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.

scholarly journals Smoothable del Pezzo surfaces with quotient singularities

2009 ◽  
Vol 146 (1) ◽  
pp. 169-192 ◽  
Author(s):  
Paul Hacking ◽  
Yuri Prokhorov
Keyword(s):  
Minimal Model ◽  
Del Pezzo Surfaces ◽  
Model Program ◽  
Minimal Model Program ◽  
Rank One ◽  
Quotient Singularities ◽  
Del Pezzo ◽  
Moduli Problems

AbstractWe classify del Pezzo surfaces with quotient singularities and Picard rank one which admit a ℚ-Gorenstein smoothing. These surfaces arise as singular fibres of del Pezzo fibrations in the 3-fold minimal model program and also in moduli problems.

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 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 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

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 $B$.

scholarly journals Relations in the Sarkisov program

2013 ◽  
Vol 149 (10) ◽  
pp. 1685-1709 ◽  
Author(s):  
Anne-Sophie Kaloghiros
Keyword(s):  
Finite Number ◽  
Minimal Model ◽  
Model Program ◽  
Minimal Model Program ◽  
Fibre Space ◽  
Projective Varieties ◽  
Birational Maps ◽  
Birational Map ◽  
End Products

AbstractThe Sarkisov program studies birational maps between varieties that are end products of the Minimal Model Program (MMP) on nonsingular uniruled varieties. If $X$ and $Y$ are terminal $ \mathbb{Q} $-factorial projective varieties endowed with a structure of Mori fibre space, a birational map $f: X\dashrightarrow Y$ is the composition of a finite number of elementary Sarkisov links. This decomposition is in general not unique: two such define a relation in the Sarkisov program. I define elementary relations, and show they generate relations in the Sarkisov program. Roughly speaking, elementary relations are the relations among the end products of suitable relative MMPs of $Z$ over $W$ with $\rho (Z/ W)= 3$.

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