of primitive collections for non-simplicial fans and relates it to existing concepts, providing new insights into toric variety geometry.

Findings

Characterization of the nef cone via primitive collections

Extension of primitive collections to non-simplicial fans

Discussion of open problems in the field

Abstract

This paper studies Batyrev's notion of primitive collection. We use primitive collections to characterize the nef cone of a quasi-projective toric variety whose fan has convex support, a result stated without proof by Batyrev in the smooth projective case. When the fan is non-simplicial, we modify the definition of primitive collection and explain how our definition relates to primitive collections of simplicial subdivisons. The paper ends with some open problems.