Del Pezzo Surfaces Research Articles

On the topology of real Lagrangians in toric symplectic manifolds

We explore the topology of real Lagrangian submanifolds in a toric symplectic manifold which come from involutive symmetries on its moment polytope. We establish a real analog of the Delzant construction for those real Lagrangians, which says that their diffeomorphism type is determined by combinatorial data. As an application, we realize all possible diffeomorphism types of connected real Lagrangians in toric symplectic del Pezzo surfaces.

Anticanonical geometry of the blow-up of mathbb {P}^4 in 8 points and its Fano model

Building on the work of Casagrande–Codogni–Fanelli, we develop our study on the birational geometry of the Fano fourfold Y=M_{S,-K_S} which is the moduli space of semi-stable rank-two torsion-free sheaves with c_1=-K_S and c_2=2 on a polarised degree-one del Pezzo surface (S,-K_S). Based on the relation between Y and the blow-up of mathbb {P}^4 in 8 points, we describe completely the base scheme of the anticanonical system |{-}K_Y|. We also prove that the Bertini involution iota _Y of Y, induced by the Bertini involution iota _S of S, preserves every member in |{-}K_Y|. In particular, we establish the relation between iota _Y and the anticanonical map of Y, and we describe the action of iota _Y by analogy with the action of iota _S on S.

Classification of strongly asymptotically log del Pezzo flags and surfaces

We introduce the notion of strongly asymptotically log del Pezzo flags, and classify such flags under the assumption that their zero-dimensional part lies in the boundary. We use this result to give a new and conceptual proof of the classification of strongly asymptotically log del Pezzo surfaces, originally due to Cheltsov and Rubinstein.

Concurrent lines on del Pezzo surfaces of degree one

Let X X be a del Pezzo surface of degree one over an algebraically closed field, and K X K_X its canonical divisor. The morphism φ \varphi induced by | − 2 K X | |-2K_X| realizes X X as a double cover of a cone in P 3 \mathbb {P}^3 , ramified over a smooth sextic curve. The surface X X contains 240 exceptional curves. We prove the following statements. For a point P P on the ramification curve of φ \varphi , at most sixteen exceptional curves contain P P in characteristic 2 2 , and at most ten in all other characteristics. Moreover, for a point Q Q outside the ramification curve, at most twelve exceptional curves contain Q Q in characteristic 3 3 , and at most ten in all other characteristics. We show that these upper bounds are sharp, except possibly in characteristic 5 outside the ramification curve.

On Enriques-Fano threefolds and a conjecture of Castelnuovo

Let Wsubset mathbb {P}^{13} be the image of the rational map defined by the linear system of the sextic surfaces of mathbb {P}^3 having double points along the edges of a tetrahedron. Let mathcal {L} be the linear system of the hyperplane sections of W. It is known that a general Sin mathcal {L} is an Enriques surface. The aim of this paper is to study the sublinear system mathcal {L}_{bullet }subset mathcal {L} of the hyperplane sections of W having a triple point at a general point w in W. We will show that a general element of mathcal {L}_{bullet } is birational to an elliptic ruled surface and that the image of W via the rational map defined by mathcal {L}_{bullet } is a cubic Del Pezzo surface Delta subset mathbb {P}^3 with 4 nodes. Interestingly, this fact appears to be related to a conjecture of Castelnuovo.

The discriminant of a hypersurface in weighted projective space

In this paper, we study the discriminant of a hypersurface in a weighted projective space with isolated singularities. We define the discriminant of a hypersurface in [Formula: see text] with [Formula: see text], and show that it satisfies a smoothness criterion over any commutative ring. A smooth hypersurface of degree [Formula: see text] in the weighted projective three-fold [Formula: see text] is a del Pezzo surface of degree [Formula: see text]. We show that the determinant of the Galois action on the [Formula: see text]-lattice arising from its exceptional curves can be computed via the square roots of the discriminant.

