Toric Del Pezzo Surfaces Research Articles

Arithmetic counts of tropical plane curves and their properties

Abstract Recently, the first and third author proved a correspondence theorem which recovers the Levine- Welschinger invariants of toric del Pezzo surfaces as a count of tropical curves weighted with arithmetic multiplicities. In this paper, we study properties of the arithmetic count ofplane tropical curves satisfying point conditions. We prove that this count is independent of the configuration of point conditions. Moreover, a Caporaso- Harris formula for the arithmetic count of plane tropical curves is obtained by moving one point to the very left. Repeating this process until all point conditions are stretched, we obtain an enriched count of floor diagrams which coincides with the tropical count. Finally, we prove polynomiality properties for the arithmetic counts using floor diagrams.

Zig-zag deformations of toric quiver gauge theories. Part I. Reflexive polytopes

We study one-parameter families of U(1)2 preserving deformations relating pairs of toric quiver gauge theories on D-branes probing local toric (pseudo) del Pezzo surfaces. The superpotential deformations are defined by zig-zag paths in the brane tiling and are non-trivial in the chiral ring if the geometry has a non-isolated singularity. In the dual (p, q) web, the deformation is realized as a Hanany-Witten move that reverses a semi-infinite fivebrane. We use these deformations to find RG flows between 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 SCFTs on D3-branes probing local toric (pseudo) del Pezzo surfaces of the same degree, and briefly comment on the interpretation for BPS quivers of rank one 5d SCFTs on S1.

Cohomological χ–independence for moduli of one-dimensional sheaves and moduli of Higgs bundles

We prove that the intersection cohomology (together with the perverse and the Hodge filtrations) for the moduli space of one-dimensional semistable sheaves supported in an ample curve class on a toric del Pezzo surface is independent of the Euler characteristic of the sheaves. We also prove an analogous result for the moduli space of semistable Higgs bundles with respect to an effective divisor $D$ of degree $\mathrm{deg}(D)>2g-2$. Our results confirm the cohomological $\chi$-independence conjecture by Bousseau for $\mathbb{P}^2$, and verify Toda's conjecture for Gopakumar-Vafa invariants for certain local curves and local surfaces. For the proof, we combine a generalized version of Ng\^o's support theorem, a dimension estimate for the stacky Hilbert-Chow morphism, and a splitting theorem for the morphism from the moduli stack to the good GIT quotient.

Twistor geometry of the Flag manifold

A study is made of algebraic curves and surfaces in the flag manifold $$\mathbb {F}=SU(3)/T^2$$ , and their configuration relative to the twistor projection $$\pi $$ from $$\mathbb {F}$$ to the complex projective plane $$\mathbb {P}^{2}$$ , defined with the help of an anti-holomorphic involution $$j$$ . This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space $$\mathbb {P}^3$$ of the 4-dimensional sphere $$S^4$$ . Deformations of twistor fibers project to real surfaces in $$\mathbb {P}^{2}$$ , whose metric geometry is investigated. Attention is then focussed on toric del Pezzo surfaces that are the simplest type of surfaces in $$\mathbb {F}$$ of bidegree $$(1,1)$$ . These surfaces define orthogonal complex structures on specified dense open subsets of $$\mathbb {P}^{2}$$ relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree $$(1,1)$$ are determined, and bounds given on the number of twistor fibers that are contained in more general algebraic surfaces in $$\mathbb {F}$$ .

Classification of Toric Log Del Pezzo Surfaces with Few Singular Points


 
 
 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.
 
 
 

Tropical Lagrangians in toric del-Pezzo surfaces

We look at how one can construct from the data of a dimer model a Lagrangian submanifold in (mathbb {C}^*)^n whose valuation projection approximates a tropical hypersurface. Each face of the dimer corresponds to a Lagrangian disk with boundary on our tropical Lagrangian submanifold, forming a Lagrangian mutation seed. Using this we find tropical Lagrangian tori L_{T^2} in the complement of a smooth anticanonical divisor of a toric del-Pezzo whose wall-crossing transformations match those of monotone SYZ fibers. An example is worked out for the mirror pair (mathbb {CP}^2{setminus } E, W), {check{X}}_{9111}. We find a symplectomorphism of mathbb {CP}^2{setminus } E interchanging L_{T^2} and a SYZ fiber. Evidence is provided that this symplectomorphism is mirror to fiberwise Fourier–Mukai transform on {check{X}}_{9111}.

On the Twelve-Point Theorem for $\ell$-Reflexive Polygons

It is known that, adding the number of lattice points lying on the boundary of a reflexive polygon and the number of lattice points lying on the boundary of its polar, always yields 12. Generalising appropriately the notion of reflexivity, one shows that this remains true for $\ell$-reflexive polygons. In particular, there exist (for this reason) infinitely many (lattice inequivalent) lattice polygons with the same property. The first proof of this fact is due to Kasprzyk and Nill. The present paper contains a second proof (which uses tools only from toric geometry) as well as the description of complementary properties of these polygons and of the invariants of the corresponding toric log del Pezzo surfaces.

