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 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 Cuspidal curves, minimal models and Zaidenberg’s finiteness conjecture

2019 ◽  
Vol 2019 (747) ◽  
pp. 147-174 ◽  
Author(s):  
Karol Palka
Keyword(s):  
Minimal Model ◽  
Resolution Of Singularities ◽  
Model Program ◽  
Minimal Models ◽  
Minimal Model Program ◽  
Effective Version

Abstract Let {E\subseteq\mathbb{P}^{2}} be a complex rational cuspidal curve and let {(X,D)\to(\mathbb{P}^{2},E)} be the minimal log resolution of singularities. We prove that E has at most six cusps and we establish an effective version of the Zaidenberg finiteness conjecture (1994) concerning Eisenbud–Neumann diagrams of E. This is done by analyzing the Minimal Model Program run for the pair {(X,\frac{1}{2}D)} . Namely, we show that {\mathbb{P}^{2}\setminus E} is {\mathbb{C}^{**}} -fibred or for the log resolution of the minimal model the Picard rank, the number of boundary components and their self-intersections are bounded.

scholarly journals On existence of log minimal models

2010 ◽  
Vol 146 (4) ◽  
pp. 919-928 ◽  
Author(s):  
Caucher Birkar
Keyword(s):  
Minimal Model ◽  
Kodaira Dimension ◽  
Reduction Theorem ◽  
Model Program ◽  
Minimal Models ◽  
Minimal Model Program ◽  
Log Minimal Model Program ◽  
Negative Kodaira Dimension

AbstractIn this paper, we prove that the log minimal model program in dimension d−1 implies the existence of log minimal models for effective lc pairs (e.g. of non-negative Kodaira dimension) in dimension d. In fact, we prove that the same conclusion follows from a weaker assumption, namely, the log minimal model program with scaling in dimension d−1. This enables us to prove that effective lc pairs in dimension five have log minimal models. We also give new proofs of the existence of log minimal models for effective lc pairs in dimension four and of the Shokurov reduction theorem.

scholarly journals New directions in the Minimal Model Program

2020 ◽  
Author(s):  
Paolo Cascini
Keyword(s):  
Minimal Model ◽  
Positive Characteristic ◽  
Closed Field ◽  
Model Program ◽  
Minimal Model Program ◽  
Algebraically Closed Field ◽  
New Directions ◽  
Field Of Positive Characteristic

Abstract We survey some recents developments in the Minimal Model Program. After an elementary introduction to the program, we focus on its generalisations to the category of foliated varieties and the category of varieties defined over any algebraically closed field of positive characteristic.

scholarly journals A BOUND ON EMBEDDING DIMENSIONS OF GEOMETRIC GENERIC FIBERS

2016 ◽  
Vol 4 ◽  
Author(s):  
ZACHARY MADDOCK
Keyword(s):  
Minimal Model ◽  
Positive Characteristic ◽  
Small Dimension ◽  
Del Pezzo Surfaces ◽  
Model Program ◽  
Embedding Dimension ◽  
Minimal Model Program ◽  
Field Extensions ◽  
Del Pezzo ◽  
Rule Out

The author finds a limit on the singularities that arise in geometric generic fibers of morphisms between smooth varieties of positive characteristic by studying changes in embedding dimension under inseparable field extensions. This result is then used in the context of the minimal model program to rule out the existence of smooth varieties fibered by certain nonnormal del Pezzo surfaces over bases of small dimension.

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 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.

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