Del Pezzo Surfaces Research Articles

Gamma conjecture I for del Pezzo surfaces

Gamma conjecture I and the underlying Conjecture O for Fano manifolds were proposed by Galkin, Golyshev and Iritani recently. We show that both conjectures hold for all two-dimensional Fano manifolds. We prove Conjecture O by deriving a generalized Perron-Frobenius theorem on eigenvalues of real matrices and a vanishing result of certain Gromov-Witten invariants for del Pezzo surfaces. We prove Gamma conjecture I by applying mirror techniques proposed by Galkin-Iritani together with the study of Gamma conjecture I for weighted projective spaces. We also provide applications of our generalized Perron-Frobenius theorem on Conjecture O for two Fano manifolds of higher dimensions.

Generic pointed quartic curves in {mathbb {R}}{mathbb {P}}^{2} and uninodal dessins

In this article we obtain a rigid isotopy classification of generic pointed quartic curves (A, p) in {mathbb {R}}{mathbb {P}}^{2} by studying the combinatorial properties of dessins. The dessins are real versions, proposed by Orevkov (Ann Fac Sci Toulouse 12(4):517–531, 2003), of Grothendieck’s dessins d’enfants. This classification contains 20 classes determined by the number of ovals of A, the parity of the oval containing the marked point p, the number of ovals that the tangent line T_p A intersects, the nature of connected components of Asetminus T_p A adjacent to p, and in the maximal case, on the convexity of the position of the connected components of Asetminus T_p A. We study the combinatorial properties and decompositions of dessins corresponding to real uninodal trigonal curves in real ruled surfaces. Uninodal dessins in any surface with non-empty boundary can be decomposed in blocks corresponding to cubic dessins in the disk {mathbf {D}}^2, which produces a classification of these dessins. The classification of dessins under consideration leads to a rigid isotopy classification of generic pointed quartic curves in {mathbb {R}}{mathbb {P}}^{2}. This classification was first obtained in Rieken (Geometr Ded 185(1):171–203, 2016) based on the relation between quartic curves and del Pezzo surfaces.

On the Kodaira vanishing theorem for log del Pezzo surfaces in positive characteristic

We investigate the vanishing of H^1(X,mathcal {O}_X(-D)) for a big and nef mathbb {Q}-Cartier mathbb {Z}-divisor D on a log del Pezzo pair (X,Delta ) over a perfect field of positive characteristic p.

Special Lagrangian submanifolds of log Calabi–Yau manifolds

We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian and Yau. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and D∈|−KY| is a smooth divisor with D2=d, then X=Y∖D admits a special Lagrangian torus fibration, as conjectured by Strominger–Yau–Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or Y= P2, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kähler rotation, X can be compactified to the complement of a Kodaira type Id fiber appearing as a singular fiber in a rational elliptic surface πˇ:Yˇ→P1.

Del Pezzo surfaces with infinite automorphism groups

Del Pezzo surfaces with infinite automorphism groups

Noncommutative Homological Mirror Functor

We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer theory. The construction comes with a natural functor from the Fukaya category to the category of matrix factorizations of the constructed Landau-Ginzburg model. As applications, it is applied to elliptic orbifolds, punctured Riemann surfaces and certain non-compact Calabi-Yau threefolds to construct their mirrors and functors. In particular it recovers and strengthens several interesting results of Etingof-Ginzburg, Bocklandt and Smith, and gives a unified understanding of their results in terms of mirror symmetry and symplectic geometry. As an interesting application, we construct an explicit global deformation quantization of an affine del Pezzo surface as a noncommutative mirror to an elliptic orbifold.

On Kawamata-Viehweg type vanishing for three dimensional Mori fiber spaces in positive characteristic

In this paper, we prove a Kawamata--Viehweg type vanishing theorem for smooth Fano threefolds, canonical del Pezzo surfaces and del Pezzo fibrations in positive characteristic.

On semistable degenerations of Fano varieties

Consider a family of Fano varieties \(\pi :X \rightarrow B\ni o\) over a curve germ with a smooth total space X. Assume that the generic fiber is smooth and the special fiber \(F=\pi ^{-1}(o)\) has simple normal crossings. Then F is called a semistable degeneration of Fano varieties. We show that the dual complex of F is a simplex of dimension \(\hbox {\,\char 054\,}\mathrm {dim}\, F\). Simplices of any admissible dimension can be realized for any dimension of the fiber. Using this result and the Minimal Model Program in dimension 3 we reproduce the classification of the semistable degenerations of del Pezzo surfaces obtained by Fujita. We also show that the maximal degeneration is unique and has trivial monodromy in dimension \(\hbox {\,\char 054\,}\,3\).

Kawamata-Viehweg vanishing fails for log del Pezzo surfaces in characteristic 3

We construct a klt del Pezzo surface in characteristic three violating the Kawamata-Viehweg vanishing theorem. As a consequence we show that there exists a Kawamata log terminal threefold singularity which is not Cohen-Macaulay in characteristic three.

