On nef subvarieties

TL;DR

This paper introduces the concept of nef subschemes, generalizing nef normal bundles, and explores their intersection-theoretic properties, including effects on divisor positivity and the structure of cones generated by nef subvarieties.

Contribution

It defines nef subschemes, proves their properties regarding divisor restriction and ampleness, and introduces the weakly movable cone, expanding the understanding of positivity in intersection theory.

Findings

Restriction of pseudoeffective divisors to nef subvarieties remains pseudoeffective.

Nefness and ampleness are transitive properties among subvarieties.

The weakly movable cone contains the movable cone and shares similar properties.

Abstract

The goal of this work is to study positivity of subvarieties with nef normal bundle in the sense of intersection theory. After Ottem's work on ample subschemes, we introduce the notion of a nef subscheme, which generalizes the notion of a subvariety with nef normal bundle. We show that restriction of a pseudoeffective (resp. big) divisor to a nef subvariety is pseudoeffective (resp. big). We also show that ampleness and nefness are transitive properties. We define the weakly movable cone as the cone generated by the pushforward of cycle classes of nef subvarieties via proper surjective maps. This cone contains the movable cone and shares similar intersection-theoretic properties with it, thanks to the aforementioned properties of nef subvarieties. On the other hand, we prove that if $Y\subset X$ is an ample subscheme of codimension $r$ and $D|_Y$ is $q$-ample, then $D$ is $(q+r)$-ample. This is analogous to a result proved by Demailly-Peternell-Schneider. We use the theory of $q$-ample divisors, as developed by Totaro, throughout the paper.