Bounds for del pezzo surfaces of degree two

Abstract In this article, we obtain an upper bound for the number of integral points on del Pezzo surfaces of degree two.

Moduli spaces on the Kuznetsov component of Fano threefolds of index 2

General hyperplane sections of a Fano threefold $Y$ of index 2 and Picard rank 1 are del Pezzo surfaces, and their Picard group is related to a root system. To the corresponding roots, we associate objects in the Kuznetsov component of $Y$ and investigate their moduli spaces, using the stability condition constructed by Bayer, Lahoz, Macr\`i, and Stellari, and the Abel--Jacobi map. We identify a subvariety of the moduli space isomorphic to $Y$ itself, and as an application we prove a (refined) categorical Torelli theorem for general quartic double solids.

Rank jumps on elliptic surfaces and the Hilbert property

Given an elliptic surface over a number field, we study the collection of fibres whose Mordell-Weil rank is greater than the generic rank. Under suitable assumptions, we show that this collection is not thin. Our results apply to quadratic twist families and del Pezzo surfaces of degree $1$.

Automorphisms of real del Pezzo surfaces and the real plane Cremona group

We study automorphism groups of real del Pezzo surfaces, concentrating on finite groups acting with invariant Picard number equal to one. As a result, we obtain a vast part of classification of finite subgroups in the real plane Cremona group.

Rational points on Del Pezzo surfaces of degree four

We study the distribution of the Brauer group and the frequency of the Brauer–Manin obstruction to the Hasse principle and weak approximation in a family of smooth del Pezzo surfaces of degree four over the rationals.

Weak Approximation for del Pezzo Surfaces of Low Degree

Abstract We prove, via an “arithmetic surjectivity” approach inspired by work of Denef, that weak weak approximation holds for surfaces with two conic fibrations satisfying a general assumption. In particular, weak weak approximation holds for general del Pezzo surfaces of degrees $1$ or $2$ with a conic fibration.

Density of rational points on a family of del Pezzo surfaces of degree one

Let k be an infinite field of characteristic 0, and X a del Pezzo surface of degree d with at least one k-rational point. Various methods from algebraic geometry and arithmetic statistics have shown the Zariski density of the set X(k) of k-rational points in X for d≥2 (under an extra condition for d=2), but fail to work in generality when the degree of X is 1, leaving a large class of del Pezzo surfaces for which the question of density of rational points is still open. In this paper, we prove the Zariski density of X(k) when X has degree 1 and is represented in the weighted projective space P(2,3,1,1) with coordinates x,y,z,w by an equation of the form y2=x3+az6+bz3w3+cw6 for a,b,c∈k with a,c non-zero, under the condition that the elliptic surface obtained by blowing up the base point of the anticanonical linear system |−KX| contains a smooth fiber above a point in P1∖{(1:0),(0:1)} with positive rank over k. When k is of finite type over Q, this condition is sufficient and necessary.

BIRATIONAL RIGIDITY OF ORBIFOLD DEGREE 2 DEL PEZZO FIBRATIONS

AbstractVarieties fibered into del Pezzo surfaces form a class of possible outputs of the minimal model program. It is known that del Pezzo fibrations of degrees$1$and$2$over the projective line with smooth total space satisfying the so-called$K^2$-condition are birationally rigid: their Mori fiber space structure is unique. This implies that they are not birational to any Fano varieties, conic bundles, or other del Pezzo fibrations. In particular, they are irrational. The families of del Pezzo fibrations with smooth total space of degree$2$are rather special, as formostfamilies a general del Pezzo fibration has the simplest orbifold singularities. We prove that orbifold del Pezzo fibrations of degree$2$over the projective line satisfying explicit generality conditions as well as a generalized$K^2$-condition are birationally rigid.

