Finiteness of Log Minimal Models and Nef curves on $3$-folds in characteristic $p>5$

TL;DR

This paper establishes finiteness of log minimal models for 3-folds in characteristic p>5, proves a version of Batyrev's conjecture on nef cones, and explores duality of curves and divisors across characteristics.

Contribution

It proves finiteness of log minimal models for 3-folds in characteristic p>5 and confirms Batyrev's conjecture on nef cones in this setting and characteristic zero.

Findings

Finiteness of log minimal models for 3-folds in characteristic p>5.

Validation of Batyrev's conjecture on nef cone structure.

Duality of movable curves and pseudo-effective divisors in arbitrary characteristic.

Abstract

In this article we prove a finiteness result on the number of log minimal models for $3$-folds in char $p>5$. We then use this result to prove a version of Batyrev's conjecture on the structure of nef cone of curves on $3$-folds in characteristic $p>5$. We also give a proof of the same conjecture in full generality in characteristic $0$. We further verify that the duality of movable curves and pseudo-effective divisors hold in arbitrary characteristic. We then give a criterion for the pseudo-effectiveness of the canonical divisor $K_X$ of a smooth projective variety in arbitrary characteristic in terms of the existence of a family of rational curves on $X$.