scholarly journals On exceptional collections of line bundles and mirror symmetry for toric Del-Pezzo surfaces

2017 ◽  
Vol 58 (3) ◽  
pp. 031704 ◽  
Author(s):  
Yochay Jerby
Keyword(s):  
Mirror Symmetry ◽  
Line Bundles ◽  
Del Pezzo Surfaces ◽  
Del Pezzo

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 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 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 ON DEL PEZZO FIBRATIONS IN POSITIVE CHARACTERISTIC

2020 ◽  
pp. 1-43
Author(s):  
Fabio Bernasconi ◽  
Hiromu Tanaka
Keyword(s):  
Positive Characteristic ◽  
Three Dimensional ◽  
Line Bundles ◽  
Del Pezzo Surfaces ◽  
Bounded Degree ◽  
Explicit Bound ◽  
Del Pezzo ◽  
Log Del Pezzo Surfaces

We establish two results on three-dimensional del Pezzo fibrations in positive characteristic. First, we give an explicit bound for torsion index of relatively torsion line bundles. Second, we show the existence of purely inseparable sections with explicit bounded degree. To prove these results, we study log del Pezzo surfaces defined over imperfect fields.

scholarly journals Universal Series for Hilbert Schemes and Strange Duality

2018 ◽  
Vol 2020 (10) ◽  
pp. 3130-3152
Author(s):  
Drew Johnson
Keyword(s):  
Combinatorial Proof ◽  
Line Bundles ◽  
Del Pezzo Surfaces ◽  
Hilbert Schemes ◽  
Euler Characteristics ◽  
Quot Scheme ◽  
Del Pezzo ◽  
Complex Projective ◽  
Strange Duality ◽  
Projective Surfaces

Abstract We show how the “finite Quot scheme method” applied to Le Potier’s strange duality on del Pezzo surfaces leads to conjectures (valid for all smooth complex projective surfaces) relating two sets of universal power series on Hilbert schemes of points on surfaces: those for top Chern classes of tautological sheaves and those for Euler characteristics of line bundles. We have verified these predictions computationally for low order. We then give an analysis of these conjectures in small ranks. We also give a combinatorial proof of a formula predicted by our conjectures: the top Chern class of the tautological sheaf on $S^{[n]}$ associated to the structure sheaf of a point is equal to $(-1)^n$ times the nth Catalan number.

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
Sign in / Sign up

Export Citation Format

Share Document