scholarly journals Cylinders in singular del Pezzo surfaces

2016 ◽  
Vol 152 (6) ◽  
pp. 1198-1224 ◽  
Author(s):  
Ivan Cheltsov ◽  
Jihun Park ◽  
Joonyeong Won
Keyword(s):  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Del Pezzo Surface ◽  
Open Set ◽  
Del Pezzo

For each del Pezzo surface $S$ with du Val singularities, we determine whether it admits a $(-K_{S})$-polar cylinder or not. If it allows one, then we present an effective $\mathbb{Q}$-divisor $D$ that is $\mathbb{Q}$-linearly equivalent to $-K_{S}$ and such that the open set $S\setminus \text{Supp}(D)$ is a cylinder. As a corollary, we classify all the del Pezzo surfaces with du Val singularities that admit non-trivial $\mathbb{G}_{a}$-actions on their affine cones defined by their anticanonical divisors.

scholarly journals On Manin's conjecture for singular del Pezzo surfaces of degree four, II

2007 ◽  
Vol 143 (3) ◽  
pp. 579-605 ◽  
Author(s):  
R. DE LA BRETÈCHE ◽  
T. D. BROWNING
Keyword(s):  
Zeta Function ◽  
Half Plane ◽  
Open Subset ◽  
Height Function ◽  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Del Pezzo Surface ◽  
Del Pezzo ◽  
Manin’S Conjecture ◽  
Height Zeta Function

AbstractThis paper establishes the Manin conjecture for a certain non-split singular del Pezzo surface of degree four$X \subset \bfP^4$. In fact, ifU⊂Xis the open subset formed by deleting the lines fromX, andHis the usual projective height function on$\bfP^4(\Q)$, then the height zeta function$ \sum_{x \in U(\Q)}{H(x)^{-s}} $is analytically continued to the half-plane ℜe(s) > 17/20.

THE BRAUER GROUP OF DEL PEZZO SURFACES

2011 ◽  
Vol 07 (02) ◽  
pp. 261-287
Author(s):  
ANDREA C. CARTER
Keyword(s):  
Number Field ◽  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Brauer Group ◽  
Nontrivial Element ◽  
Del Pezzo Surface ◽  
Del Pezzo

Let S1 be a Del Pezzo surface of degree 1 over a number field k. We establish a criterion for the existence of a nontrivial element of order 5 in the Brauer group of S1 in terms of certain Galois-stable configurations of exceptional divisors on this surface.

scholarly journals RATIONAL POINTS ON CERTAIN DEL PEZZO SURFACES OF DEGREE ONE

2008 ◽  
Vol 50 (3) ◽  
pp. 557-564 ◽  
Author(s):  
MACIEJ ULAS
Keyword(s):  
Rational Points ◽  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Zariski Topology ◽  
Del Pezzo Surface ◽  
Del Pezzo

AbstractLet$f(z)=z^5+az^3+bz^2+cz+d \in \Z[z]$and let us consider a del Pezzo surface of degree one given by the equation$\cal{E}_{f}\,{:}\,x^2-y^3-f(z)=0$. In this paper we prove that if the set of rational points on the curveEa,b:Y2=X3+ 135(2a−15)X−1350(5a+ 2b− 26) is infinite then the set of rational points on the surface ϵfis dense in the Zariski topology.

Weakly exceptional singularities of log del Pezzo surfaces

2019 ◽  
Vol 30 (01) ◽  
pp. 1950010
Author(s):  
In-Kyun Kim ◽  
Joonyeong Won
Keyword(s):  
Complete Intersection ◽  
Projective Spaces ◽  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Del Pezzo Surface ◽  
Log Canonical ◽  
Del Pezzo ◽  
Log Del Pezzo Surfaces

We complete the computation of global log canonical thresholds, or equivalently alpha invariants, of quasi-smooth well-formed complete intersection log del Pezzo surfaces of amplitude 1 in weighted projective spaces. As an application, we prove that they are weakly exceptional. And we investigate the super-rigid affine Fano 3-folds containing a log del Pezzo surface as boundary.

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 Log canonical thresholds on Gorenstein canonical del Pezzo surfaces

2010 ◽  
Vol 54 (1) ◽  
pp. 187-219 ◽  
Author(s):  
Jihun Park ◽  
Joonyeong Won
Keyword(s):  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Del Pezzo Surface ◽  
Log Canonical ◽  
Del Pezzo

AbstractWe classify all the effective anticanonical divisors on weak del Pezzo surfaces. Through this classification we obtain the smallest number among the log canonical thresholds of effective anticanonical divisors on a given Gorenstein canonical del Pezzo surface.

scholarly journals Quartic del Pezzo surfaces over function fields of curves

2014 ◽  
Vol 12 (3) ◽  
Author(s):  
Brendan Hassett ◽  
Yuri Tschinkel
Keyword(s):  
Finite Fields ◽  
Function Fields ◽  
Projective Line ◽  
Total Space ◽  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Del Pezzo Surface ◽  
Numerical Invariants ◽  
Del Pezzo ◽  
Intermediate Jacobian

AbstractWe classify quartic del Pezzo surface fibrations over the projective line via numerical invariants, giving explicit examples for small values of the invariants. For generic such fibrations, we describe explicitly the geometry of spaces of sections to the fibration, and mappings to the intermediate Jacobian of the total space. We exhibit examples where these are birational, which has applications to arithmetic questions, especially over finite fields.

scholarly journals Genus Six Curves, K3 Surfaces, and Stable Pairs

2020 ◽  
Author(s):  
J Ross Goluboff
Keyword(s):  
Smooth Curve ◽  
Explicit Description ◽  
Pezzo Surface ◽  
Del Pezzo Surfaces ◽  
Del Pezzo Surface ◽  
Double Covers ◽  
Del Pezzo ◽  
Stable Pairs ◽  
Special Genus ◽  
Surface Curve

Abstract A general smooth curve of genus six lies on a quintic del Pezzo surface. Artebani and Kondō [ 4] construct a birational period map for genus six curves by taking ramified double covers of del Pezzo surfaces. The map is not defined for special genus six curves. In this paper, we construct a smooth Deligne–Mumford stack ${\mathfrak{P}}_0$ parametrizing certain stable surface-curve pairs, which essentially resolves this map. Moreover, we give an explicit description of pairs in ${\mathfrak{P}}_0$ containing special curves.

scholarly journals Affine cones over cubic surfaces are flexible in codimension one

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Alexander Perepechko
Keyword(s):  
Automorphism Group ◽  
Open Subset ◽  
Pezzo Surface ◽  
Ample Divisor ◽  
Transitive Action ◽  
Del Pezzo Surface ◽  
Codimension One ◽  
Cubic Surfaces ◽  
Special Automorphism ◽  
Del Pezzo

AbstractLet Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on it.

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