scholarly journals ON THE FIRST STEPS OF THE MINIMAL MODEL PROGRAM FOR THE MODULI SPACE OF STABLE POINTED CURVES

2021 ◽  
pp. 1-67
Author(s):  
Giulio Codogni ◽  
Luca Tasin ◽  
Filippo Viviani
Keyword(s):  
Moduli Space ◽  
Minimal Model ◽  
Model Program ◽  
Minimal Model Program ◽  
Geometric Properties ◽  
Canonical Models ◽  
Log Canonical

Abstract The aim of this paper is to study all the natural first steps of the minimal model program for the moduli space of stable pointed curves. We prove that they admit a modular interpretation, and we study their geometric properties. As a particular case, we recover the first few Hassett–Keel log canonical models. As a by-product, we produce many birational morphisms from the moduli space of stable pointed curves to alternative modular projective compactifications of the moduli space of pointed curves.

Singularities with 𝔾 m -action and\break the log minimal model program for ℳ¯ g

2016 ◽  
Vol 2016 (721) ◽  
pp. 1-41 ◽  
Author(s):  
Jarod Alper ◽  
Maksym Fedorchuk ◽  
David Ishii Smyth
Keyword(s):  
Minimal Model ◽  
Model Program ◽  
Minimal Model Program ◽  
Canonical Models ◽  
Precise Formulation ◽  
Log Canonical ◽  
Log Minimal Model Program

AbstractWe give a precise formulation of the modularity principle for the log canonical models

scholarly journals Minimal Models and Abundance for Positive Characteristic Log Surfaces

2014 ◽  
Vol 216 ◽  
pp. 1-70 ◽  
Author(s):  
Hiromu Tanaka
Keyword(s):  
Finite Field ◽  
Minimal Model ◽  
Positive Characteristic ◽  
Algebraic Closure ◽  
Model Program ◽  
Minimal Models ◽  
Base Field ◽  
Singular Surfaces ◽  
Minimal Model Program ◽  
Log Canonical

AbstractWe discuss the birational geometry of singular surfaces in positive characteristic. More precisely, we establish the minimal model program and the abundance theorem for ℚ-factorial surfaces and for log canonical surfaces. Moreover, in the case where the base field is the algebraic closure of a finite field, we obtain the same results under much weaker assumptions.

scholarly journals Minimal Models and Abundance for Positive Characteristic Log Surfaces

2014 ◽  
Vol 216 ◽  
pp. 1-70 ◽  
Author(s):  
Hiromu Tanaka
Keyword(s):  
Finite Field ◽  
Minimal Model ◽  
Positive Characteristic ◽  
Algebraic Closure ◽  
Model Program ◽  
Minimal Models ◽  
Base Field ◽  
Singular Surfaces ◽  
Minimal Model Program ◽  
Log Canonical

AbstractWe discuss the birational geometry of singular surfaces in positive characteristic. More precisely, we establish the minimal model program and the abundance theorem for ℚ-factorial surfaces and for log canonical surfaces. Moreover, in the case where the base field is the algebraic closure of a finite field, we obtain the same results under much weaker assumptions.

scholarly journals Moduli of products of stable varieties

2013 ◽  
Vol 149 (12) ◽  
pp. 2036-2070 ◽  
Author(s):  
Bhargav Bhatt ◽  
Wei Ho ◽  
Zsolt Patakfalvi ◽  
Christian Schnell
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Minimal Model ◽  
Complex Numbers ◽  
Model Program ◽  
Minimal Model Program ◽  
Cotangent Complex ◽  
Geometric Methods ◽  
Local Results

AbstractWe study the moduli space of a product of stable varieties over the field of complex numbers, as defined via the minimal model program. Our main results are: (a) taking products gives a well-defined morphism from the product of moduli spaces of stable varieties to the moduli space of a product of stable varieties; (b) this map is always finite étale; and (c) this map very often is an isomorphism. Our results generalize and complete the work of Van Opstall in dimension$1$. The local results rely on a study of the cotangent complex using some derived algebro-geometric methods, while the global ones use some differential-geometric input.

scholarly journals Minimal model program for log canonical threefolds in positive characteristic

2020 ◽  
Vol 27 (4) ◽  
pp. 1003-1054
Author(s):  
Kenta Hashizume ◽  
Yusuke Nakamura ◽  
Hiromu Tanaka
Keyword(s):  
Minimal Model ◽  
Positive Characteristic ◽  
Model Program ◽  
Minimal Model Program ◽  
Log Canonical

THE LMMP FOR LOG CANONICAL 3-FOLDS IN CHARACTERISTIC

2017 ◽  
Vol 230 ◽  
pp. 48-71 ◽  
Author(s):  
JOE WALDRON
Keyword(s):  
Minimal Model ◽  
Effective Divisor ◽  
Model Program ◽  
Minimal Models ◽  
Minimal Model Program ◽  
Log Canonical ◽  
Contraction Theorem ◽  
Log Minimal Model Program

We prove that one can run the log minimal model program for log canonical 3-fold pairs in characteristic $p>5$. In particular, we prove the cone theorem, contraction theorem, the existence of flips and the existence of log minimal models for pairs with log divisor numerically equivalent to an effective divisor. These follow from our main results, which are that certain log minimal models are good.

scholarly journals FAMILIES OF AFFINE RULED SURFACES: EXISTENCE OF CYLINDERS

2016 ◽  
Vol 223 (1) ◽  
pp. 1-20 ◽  
Author(s):  
ADRIEN DUBOULOZ ◽  
TAKASHI KISHIMOTO
Keyword(s):  
Minimal Model ◽  
Finite Extension ◽  
Smooth Curve ◽  
Ruled Surfaces ◽  
Model Program ◽  
Minimal Model Program ◽  
Generic Fiber ◽  
Projective Model ◽  
The Family

We show that the generic fiber of a family $f:X\rightarrow S$ of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base $S$. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking $S$, such a family actually factors through an $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ over a certain $S$-scheme $Y\rightarrow S$ induced by the MRC-fibration of a relative smooth projective model of $X$ over $S$. For affine threefolds $X$ equipped with a fibration $f:X\rightarrow B$ by irrational $\mathbb{A}^{1}$-ruled surfaces over a smooth curve $B$, the induced $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ can also be recovered from a relative minimal model program applied to a smooth projective model of $X$ over $B$.

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