Discrete symmetries in dimer diagrams

We apply dimer diagram techniques to uncover discrete global symmetries in the fields theories on D3-branes at singularities given by general orbifolds of general toric Calabi-Yau threefold singularities. The discrete symmetries are discrete Heisenberg groups, with two ZN generators A, B with commutation AB = C BA, with C a central element. This fully generalizes earlier observations in particular orbifolds of C3, the conifold and Yp,q . The solution for any orbifold of a given parent theory follows from a universal structure in the infinite dimer in R2 giving the covering space of the unit cell of the parent theory before orbifolding. The generator A is realized as a shift in the dimer diagram, associated to the orbifold quantum symmetry; the action of B is determined by equations describing a 1-form in the dimer graph in the unit cell of the parent theory with twisted boundary conditions; finally, C is an element of the (mesonic and baryonic) non-anomalous U (1) symmetries, determined by geometric identities involving the elements of the dimer graph of the parent theory. These discrete global symmetries of the quiver gauge theories are holographically dual to discrete gauge symmetries from torsion cycles in the horizon, as we also briefly discuss. Our findings allow to easily construct the discrete symmetries for infinite classes of orbifolds. We provide explicit examples by constructing the discrete symmetries for the infinite classes of general orbifolds of C3, conifold, and complex cones over the toric del Pezzo surfaces, dP1, dP2 and dP3.

Asymptotic Chow stability of toric Del Pezzo surfaces

In this short note, we study the asymptotic Chow polystability of toric Del Pezzo surfaces appear in the moduli space of K\ahler-Einstein Fano varieties constructed in [OSS16].

Smoothing toric Fano surfaces using the Gross-Siebert algorithm

A toric del Pezzo surface $X_P$ with cyclic quotient singularities determines and is determined by a Fano polygon $P$. We construct an affine manifold with singularities that partially smooths the boundary of $P$; this a tropical version of a Q-Gorenstein partial smoothing of $X_P$. We implement a mild generalization of the Gross-Siebert reconstruction algorithm - allowing singularities that are not locally rigid - and thereby construct (a formal version of) this partial smoothing directly from the affine manifold. This has implications for mirror symmetry: roughly speaking, it implements half of the expected mirror correspondence between del Pezzo surfaces with cyclic quotient singularities and Laurent polynomials in two variables.

More Toda-like (0,2) mirrors

In this paper, we extend our previous work to construct (0, 2) Toda-like mirrors to A/2-twisted theories on more general spaces, as part of a program of understanding (0,2) mirror symmetry. Specifically, we propose (0, 2) mirrors to GLSMs on toric del Pezzo surfaces and Hirzebruch surfaces with deformations of the tangent bundle. We check the results by comparing correlation functions, global symmetries, as well as geometric blowdowns with the corresponding (0, 2) Toda-like mirrors. We also briefly discuss Grassmannian manifolds.

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

Let X be a toric Del-Pezzo surface and let Crit(W)⊂(ℂ*)n be the solution scheme of the Landau-Ginzburg system of equations. Denote by X° the polar variety of X. Our aim in this work is to describe a map L:Crit(W)→Fuktrop(X°) whose image under homological mirror symmetry corresponds to a full strongly exceptional collection of line bundles.

Local Donaldson–Thomas invariants of blowups of surfaces

Using the degeneration formula for Doanldson-Thomas in- variants, we proved a formula for the change of Donaldson-Thomas in- variants of local surfaces under blowing up along points. Given a smooth projective Calabi-Yau 3-fold X, the moduli space of stable sheaves on X has virtual dimension zero. Donaldson and Thomas (D-T) defined the holomorphic Casson invariant of X which essentially counts the number of stable bundles on X. However, the moduli space has positive dimension and is singular in general. Making use of virtual cycle technique (see (B-F) and (L-T)), Thomas in (Thomas) showed that one can define a virtual moduli cycle for some X including Calabi-Yau and Fano 3-folds. As a consequence, one can define Donaldson-type invariants of X which are deformation invariant. Donaldson-Thomas invariants provide a new vehicle to study the geometry and other aspects of higher-dimensional varieties. It is important to understand these invariants. It is well-known (MNOP1, MNOP2)that there is a correspondence be- tween Donaldson-Thomas invariants and Gromov-Witten invariants. Both invariants are deformation independent. On the side of Gromov-Witten in- variants, Li and Ruan in (L-R) first established the degeneration formula of Gromov-Witten invariants in symplectic geometry. J. Li proved an algebraic geometry version of this degeneration formula. In (Hu1, Hu2), the author studied the change of Gromov-Witten invariants under the blowup. The author (Hu3) also studies the change of local Gromov-Witten invariants of Fano surfaces under the blowup. In the birational geometry of 3-folds, we have blowups and flops which are semistable degenerations. In (HL) the au- thors studied how Donaldson-Thomas invariants change under the blowup at a point, some flops and extremal transitions. Local del Pezzo surface used to play an important role in physics. Local de Pezzo surfaces are usually associated to phase transitions in the Kahler mod- uli space of various string, M-theory, and F-theory compactifications.Non- toric del Pezzo surfaces seem to be related to exotic physics in four, five and

