Curves disjoint from a nef divisor

TL;DR

This paper demonstrates that in higher dimensions, the set of curves orthogonal to a nef line bundle can be countably infinite, contrasting the well-known dichotomy in surfaces, and constructs examples illustrating this phenomenon.

Contribution

It constructs a nef line bundle on a threefold with infinitely many orthogonal curves, answering a question of Totaro and providing new insights into nef line bundles in higher dimensions.

Findings

Counterexample to the surface dichotomy in higher dimensions

Existence of a nef line bundle trivial on infinitely many curves

A quasi-projective variety with countably infinite subvarieties

Abstract

On a projective surface it is well-known that the set of curves orthogonal to a nef line bundle is either finite or uncountable. We show that this dichotomy fails in higher dimension by constructing a nef line bundle on a threefold which is trivial on countably infinitely many curves. This answers a question of Totaro. As a pleasant corollary, we exhibit a quasi-projective variety with only a countably infinite set of complete, positive-dimensional subvarieties.