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 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 Polarized Rigid Del Pezzo Surfaces in Low Codimension

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1567
Author(s):  
Muhammad Imran Qureshi
Keyword(s):  
Computer Algebra System ◽  
Symmetric Matrix ◽  
Single Equation ◽  
Complete Classification ◽  
Computer Assisted ◽  
Del Pezzo Surfaces ◽  
Linearly Independent ◽  
Del Pezzo ◽  
Detailed Computer

We provide explicit graded constructions of orbifold del Pezzo surfaces with rigid orbifold points of type ki×1ri(1,ai):3≤ri≤10,ki∈Z≥0 as well-formed and quasismooth varieties embedded in some weighted projective space. In particular, we present a collection of 147 such surfaces such that their image under their anti-canonical embeddings can be described by using one of the following sets of equations: a single equation, two linearly independent equations, five maximal Pfaffians of 5×5 skew symmetric matrix, and nine 2×2 minors of size 3 square matrix. This is a complete classification of such surfaces under certain carefully chosen bounds on the weights of ambient weighted projective spaces and it is largely based on detailed computer-assisted searches by using the computer algebra system MAGMA.

scholarly journals Chern-Simons: Fano and Calabi-Yau

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Amihay Hanany ◽  
Yang-Hui He
Keyword(s):  
Algebraic Geometry ◽  
Complete Classification ◽  
Simons Theory ◽  
Del Pezzo Surfaces ◽  
Fano Threefolds ◽  
Del Pezzo ◽  
Chern Simons ◽  
Chern Simons Theory

We present the complete classification of smooth toric Fano threefolds, known to the algebraic geometry literature, and perform some preliminary analyses in the context of brane tilings and Chern-Simons theory on M2-branes probing Calabi-Yau fourfold singularities. We emphasise that these 18 spaces should be as intensely studied as their well-known counterparts: the del Pezzo surfaces.

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 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.

scholarly journals Toric systems and mirror symmetry

2013 ◽  
Vol 149 (11) ◽  
pp. 1839-1855 ◽  
Author(s):  
Raf Bocklandt
Keyword(s):  
Mirror Symmetry ◽  
Line Bundles ◽  
Del Pezzo Surfaces ◽  
Toric Surface ◽  
Exceptional Sequence ◽  
Del Pezzo Surface ◽  
Rational Surfaces ◽  
Exceptional Sequences ◽  
Del Pezzo ◽  
Compositio Math

AbstractIn their paper [Exceptional sequences of invertible sheaves on rational surfaces, Compositio Math. 147 (2011), 1230–1280], Hille and Perling associate to every cyclic full strongly exceptional sequence of line bundles on a toric weak del Pezzo surface a toric system, which defines a new toric surface. We interpret this construction as an instance of mirror symmetry and extend it to a duality on the set of toric weak del Pezzo surfaces equipped with a cyclic full strongly exceptional sequence.

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