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 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 Maximally mutable Laurent polynomials

2021 ◽  
Vol 477 (2254) ◽  
Author(s):  
Tom Coates ◽  
Alexander M. Kasprzyk ◽  
Giuseppe Pitton ◽  
Ketil Tveiten
Keyword(s):  
Mirror Symmetry ◽  
Three Dimensional ◽  
Computer Assisted ◽  
Del Pezzo Surfaces ◽  
Fano Varieties ◽  
Laurent Polynomials ◽  
Fano Manifold ◽  
One To One ◽  
Del Pezzo

We introduce a class of Laurent polynomials, called maximally mutable Laurent polynomials (MMLPs), which we believe correspond under mirror symmetry to Fano varieties. A subclass of these, called rigid, are expected to correspond to Fano varieties with terminal locally toric singularities. We prove that there are exactly 10 mutation classes of rigid MMLPs in two variables; under mirror symmetry these correspond one-to-one with the 10 deformation classes of smooth del Pezzo surfaces. Furthermore, we give a computer-assisted classification of rigid MMLPs in three variables with reflexive Newton polytope; under mirror symmetry these correspond one-to-one with the 98 deformation classes of three-dimensional Fano manifolds with very ample anti-canonical bundle. We compare our proposal to previous approaches to constructing mirrors to Fano varieties, and explain why mirror symmetry in higher dimensions necessarily involves varieties with terminal singularities. Every known mirror to a Fano manifold, of any dimension, is a rigid MMLP.

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

Primitive Genus Two Groups of Meta Type

2018 ◽  
Vol 7 (1-2) ◽  
pp. 1
Author(s):  
Haval Mohammed Salih
Keyword(s):  
Computer Algebra ◽  
Computer Algebra System ◽  
Complete Classification ◽  
Permutation Groups ◽  
Affine Groups ◽  
Two Systems ◽  
Genus Two

This paper is a contribution to the classification of the finite primitive permutation groups of genus two. We consider the case of affine groups. Our main result, Lemma 3.10 gives a complete classification of genus two systems when . We achieve this classification with the aid of the computer algebra system GAP.

scholarly journals A Characterization of Some Fano 4-folds Through Conic Fibrations

2019 ◽  
Author(s):  
Pedro Montero ◽  
Eleonora Anna Romano
Keyword(s):  
Del Pezzo Surfaces ◽  
Conic Bundle ◽  
Conic Bundles ◽  
Del Pezzo ◽  
Bundle Structure

Abstract We find a characterization for Fano 4-folds $X$ with Lefschetz defect $\delta _{X}=3$: besides the product of two del Pezzo surfaces, they correspond to varieties admitting a conic bundle structure $f\colon X\to Y$ with $\rho _{X}-\rho _{Y}=3$. Moreover, we observe that all of these varieties are rational. We give the list of all possible targets of such contractions. Combining our results with the classification of toric Fano $4$-folds due to Batyrev and Sato we provide explicit examples of Fano conic bundles from toric $4$-folds with $\delta _{X}=3$.

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