On the equations for universal torsors over del Pezzo surfaces

2009 ◽  
Vol 9 (1) ◽  
pp. 203-223 ◽  
Author(s):  
Vera V. Serganova ◽  
Alexei. N. Skorobogatov
Keyword(s):  
Homogeneous Space ◽  
Del Pezzo Surfaces ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Del Pezzo

AbstractWe describe equations of the universal torsors over del Pezzo surfaces of degrees from 2 to 5 over an algebraically closed field in terms of the equations of the corresponding homogeneous space G/P. We also give a generalization for fields that are not algebraically closed.

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 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.

HOMOGENEOUS SPACE FIBRATIONS OVER SURFACES

2017 ◽  
Vol 18 (2) ◽  
pp. 293-327 ◽  
Author(s):  
Yi Zhu
Keyword(s):  
Homogeneous Space ◽  
Function Field ◽  
Algebraic Surface ◽  
Rational Point ◽  
Homogeneous Spaces ◽  
Rational Curves ◽  
Closed Field ◽  
Generic Fiber ◽  
Algebraically Closed Field ◽  
Simple Connectedness

By studying the theory of rational curves, we introduce a notion of rational simple connectedness for projective homogeneous spaces. As an application, we prove that over a function field of an algebraic surface over an algebraically closed field, a variety whose geometric generic fiber is a projective homogeneous space admits a rational point if and only if the elementary obstruction vanishes.

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 The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Main Application ◽  
Finite Dimensional ◽  
Simple Connectedness ◽  
Tame Algebras

AbstractWe study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

scholarly journals Cox rings of degree one del Pezzo surfaces

2009 ◽  
Vol 3 (7) ◽  
pp. 729-761 ◽  
Author(s):  
Damiano Testa ◽  
Anthony Várilly-Alvarado ◽  
Mauricio Velasco
Keyword(s):  
Del Pezzo Surfaces ◽  
Cox Rings ◽  
Del Pezzo

scholarly journals On Unramified Separable Abelian p-Extensions of Function Fields I

1959 ◽  
Vol 14 ◽  
pp. 223-234 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Galois Group ◽  
Function Field ◽  
Function Fields ◽  
Abelian Extension ◽  
Closed Field ◽  
Jacobian Variety ◽  
Algebraically Closed Field

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.

scholarly journals Tropicalization of del Pezzo surfaces

2016 ◽  
Vol 300 ◽  
pp. 156-189 ◽  
Author(s):  
Qingchun Ren ◽  
Kristin Shaw ◽  
Bernd Sturmfels
Keyword(s):  
Del Pezzo Surfaces ◽  
Del Pezzo
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