Unirationality of RDP Del Pezzo surfaces of degree 2

We study unirationality of a Del Pezzo surface of degree two over a given (non algebraically closed) field, under the assumption that it admits at least one rational double point over an algebraic closure of the base field. As corollaries of our main results, we find that over a finite field, it is unirational if the cardinality of the field is greater than or equal to nine and we also find that over an infinite field, which is not necessarily perfect, it is unirational if and only if the rational points are Zariski dense over the field.

Delta-invariants of complete intersection log del Pezzo surfaces

We show that complete intersection log del Pezzo surfaces with amplitude one in weighted projective spaces are uniformly $K$ -stable. As a result, they admit an orbifold Kähler–Einstein metric.

Character Varieties and Algebraic Surfaces for the Topology of Quantum Computing

It is shown that the representation theory of some finitely presented groups thanks to their SL2(C) character variety is related to algebraic surfaces. We make use of the Enriques–Kodaira classification of algebraic surfaces and related topological tools to make such surfaces explicit. We study the connection of SL2(C) character varieties to topological quantum computing (TQC) as an alternative to the concept of anyons. The Hopf link H, whose character variety is a Del Pezzo surface fH (the trace of the commutator), is the kernel of our view of TQC. Qutrit and two-qubit magic state computing, derived from the trefoil knot in our previous work, may be seen as TQC from the Hopf link. The character variety of some two-generator Bianchi groups, as well as that of the fundamental group for the singular fibers E˜6 and D˜4 contain fH. A surface birationally equivalent to a K3 surface is another compound of their character varieties.

CYLINDERS IN WEAK DEL PEZZO FIBRATIONS

In this article, we shall look into the existence of vertical cylinders contained in a weak del Pezzo fibration as a generalization of the former work due to Dubouloz and Kishimoto in which they observed that of vertical cylinders found in del Pezzo fibrations. The essence lying in the existence of a cylinder in the generic fiber, we devote mainly ourselves into a geometry of minimal weak del Pezzo surfaces defined over a field of characteristic zero from the point of view of cylinders. As a result, we give the classification of minimal weak del Pezzo surfaces defined over a field of characteristic zero, moreover, we show that weak del Pezzo fibrations containing vertical cylinders are quite restrictive.

From β to η: a new cohomology for deformed Sasaki-Einstein manifolds

We discuss in detail the different analogues of Dolbeault cohomology groups on Sasaki-Einstein manifolds and prove a new vanishing result for the transverse Dolbeault cohomology groups {H}_{overline{partial}}^{left(p,0right)}(k) graded by their charge under the Reeb vector. We then introduce a new cohomology, η-cohomology, which is defined by a CR structure and a holomorphic function f with non-vanishing η ≡ df. It is the natural cohomology associated to a class of supersymmetric type IIB flux backgrounds that generalise the notion of a Sasaki-Einstein manifold. These geometries are dual to finite deformations of the 4d mathcal{N} = 1 SCFTs described by conventional Sasaki-Einstein manifolds. As such, they are associated to Calabi-Yau algebras with a deformed superpotential. We show how to compute the η-cohomology in terms of the transverse Dolbeault cohomology of the undeformed Sasaki-Einstein space. The gauge-gravity correspondence implies a direct relation between the cyclic homologies of the Calabi-Yau algebra, or equivalently the counting of short multiplets in the deformed SCFT, and the η-cohomology groups. We verify that this relation is satisfied in the case of S5, and use it to predict the reduced cyclic homology groups in the case of deformations of regular Sasaki-Einstein spaces. The corresponding Calabi-Yau algebras describe non-commutative deformations of ℙ2, ℙ1× ℙ1 and the del Pezzo surfaces.

On the Kawamata–Viehweg vanishing theorem for log del Pezzo surfaces in positive characteristic

We prove the Kawamata–Viehweg vanishing theorem for surfaces of del Pezzo type over perfect fields of positive characteristic$p>5$. As a consequence, we show that klt threefold singularities over a perfect base field of characteristic$p>5$are rational. We show that these theorems are sharp by providing counterexamples in characteristic$5$.