On Landau–Ginzburg systems, quivers and monodromy

Let X be a toric Fano manifold and denote by C r i t ( f X ) ⊂ ( C ∗ ) n the solution scheme of the corresponding Landau–Ginzburg system of equations. For toric Del-Pezzo surfaces and various toric Fano threefolds we define a map L : C r i t ( f X ) → P i c ( X ) such that E L ( X ) : = L ( C r i t ( f X ) ) ⊂ P i c ( X ) is a full strongly exceptional collection of line bundles. We observe the existence of a natural monodromy map M : π 1 ( L ( X ) ∖ R X , f X ) → A u t ( C r i t ( f X ) ) where L ( X ) is the space of all Laurent polynomials whose Newton polytope is equal to the Newton polytope of f X , the Landau–Ginzburg potential of X , and R X ⊂ L ( X ) is the space of all elements whose corresponding solution scheme is reduced. We show that monodromies of C r i t ( f X ) admit non-trivial relations to quiver representations of the exceptional collection E L ( X ) . We refer to this property as the M -aligned property of the maps L : C r i t ( f X ) → P i c ( X ) . We discuss possible applications of the existence of such M -aligned exceptional maps to various aspects of mirror symmetry of toric Fano manifolds.

Welschinger invariants of small non-toric Del Pezzo surfaces

We give a recursive formula for purely real Welschinger invariants of the following real Del Pezzo surfaces: the projective plane blown up at q real and s \leq 1 pairs of conjugate imaginary points, where q+2s\le 5 , and the real quadric blown up at s \leq 1 pairs of conjugate imaginary points and having non-empty real part. The formula is similar to Vakil's recursive formula [22] for Gromov–Witten invariants of these surfaces and generalizes our recursive formula [12] for purely real Welschinger invariants of real toric Del Pezzo surfaces. As a consequence, we prove the positivity of the Welschinger invariants under consideration and their logarithmic asymptotic equivalence to genus zero Gromov–Witten invariants.

Integrality of relative BPS state counts of toric del Pezzo surfaces

Relative Bogomolny-Prasad-Sommerfield (BPS) state counts for log Calabi-Yau surface pairs were introduced by Gross-Pandharipande-Siebert in [4] and conjectured by the authors to be integers. For toric del Pezzo surfaces, we provide an arithmetic proof of this conjecture, by relating these invariants to the local BPS state counts of the surfaces. The latter were shown to be integers by Peng in [15]; and more generally for toric Calabi-Yau three-folds by Konishi in [8].

Homological mirror symmetry for toric orbifolds of toric del Pezzo surfaces

We formulate a conjecture which describes the Fukaya category of an exact Lefschetz fibration defined by a Laurent polynomial in two variables in terms of a pair consisting of a consistent dimer model and a perfect matching on it. We prove this conjecture in some cases, and obtain homological mirror symmetry for quotient stacks of toric del Pezzo surfaces by finite subgroups of the torus as a corollary.

Multiplier ideal sheaves and integral invariants on toric Fano manifolds

We extend Nadel’s results on some conditions for the multiplier ideal sheaves to satisfy which are described in terms of an obstruction defined by the first author. Applying our extension we can determine the multiplier ideal subvarieties on toric del Pezzo surfaces which do not admit Kahler–Einstein metrics. We also show that one can define multiplier ideal sheaves for Kahler–Ricci solitons and extend the result of Nadel using the holomorphic invariant defined by Tian and Zhu.

Topological open string amplitudes on local toric del Pezzo surfaces via remodeling the B-model

We study topological strings on local toric del Pezzo surfaces by a method called remodeling the B-model which was recently proposed by Bouchard, Klemm, Mariño and Pasquetti. For a large class of local toric del Pezzo surfaces we prove a functional formula of the Bergman kernel which is the basic constituent of the topological string amplitudes by the topological recursion relation of Eynard and Orantin. Because this formula is written as a functional of the period, we can obtain the topological string amplitudes at any point of the moduli space by a simple change of variables of the Picard–Fuchs equations for the period. By this formula and mirror symmetry we compute the A-model amplitudes on K F 2 , and predict the open orbifold Gromov–Witten invariants of C 3 / Z 4 .

A Caporaso–Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces

We define a series of relative tropical Welschinger-type invariants of real toric surfaces. In the Del Pezzo case, these invariants can be seen as real tropical analogs of relative Gromov–Witten invariants, and are subject to a recursive formula. As application we obtain new formulas for Welschinger invariants of real toric Del Pezzo surfaces.