Classification of Toric log Del Pezzo Surfaces having Picard Number 1 and Index ≤ 3

2009 ◽  
Vol 54 (3-4) ◽  
pp. 219-252 ◽  
Author(s):  
Dimitrios I. Dais
Keyword(s):  
Del Pezzo Surfaces ◽  
Picard Number ◽  
Del Pezzo ◽  
Log Del Pezzo Surfaces

scholarly journals Quotients of del Pezzo surfaces

2019 ◽  
Vol 30 (12) ◽  
pp. 1950068
Author(s):  
Andrey Trepalin
Keyword(s):  
Complete Classification ◽  
Finite Subgroup ◽  
Del Pezzo Surfaces ◽  
Picard Number ◽  
Del Pezzo Surface ◽  
Cyclic Groups ◽  
Trivial Group ◽  
Quotient Surface ◽  
Del Pezzo

Let [Formula: see text] be any field of characteristic zero, [Formula: see text] be a del Pezzo surface and [Formula: see text] be a finite subgroup in [Formula: see text]. In this paper, we study when the quotient surface [Formula: see text] can be non-rational over [Formula: see text]. Obviously, if there are no smooth [Formula: see text]-points on [Formula: see text] then it is not [Formula: see text]-rational. Therefore, under assumption that the set of smooth [Formula: see text]-points on [Formula: see text] is not empty we show that there are few possibilities for non-[Formula: see text]-rational quotients. The quotients of del Pezzo surfaces of degree [Formula: see text] and greater are considered in the author’s previous papers. In this paper, we study the quotients of del Pezzo surfaces of degree [Formula: see text]. We show that they can be non-[Formula: see text]-rational only for the trivial group or cyclic groups of order [Formula: see text], [Formula: see text] and [Formula: see text]. For the trivial group and the group of order [Formula: see text], we show that both [Formula: see text] and [Formula: see text] are not [Formula: see text]-rational if the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. For the groups of order [Formula: see text] and [Formula: see text], we construct examples of both [Formula: see text]-rational and non-[Formula: see text]-rational quotients of both [Formula: see text]-rational and non-[Formula: see text]-rational del Pezzo surfaces of degree [Formula: see text] such that the [Formula: see text]-invariant Picard number of [Formula: see text] is [Formula: see text]. As a result of complete classification of non-[Formula: see text]-rational quotients of del Pezzo surfaces we classify surfaces that are birationally equivalent to quotients of [Formula: see text]-rational surfaces, and obtain some corollaries concerning fields of invariants of [Formula: see text].

scholarly journals Classification of log del Pezzo surfaces of index three

2017 ◽  
Vol 69 (1) ◽  
pp. 163-225 ◽  
Author(s):  
Kento FUJITA ◽  
Kazunori YASUTAKE
Keyword(s):  
Del Pezzo Surfaces ◽  
Del Pezzo ◽  
Log Del Pezzo Surfaces

scholarly journals BOUNDING THE VOLUMES OF SINGULAR WEAK LOG DEL PEZZO SURFACES

2013 ◽  
Vol 24 (13) ◽  
pp. 1350110 ◽  
Author(s):  
CHEN JIANG
Keyword(s):  
Upper Bound ◽  
General Position ◽  
Del Pezzo Surfaces ◽  
Minimal Resolution ◽  
Picard Number ◽  
Hirzebruch Surface ◽  
Del Pezzo Surface ◽  
Blowing Up ◽  
Del Pezzo ◽  
Log Del Pezzo Surfaces

We give an optimal upper bound for the anti-canonical volume of an ϵ-lc weak log del Pezzo surface. Moreover, we consider the relation between the bound of the volume and the Picard number of the minimal resolution of the surface. Furthermore, we consider blowing up several points on a Hirzebruch surface in general position and give some examples of smooth weak log del Pezzo surfaces.

scholarly journals On the combinatorial classification of toric log del Pezzo surfaces

2010 ◽  
Vol 13 ◽  
pp. 33-46 ◽  
Author(s):  
Alexander M. Kasprzyk ◽  
Maximilian Kreuzer ◽  
Benjamin Nill
Keyword(s):  
Upper Bounds ◽  
Lattice Points ◽  
Del Pezzo Surfaces ◽  
Convex Lattice ◽  
Combinatorial Classification ◽  
Del Pezzo ◽  
Convex Lattice Polygons ◽  
Log Del Pezzo Surfaces ◽  
Lattice Polygons

AbstractToric log del Pezzo surfaces correspond to convex lattice polygons containing the origin in their interior and having only primitive vertices. Upper bounds on the volume and on the number of boundary lattice points of these polygons are derived in terms of the indexℓ. Techniques for classifying these polygons are also described: a direct classification for index two is given, and a classification for allℓ≤16 is obtained.

scholarly journals Classification of Toric Log Del Pezzo Surfaces with Few Singular Points

10.37236/9056 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Yusuke Suyama
Keyword(s):  
Singular Points ◽  
Del Pezzo Surfaces ◽  
Del Pezzo ◽  
Log Del Pezzo Surfaces

We give a classification of toric log del Pezzo surfaces with two or three singular points. Our proofs are purely combinatorial, relying on the bijection between toric log del Pezzo surfaces and the so-called LDP-polygons introduced by Dais and Nill.  

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