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 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 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 Anticanonical codes from del Pezzo surfaces with Picard rank one

2020 ◽  
Vol 373 (8) ◽  
pp. 5371-5393 ◽  
Author(s):  
Régis Blache ◽  
Alain Couvreur ◽  
Emmanuel Hallouin ◽  
David Madore ◽  
Jade Nardi ◽  
...  
Keyword(s):  
Del Pezzo Surfaces ◽  
Rank One ◽  
Del Pezzo

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 $\rho (Z/ W)= 3$.

A Non-vanishing Theorem of Del Pezzo Surfaces

2016 ◽  
Vol 59 (2) ◽  
pp. 463-472
Author(s):  
Chin-Yi Lin
Keyword(s):  
Del Pezzo Surfaces ◽  
Vanishing Theorem ◽  
Quotient Singularities ◽  
Del Pezzo

AbstractWe develop a new non-vanishing theorem for del Pezzo surfaces with quotient singularities.

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 Families of canonically polarized manifolds over log Fano varieties

2013 ◽  
Vol 149 (6) ◽  
pp. 1019-1040
Author(s):  
Daniel Lohmann
Keyword(s):  
Minimal Model ◽  
Projective Variety ◽  
Fano Varieties ◽  
Model Program ◽  
Minimal Model Program ◽  
Smooth Family ◽  
Family Of Curves ◽  
Moving Cone ◽  
Extremal Ray

AbstractLet $(X,D)$ be a dlt pair, where $X$ is a normal projective variety. We show that any smooth family of canonically polarized varieties over $X\setminus \,{\rm Supp}\lfloor D \rfloor $ is isotrivial if the divisor $-(K_X+D)$ is ample. This result extends results of Viehweg–Zuo and Kebekus–Kovács. To prove this result we show that any extremal ray of the moving cone is generated by a family of curves, and these curves are contracted after a certain run of the minimal model program. In the log Fano case, this generalizes a theorem by Araujo from the klt to the dlt case. In order to run the minimal model program, we have to switch to a $\mathbb Q$-factorialization of $X$. As $\mathbb Q$-factorializations are generally not unique, we use flops to pass from one $\mathbb Q$-factorialization to another, proving the existence of a $\mathbb Q$-factorialization suitable for our purposes.

Sign in / Sign up

Export Citation Format

Share Document