[Paper Review] About a Minimal Model Program without flips

Alberto Chiecchio|arXiv (Cornell University)|Oct 23, 2014

Algebraic Geometry and Number Theory16 references3 citations

TL;DR

This paper introduces a Weil Néron-Severi space and associated nef cone and cone of Weil curves for normal projective varieties with log terminal singularities, enabling a Minimal Model Program without flips. By establishing a new Cone Theorem and contraction theorem in this framework, the authors construct a program that avoids flips by concluding with a Q-factorialization, offering a parallel path to the standard MMP with the same outcome for Q-factorial models.

ABSTRACT

We introduce a new vector space associated to projective variety, the Weil Neron-Severi space, which we show is finitely generated and contains the usual Neron-Severi space as a subspace. We define the Nef cone of Weil divisor and the cone of Weil curves. We study these cones, and prove a new Cone theorem. We use this theorem to propose a Minimal Model Program without flips.

Motivation & Objective

• To develop a Minimal Model Program (MMP) that avoids flips by working with Weil divisors on non-Q-factorial varieties.
• To define a new vector space, the Weil Néron-Severi space, which is finitely generated and contains the classical Néron-Severi space as a subspace.
• To establish a Cone Theorem and contraction theorem for Weil divisors on varieties with log terminal singularities.
• To show that the resulting program produces models with Q-factorial singularities and log Fano fibers, matching the outcome of the standard MMP.
• To provide a parallel framework to the standard MMP, not an alternative, relying on foundational results from [BCHM10] and [CU13].

Proposed method

• Define the Weil Néron-Severi space NS(X)W as the space of Weil divisors modulo numerically trivial Weil divisors, proving its finite-dimensionality.
• Introduce the Weil nef cone Nef(X)W and the cone of Weil curves NE(X)W via duality, with NE(X)W consisting of virtual curves.
• Prove a key technical result: a Weil R-divisor D on Y is nef if and only if it is f-nef and f*D is nef, where f:Y→X is a small, projective, birational morphism.
• Establish Kleiman’s criterion for ampleness in the Weil setting: D is ample iff NE(X)W∖{0} ⊆ D>0.
• Prove a new Cone Theorem: NE(X)W = NE(X)W,KX≥0 + ∑ℝ≥0·Cj, with Cj not accumulating in the KX<0 half-space.
• Extend the theory to relative settings, proving relative Cone and Contraction theorems for projective morphisms over a base U.

Experimental results

Research questions

• RQ1 Can a Minimal Model Program be constructed without flips by working with Weil divisors on non-Q-factorial varieties?
• RQ2 Is there a well-behaved cone theory for Weil divisors that supports a cone and contraction theorem in the absence of Q-factoriality?
• RQ3 How does the Weil Néron-Severi space relate to the classical Néron-Severi space, and is it finitely generated?
• RQ4 Can the standard MMP outcomes—Q-factorial models and log Fano fibers—be achieved without flips via this new framework?
• RQ5 What is the relationship between the models produced by this new program and those from the standard MMP?

Key findings

• The Weil Néron-Severi space NS(X)W is finite-dimensional and contains the classical Néron-Severi space as a subspace.
• The Weil nef cone Nef(X)W and the cone of Weil curves NE(X)W are well-defined and satisfy duality.
• A new Cone Theorem holds: NE(X)W decomposes as the sum of the KX≥0 part and countably many rays Cj not accumulating in the KX<0 region.
• The contraction theorem is established: for any KX-negative extremal face F, there exists a small, projective, birational morphism f: X̃ → X with ρ(X̃) ≤ ρ(X)+1, inducing a contraction of F.
• The relative versions of the Cone and Contraction theorems hold for projective morphisms over a base, with analogous statements for nefness, ampleness, and pushforwards.
• The program avoids flips by concluding with a Q-factorialization, yielding models with Q-factorial singularities and log Fano fibers, matching the standard MMP outcome.