Counterexamples to the MMP for 1-foliations in positive characteristic
type: Item
instance of: scholarly article
title: Counterexamples to the MMP for 1-foliations in positive characteristic
zbMATH Open document ID: 1545.14024
author: Fabio Bernasconi
published in: Annali dell'Università di Ferrara. Sezione VII. Scienze Matematiche
publication date: 2024-08-26T00:00:00Z
review text: The minimal model program (for varieties) is expected to exist and have nice properties both in characteristic zero and in characteristic \(p > 0\) (in suitable formulations). If we consider foliations instead (that is, subsheaves \(\mathcal F\) of the tangent sheaf closed under Lie bracket), we can still associate to \(\mathcal F\) a canonical divisor \(K_{\mathcal F}\) and therefore formulate conjectures analogous to the MMP. This is known as the MMP for foliations.\N\NA lot is already known about the MMP for foliations in characteristic zero. In the present paper, the author shows that even the most basic statements, such as the Cone Theorem in dimension two, fail for foliations if the characteristic is positive. He also speculates that the correct framework for these questions might be that of \(\infty\)-foliations, but that is not the subject of this paper.
reviewed by: Patrick Graf
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 14G17
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 7901281
zbMATH Keywords: minimal model program
zbMATH Keywords: foliations
zbMATH Keywords: positive characteristic ground fields
MaRDI profile type: Publication
cites work: On the cone of curves of an abelian variety
cites work: Existence of minimal models for varieties of log general type
cites work: Rational Curves on Foliated Varieties
cites work: Globally \(+\)-regular varieties and the minimal model program for threefolds in mixed characteristic
cites work: Birational Geometry of Foliations
cites work: MMP for co-rank one foliations on threefolds
cites work: Higher-dimensional algebraic geometry
cites work: A decomposition theorem for singular spaces with trivial canonical class of dimension at most five
cites work: Q3808225
cites work: Differential forms on log canonical spaces in positive characteristic
cites work: On the three dimensional minimal model program in positive characteristic
cites work: Galois Theory of Purely Inseparable Fields of Exponent One
cites work: Q4693118
cites work: Basepoint freeness for nef and big line bundles in positive characteristic
cites work: Q4212603
cites work: Introduction
cites work: Nonrational Hypersurfaces
cites work: Generic positivity and foliations in positive characteristic
cites work: On Smooth Projective D-Affine Varieties
cites work: Singular foliations with trivial canonical class
cites work: Canonical models of foliations
cites work: Formal groupoids
cites work: Diophantine approximations and foliations
cites work: Singularities of general fibers and the LMMP
cites work: Higher-dimensional foliated Mori theory
cites work: Local and global applications of the minimal model program for co-rank 1 foliations on threefolds
cites work: Minimal Models and Abundance for Positive Characteristic Log Surfaces
cites work: Minimal model program for excellent surfaces
cites work: Minimal model program for semi-stable threefolds in mixed characteristic
DOI: 10.1007/S11565-024-00488-7
description: scientific article; zbMATH DE number 7901281