scholarly journals MINIMALITY AND MUTATION-EQUIVALENCE OF POLYGONS

2017 ◽  
Vol 5 ◽  
Author(s):  
ALEXANDER KASPRZYK ◽  
BENJAMIN NILL ◽  
THOMAS PRINCE
Keyword(s):  
Mirror Symmetry ◽  
Smooth Surface ◽  
Equivalence Classes ◽  
Del Pezzo Surfaces ◽  
Toric Surface ◽  
Del Pezzo

We introduce a concept of minimality for Fano polygons. We show that, up to mutation, there are only finitely many Fano polygons with given singularity content, and give an algorithm to determine representatives for all mutation-equivalence classes of such polygons. This is a key step in a program to classify orbifold del Pezzo surfaces using mirror symmetry. As an application, we classify all Fano polygons such that the corresponding toric surface is qG-deformation-equivalent to either (i) a smooth surface; or (ii) a surface with only singularities of type$1/3(1,1)$.

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.

scholarly journals Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

2006 ◽  
Vol 166 (3) ◽  
pp. 537-582 ◽  
Author(s):  
Denis Auroux ◽  
Ludmil Katzarkov ◽  
Dmitri Orlov
Keyword(s):  
Mirror Symmetry ◽  
Del Pezzo Surfaces ◽  
Coherent Sheaves ◽  
Vanishing Cycles ◽  
Del Pezzo

scholarly journals ALGORITHMS FOR SINGULARITIES AND REAL STRUCTURES OF WEAK DEL PEZZO SURFACES

2014 ◽  
Vol 13 (05) ◽  
pp. 1350158
Author(s):  
NIELS LUBBES
Keyword(s):  
Equivalence Classes ◽  
Point Of View ◽  
Alternative Proof ◽  
Del Pezzo Surfaces ◽  
Isolated Singularities ◽  
Classification Of Singularities ◽  
Del Pezzo ◽  
Real Forms

In this paper, we consider the classification of singularities [P. Du Val, On isolated singularities of surfaces which do not affect the conditions of adjunction. I, II, III, Proc. Camb. Philos. Soc.30 (1934) 453–491] and real structures [C. T. C. Wall, Real forms of smooth del Pezzo surfaces, J. Reine Angew. Math.1987(375/376) (1987) 47–66, ISSN 0075-4102] of weak Del Pezzo surfaces from an algorithmic point of view. It is well-known that the singularities of weak Del Pezzo surfaces correspond to root subsystems. We present an algorithm which computes the classification of these root subsystems. We represent equivalence classes of root subsystems by unique labels. These labels allow us to construct examples of weak Del Pezzo surfaces with the corresponding singularity configuration. Equivalence classes of real structures of weak Del Pezzo surfaces are also represented by root subsystems. We present an algorithm which computes the classification of real structures. This leads to an alternative proof of the known classification for Del Pezzo surfaces and extends this classification to singular weak Del Pezzo surfaces. As an application we classify families of real conics on cyclides.

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 On the quantum periods of del Pezzo surfaces with ⅓ (1, 1) singularities

2018 ◽  
Vol 18 (3) ◽  
pp. 303-336
Author(s):  
Alessandro Oneto ◽  
Andrea Petracci
Keyword(s):  
Strong Evidence ◽  
Mirror Symmetry ◽  
Broad Class ◽  
Del Pezzo Surfaces ◽  
Laurent Polynomials ◽  
Joint Work ◽  
Toric Degeneration ◽  
Quotient Singularities ◽  
Del Pezzo

AbstractIn earlier joint work with collaborators we gave a conjectural classification of a broad class of orbifold del Pezzo surfaces, using Mirror Symmetry. We proposed that del Pezzo surfaces X with isolated cyclic quotient singularities such that X admits a ℚ-Gorenstein toric degeneration correspond via Mirror Symmetry to maximally mutable Laurent polynomials f in two variables, and that the quantum period of such a surface X, which is a generating function for Gromov–Witten invariants of X, coincides with the classical period of its mirror partner f.In this paper we give strong evidence for this conjecture. Contingent on conjectural generalisations of the Quantum Lefschetz theorem and the Abelian/non-Abelian correspondence, we compute many quantum periods for del Pezzo surfaces with $\begin{array}{} \frac{1}{3} \end{array} $(1, 1) singularities. Our computations also give strong evidence for the extension of these two principles to the orbifold setting.

scholarly journals Mirror symmetry and the classification of orbifold del Pezzo surfaces

2015 ◽  
Vol 144 (2) ◽  
pp. 513-527 ◽  
Author(s):  
Mohammad Akhtar ◽  
Tom Coates ◽  
Alessio Corti ◽  
Liana Heuberger ◽  
Alexander Kasprzyk ◽  
...  
Keyword(s):  
Mirror Symmetry ◽  
Del Pezzo Surfaces ◽  
Del Pezzo

scholarly journals CLOSED SUB-MONODROMY PROBLEMS, LOCAL MIRROR SYMMETRY AND BRANES ON ORBIFOLDS

2001 ◽  
Vol 13 (06) ◽  
pp. 675-715 ◽  
Author(s):  
KENJI MOHRI ◽  
YOKO ONJO ◽  
SUNG-KIL YANG
Keyword(s):  
Physical Properties ◽  
Exponential Growth ◽  
Mirror Symmetry ◽  
Full Detail ◽  
Del Pezzo Surfaces ◽  
Special Value ◽  
Del Pezzo ◽  
Local Mirror Symmetry

We study D-branes wrapping an exceptional four-cycle P(1,a,b) in a blown-up C3/Zm non-compact Calabi–Yau threefold with (m;a,b)=(3;1,1), (4;1,2) and (6; 2, 3). In applying the method of local mirror symmetry we find that the Picard–Fuchs equations for the local mirror periods in the Z3,4,6 orbifolds take the same form as the ones in the local E6,7,8 del Pezzo models, respectively. It is observed, however, that the orbifold models and the del Pezzo models possess different physical properties because the background NS B-field is turned on in the case of Z3,4,6 orbifolds. This is shown by analyzing the periods and their monodromies in full detail with the help of Meijer G-functions. We use the results to discuss D-brane configurations on P(1,a,b) as well as on del Pezzo surfaces. We also discuss the number theoretic aspect of local mirror symmetry and observe that the exponent which governs the exponential growth of the Gromov–Witten invariants is determined by the special value of the Dirichlet L-function.

Sign in / Sign up

Export Citation Format

Share Document