Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Log Canonical Minimal Model Program for corank one foliations on Threefolds

Priyankur Chaudhuri, Roktim Mascharak

TL;DR

The paper develops a comprehensive log canonical MMP for corank one foliations on threefolds with klt-type boundaries and extends it to NQC generalized foliated quadruples. It establishes the cone and contraction theorems, proves the existence of flips, and shows termination of foliated MMP in dimension three, then extends the framework to gfqs including a basepoint free theorem and finite log geography. A key outcome is that any two foliated minimal models can be connected by a sequence of flops, and the number of minimal/log canonical models in a boundary-polarized setting is finite. An appendix broadens contraction theory to algebraic spaces without assuming a klt boundary. This work advances birational geometry of foliations and provides toolsets for moduli and boundedness questions in foliated settings.

Abstract

If $(X, \mcF, \D)$ is a projective rank two foliated log canonical triple such that $(X,B)$ is klt for some $0 \leq B \leq \D$, we show that we can run a $(K_\mcF +Δ)$-MMP and any such MMP terminates with either a minimal model or Mori fiber space. Next, we establish a Bertini type lemma and adjunction for generalized foliated quadruples. Using these, we extend the full log canonical MMP to the setting of rank two NQC generalized foliated quadruples. Finally, we apply the generalized MMP to study the relation between different minimal models, namely, any two minimal models of a given foliated log canonical triple can be connected by a sequence of flops and in the boundary polarized case, the minimal models are good and only finitely many in number.

Log Canonical Minimal Model Program for corank one foliations on Threefolds

TL;DR

The paper develops a comprehensive log canonical MMP for corank one foliations on threefolds with klt-type boundaries and extends it to NQC generalized foliated quadruples. It establishes the cone and contraction theorems, proves the existence of flips, and shows termination of foliated MMP in dimension three, then extends the framework to gfqs including a basepoint free theorem and finite log geography. A key outcome is that any two foliated minimal models can be connected by a sequence of flops, and the number of minimal/log canonical models in a boundary-polarized setting is finite. An appendix broadens contraction theory to algebraic spaces without assuming a klt boundary. This work advances birational geometry of foliations and provides toolsets for moduli and boundedness questions in foliated settings.

Abstract

If is a projective rank two foliated log canonical triple such that is klt for some , we show that we can run a -MMP and any such MMP terminates with either a minimal model or Mori fiber space. Next, we establish a Bertini type lemma and adjunction for generalized foliated quadruples. Using these, we extend the full log canonical MMP to the setting of rank two NQC generalized foliated quadruples. Finally, we apply the generalized MMP to study the relation between different minimal models, namely, any two minimal models of a given foliated log canonical triple can be connected by a sequence of flops and in the boundary polarized case, the minimal models are good and only finitely many in number.
Paper Structure (14 sections, 25 theorems, 6 equations)

This paper contains 14 sections, 25 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cone Theorem
  4. Contraction Theorem
  5. Existence of flips
  6. Running the MMP
  7. Termination
  8. Minimal model program for generalized foliated quaduples
  9. Minimal model program for dlt gfqs
  10. Minimal model program for lc gfqs
  11. A basepoint free theorem
  12. Flop connections between minimal models
  13. Log geography of minimal models
  14. Appendix: Towards a general contraction theorem

Key Result

Theorem 1.1

Let $(X, \mathcal{F}, \Delta)/U$ be a rank two projective lc foliated triple, where $\pi:X \to U$ is a projective morphism such that $\operatorname{dim} X =3$ and $(X,B)$ is klt for some $0 \leq B \leq \Delta$. Then we can run a $(K_\mathcal{F}+\Delta)$-MMP over $U$. Moreover, any such MMP $(X,\math

Theorems & Definitions (55)

  • Theorem 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1: Basics on Foliations; see Spi, Dru
  • Definition 2.2: Generalized foliated quadruples and their singularities
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 45 more