scholarly journals Automorphisms of cubic surfaces in positive characteristic

2019 ◽  
Vol 83 (3) ◽  
pp. 15-92
Author(s):  
Игорь Владимирович Долгачев ◽  
Igor Vladimirovich Dolgachev ◽  
Alexander Duncan ◽  
Alexander Duncan
Keyword(s):  
Isomorphism Class ◽  
Normal Forms ◽  
Positive Characteristic ◽  
Automorphism Groups ◽  
Del Pezzo Surfaces ◽  
Closed Field ◽  
Intermediate Step ◽  
Cubic Surface ◽  
Cubic Surfaces ◽  
Del Pezzo

We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely characterize when a smooth cubic surface in positive characteristic, together with a group action, can be lifted to characteristic zero. Bibliography: 29 titles.

scholarly journals Which rational double points occur on del Pezzo surfaces?

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Claudia Stadlmayr
Keyword(s):  
Positive Characteristic ◽  
Del Pezzo Surfaces ◽  
Closed Field ◽  
Arbitrary Degree ◽  
Algebraically Closed Field ◽  
Double Points ◽  
Arbitrary Characteristic ◽  
Classical Work ◽  
Del Pezzo

We determine all configurations of rational double points that occur on RDP del Pezzo surfaces of arbitrary degree and Picard rank over an algebraically closed field $k$ of arbitrary characteristic ${\rm char}(k)=p \geq 0$, generalizing classical work of Du Val to positive characteristic. Moreover, we give simplified equations for all RDP del Pezzo surfaces of degree $1$ containing non-taut rational double points.

scholarly journals Del Pezzo surfaces over finite fields and their Frobenius traces

2018 ◽  
Vol 167 (01) ◽  
pp. 35-60 ◽  
Author(s):  
BARINDER BANWAIT ◽  
FRANCESC FITÉ ◽  
DANIEL LOUGHRAN
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Del Pezzo Surfaces ◽  
Cubic Surface ◽  
Inverse Galois Problem ◽  
Complete Answer ◽  
Cubic Surfaces ◽  
Analogous Question ◽  
Special Cases ◽  
Del Pezzo

AbstractLet S be a smooth cubic surface over a finite field $\mathbb{F}$q. It is known that #S($\mathbb{F}$q) = 1 + aq + q2 for some a ∈ {−2, −1, 0, 1, 2, 3, 4, 5, 7}. Serre has asked which values of a can arise for a given q. Building on special cases treated by Swinnerton–Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton–Dyer's tables on cubic surfaces over finite fields.

scholarly journals Del Pezzo surfaces and Mori fiber spaces in positive characteristic

2019 ◽  
Vol 373 (3) ◽  
pp. 1775-1843 ◽  
Author(s):  
Andrea Fanelli ◽  
Stefan Schröer
Keyword(s):  
Positive Characteristic ◽  
Del Pezzo Surfaces ◽  
Del Pezzo

scholarly journals ON DEL PEZZO FIBRATIONS IN POSITIVE CHARACTERISTIC

2020 ◽  
pp. 1-43
Author(s):  
Fabio Bernasconi ◽  
Hiromu Tanaka
Keyword(s):  
Positive Characteristic ◽  
Three Dimensional ◽  
Line Bundles ◽  
Del Pezzo Surfaces ◽  
Bounded Degree ◽  
Explicit Bound ◽  
Del Pezzo ◽  
Log Del Pezzo Surfaces

We establish two results on three-dimensional del Pezzo fibrations in positive characteristic. First, we give an explicit bound for torsion index of relatively torsion line bundles. Second, we show the existence of purely inseparable sections with explicit bounded degree. To prove these results, we study log del Pezzo surfaces defined over imperfect fields.

p-Subgroups in Automorphism Groups of Real del Pezzo Surfaces

2018 ◽  
Vol 97 (2) ◽  
pp. 129-130
Author(s):  
E. A. Yasinsky
Keyword(s):  
Automorphism Groups ◽  
Del Pezzo Surfaces ◽  
Del Pezzo

scholarly journals On log del Pezzo surfaces in large characteristic

2017 ◽  
Vol 153 (4) ◽  
pp. 820-850 ◽  
Author(s):  
Paolo Cascini ◽  
Hiromu Tanaka ◽  
Jakub Witaszek
Keyword(s):  
Characteristic Zero ◽  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Closed Field ◽  
Vanishing Theorem ◽  
Del Pezzo Surface ◽  
Algebraically Closed Field ◽  
Del Pezzo ◽  
Log Del Pezzo Surfaces

We show that any Kawamata log terminal del Pezzo surface over an algebraically closed field of large characteristic is globally $F$-regular or it admits a log resolution which lifts to characteristic zero. As a consequence, we prove the Kawamata–Viehweg vanishing theorem for klt del Pezzo surfaces of large characteristic.

scholarly journals Automorphisms of cubic surfaces without points

2020 ◽  
Vol 31 (11) ◽  
pp. 2050083
Author(s):  
Constantin Shramov
Keyword(s):  
Automorphism Group ◽  
Finite Groups ◽  
Characteristic Zero ◽  
Automorphism Groups ◽  
Sharp Bound ◽  
Cubic Surface ◽  
Birational Transformations ◽  
Cubic Surfaces ◽  
Birational Automorphism

We classify finite groups acting by birational transformations of a nontrivial Severi–Brauer surface over a field of characteristic zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field [Formula: see text] of characteristic zero that has no [Formula: see text]-points is abelian, and find a sharp bound for the Jordan constants of birational automorphism groups of such cubic surfaces.

scholarly journals Automorphism Groups of Quartic del Pezzo Surfaces

1996 ◽  
Vol 185 (2) ◽  
pp. 374-389 ◽  
Author(s):  
Toshio Hosoh
Keyword(s):  
Automorphism Groups ◽  
Del Pezzo Surfaces ◽  
Del Pezzo

scholarly journals Log del Pezzo Surfaces with Simple Automorphism Groups

2014 ◽  
Vol 58 (1) ◽  
pp. 33-52 ◽  
Author(s):  
Grigory Belousov
Keyword(s):  
Simple Group ◽  
Finite Simple Group ◽  
Automorphism Groups ◽  
Del Pezzo Surfaces ◽  
Del Pezzo ◽  
Log Del Pezzo Surfaces

AbstractIn the present paper we classify del Pezzo surfaces with log terminal singularities admitting an action of a finite simple group.

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.

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