A cone theorem for nef curves

TL;DR

This paper establishes a cone theorem for nef curves, extending Batyrev's ideas, and explores its implications for the minimal model program in algebraic geometry.

Contribution

It proves an analogue of the Cone Theorem for nef curves and clarifies the relationship between K_X-negative faces and minimal model program outcomes.

Findings

Enlargement of the nef cone includes countably many K_X-negative coextremal rays.

An example demonstrates the necessity of the cone enlargement.

Connection between K_X-negative faces and minimal model program results.

Abstract

Following ideas of V. Batyrev, we prove an analogue of the Cone Theorem for the closed cone of nef curves: an enlargement of the cone of nef curves is the closure of the sum of a K_X-non-negative portion and countably many K_X-negative coextremal rays. An example shows that this enlargement is necessary. We also describe the relationship between K_X-negative faces of this cone and the possible outcomes of the minimal model program.