Geometry of the moduli of parabolic bundles on elliptic curves

The goal of this paper is the study of simple rank 2 parabolic vector bundles over a 2 2 -punctured elliptic curve C C . We show that the moduli space of these bundles is a non-separated gluing of two charts isomorphic to P 1 × P 1 \mathbb {P}^1 \times \mathbb {P}^1 . We also showcase a special curve Γ \Gamma isomorphic to C C embedded in this space, and this way we prove a Torelli theorem. This moduli space is related to the moduli space of semistable parabolic bundles over P 1 \mathbb {P}^1 via a modular map which turns out to be the 2:1 cover ramified in Γ \Gamma . We recover the geometry of del Pezzo surfaces of degree 4 and we reconstruct all their automorphisms via elementary transformations of parabolic vector bundles.

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.
 
 
 

Existence of semistable sheaves on Hirzebruch surfaces

Let Fe denote the Hirzebruch surface P(OP1⊕OP1(e)), and let H be any ample divisor. In this paper, we algorithmically determine when the moduli space of semistable sheaves MFe,H(r,c1,c2) is nonempty. Our algorithm relies on certain stacks of prioritary sheaves. We first solve the existence problem for these stacks and then algorithmically determine the Harder-Narasimhan filtration of the general sheaf in the stack. In particular, semistable sheaves exist if and only if the Harder-Narasimhan filtration has length one.We then study sharp Bogomolov inequalities Δ≥δH(c1/r) for the discriminants of stable sheaves which take the polarization and slope into account; these inequalities essentially completely describe the characters of stable sheaves. The function δH(c1/r) can be computed to arbitrary precision by a limiting procedure. In the case of an anticanonically polarized del Pezzo surface, exceptional bundles are always stable and δH(c1/r) is computed by exceptional bundles. More generally, we show that for an arbitrary polarization there are further necessary conditions for the existence of stable sheaves beyond those provided by stable exceptional bundles. We compute δH(c1/r) exactly in some of these cases. Finally, solutions to the existence problem have immediate applications to the birational geometry of MFe,H(v).

Unstable singular del Pezzo hypersurfaces with lower index

We give examples of K-unstable singular del Pezzo surfaces which are weighted hypersurfaces with index 2.

A two-dimensional family of surfaces of general type with pg = 0 and K2 = 7

We study the construction of complex minimal smooth surfaces S of general type with pg(S)=0 and KS2=7. Inoue constructed the first examples of such surfaces, which can be described as Galois Z/2Z×Z/2Z-covers over the four-nodal cubic surface. Later the first named author constructed more examples as Galois Z/2Z×Z/2Z-covers over certain six-nodal del Pezzo surfaces of degree one.In this paper we construct a two-dimensional family of minimal smooth surfaces of general type with pg=0 and K2=7, as Galois Z/2Z×Z/2Z-covers of certain rational surfaces with Picard number three, with eight nodes and with two elliptic fibrations. This family is different from the previous ones.

Semistable sheaves with symmetric $$c_{1}$$ on Del Pezzo surfaces of degree 5 and 6

Semistable sheaves with symmetric $$c_{1}$$ on Del Pezzo surfaces of degree 5 and 6

Derived categories of quintic del Pezzo fibrations

We provide a semiorthogonal decomposition for the derived category of fibrations of quintic del Pezzo surfaces with rational Gorenstein singularities. There are three components, two of which are equivalent to the derived categories of the base and the remaining non-trivial component is equivalent to the derived category of a flat and finite of degree 5 scheme over the base. We introduce two methods for the construction of the decomposition. One is the moduli space approach following the work of Kuznetsov on the sextic del Pezzo fibrations and the components are given by the derived categories of fine relative moduli spaces. The other approach is that one can realize the fibration as a linear section of a Grassmannian bundle and apply homological projective duality.

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}.

Topological Formulae for the Zeroth Cohomology of Line Bundles on del Pezzo and Hirzebruch Surfaces

Abstract We show that the zeroth cohomology of effective line bundles on del Pezzo and Hirzebruch surfaces can always be computed in terms of a topological index.

On subregular slices of the elliptic Grothendieck–Springer resolution

In \cite{davis19}, the author constructed an elliptic version of the Grothendieck-Springer resolution for the stack $\mathrm{Bun}_G$ of principal bundles under a simply connected simple group $G$ on an elliptic curve $E$. This is a simultaneous log resolution of a map from $\mathrm{Bun}_G$ to the union of the coarse moduli space of semistable $G$-bundles and a single stacky point. In this paper, we study singularities, resolutions and deformations coming from subregular slices of this elliptic Grothendieck-Springer resolution. More precisely, we construct explicit slices of $\mathrm{Bun}_G$ through all subregular unstable bundles, for every $G$. For $G \neq SL_2$, we describe the pullbacks of the elliptic Grothendieck-Springer resolution to these slices as concrete varieties, extending and refining earlier work of I. Grojnowski and N. Shepherd-Barron, who related these varieties to del Pezzo surfaces in type $E$. We use the resolutions to identify the singularities of the unstable locus of the subregular slices, and prove that that the extended coarse moduli space map gives deformations that are miniversal among torus-equivariant deformations with appropriate weights.

Affine cones over cubic surfaces are flexible in codimension one

Abstract Let Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on it.