Del Pezzo Surfaces Research Articles

Non-split Toric Codes

We introduce a new wide class of error-correcting codes, called non-split toric codes. These codes are a natural generalization of toric codes where non-split algebraic tori are taken instead of usual (i.e., split) ones. The main advantage of the new codes is their cyclicity; hence, they can possibly be decoded quite fast. Many classical codes, such as (doubly-extended) Reed-Solomon and (projective) Reed-Muller codes, are contained (up to equivalence) in the new class. Our codes are explicitly described in terms of algebraic and toric geometries over finite fields; therefore, they can easily be constructed in practice. Finally, we obtain new cyclic reversible codes, namely non-split toric codes on the del Pezzo surface of degree 6 and Picard number 1. We also compute their parameters, which prove to attain current lower bounds at least for small finite fields.

Index of fibrations and Brauer classes that never obstruct the Hasse principle

Let X be a smooth projective variety over a number field with a fibration into varieties that either satisfy a certain condition. We study the classes in the Brauer group of X that never obstruct the Hasse principle for X. We prove that if the generic fiber has a zero-cycle of degree d over the generic point, then the Brauer classes whose orders are prime to d do not play a role in the Brauer–Manin obstruction. As a result we show that the odd torsion Brauer classes never obstruct the Hasse principle for del Pezzo surfaces of degree 2, certain K3 surfaces, and Kummer varieties.

Voevodsky's conjecture for cubic fourfolds and Gushel–Mukai fourfolds via noncommutative K3 surfaces

In the first part of this paper we will prove the Voevodsky’s nilpotence conjecture for smooth cubic fourfolds and ordinary generic Gushel–Mukai fourfolds. Then, making use of noncommutative motives, we will prove the Voevodsky’s nilpotence conjecture for generic Gushel–Mukai fourfolds containing a \tau -plane Gr(2, 3) and for ordinary Gushel–Mukai fourfolds containing a quintic del Pezzo surface.

On the algebraic Brauer classes on open degree four del Pezzo surfaces

We study the algebraic Brauer classes on open del Pezzo surfaces of degree 4. I.e., on the complements of geometrically irreducible hyperplane sections of del Pezzo surfaces of degree 4. We show that the 2-torsion part is generated by classes of two different types. Moreover, there are two types of 4-torsion classes. For each type, we discuss methods for the evaluation of such a class at a rational point over a p-adic field.

S1‐invariant symplectic hypersurfaces in dimension 6 and the Fano condition

We prove that any symplectic Fano $6$-manifold $M$ with a Hamiltonian $S^1$-action is simply connected and satisfies $c_1 c_2(M)=24$. This is done by showing that the fixed submanifold $M_{\min}\subseteq M$ on which the Hamiltonian attains its minimum is diffeomorphic to either a del Pezzo surface, a $2$-sphere or a point. In the case when $\dim(M_{\min})=4$, we use the fact that symplectic Fano $4$-manifolds are symplectomorphic to del Pezzo surfaces. The case when $\dim(M_{\min})=2$ involves a study of $6$-dimensional Hamiltonian $S^1$-manifolds with $M_{\min}$ diffeomorphic to a surface of positive genus. By exploiting an analogy with the algebro-geometric situation we construct in each such $6$-manifold an $S^1$-invariant symplectic hypersurface ${\cal F}(M)$ playing the role of a smooth fibre of a hypothetical Mori fibration over $M_{\min}$. This relies upon applying Seiberg-Witten theory to the resolution of symplectic $4$-orbifolds occurring as the reduced spaces of $M$.

Automorphisms of cubic surfaces in positive characteristic

We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely characterize when a smooth cubic surface in positive characteristic, together with a group action, can be lifted to characteristic zero.

THE POWER‐SAVING MANIN–PEYRE CONJECTURE FOR A SENARY CUBIC

Using recent work of the first author~\cite{Bet}, we prove a strong version of the Manin-Peyre's conjectures with a full asymptotic and a power-saving error term for the two varieties respectively in $\mathbb{P}^2 \times \mathbb{P}^2$ with bihomogeneous coordinates $[x_1:x_2:x_3],[y_1:y_2,y_3]$ and in $\mathbb{P}^1\times \mathbb{P}^1 \times \mathbb{P}^1$ with multihomogeneous coordinates $[x_1:y_1],[x_2:y_2],[x_3:y_3]$ defined by the same equation $x_1y_2y_3+x_2y_1y_3+x_3y_1y_2=0$. We thus improve on recent work of Blomer, Br\"udern and Salberger \cite{BBS} and provide a different proof based on a descent on the universal torsor of the conjectures in the case of a del Pezzo surface of degree 6 with singularity type $\mathbf{A}_1$ and three lines (the other existing proof relying on harmonic analysis \cite{CLT}). Together with~\cite{Blomer2014} or with recent work of the second author \cite{Dest2}, this settles the study of the Manin-Peyre's conjectures for this equation.

Laurent inversion

We describe a practical and effective method for reconstructing the deformation class of a Fano manifold X from a Laurent polynomial f that corresponds to X under Mirror Symmetry. We explore connections to nef partitions, the smoothing of singular toric varieties, and the construction of embeddings of one (possibly-singular) toric variety in another. In particular, we construct degenerations from Fano manifolds to singular toric varieties; in the toric complete intersection case, these degenerations were constructed previously by Doran--Harder. We use our method to find models of orbifold del Pezzo surfaces as complete intersections and degeneracy loci, and to construct a new four-dimensional Fano manifold.

Weakly exceptional singularities of log del Pezzo surfaces

We complete the computation of global log canonical thresholds, or equivalently alpha invariants, of quasi-smooth well-formed complete intersection log del Pezzo surfaces of amplitude 1 in weighted projective spaces. As an application, we prove that they are weakly exceptional. And we investigate the super-rigid affine Fano 3-folds containing a log del Pezzo surface as boundary.

Cubic fourfolds fibered in sextic del Pezzo surfaces

We exhibit new examples of rational cubic fourfolds, parametrized by a countably infinite union of codimension-two subvarieties in the moduli space. Our examples are fibered in sextic del Pezzo surfaces over the projective plane; they are rational whenever the fibration has a rational section.

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].

Troisi\`eme groupe de cohomologie non ramifi\'ee des torseurs universels sur les surfaces rationnelles

Let $k$ a field of characteristic zero. Let $X$ be a smooth, projective, geometrically rational $k$-surface. Let $\mathcal{T}$ be a universal torsor over $X$ with a $k$-point et $\mathcal{T}^c$ a smooth compactification of $\mathcal{T}$. There is an open question: is $\mathcal{T}^c$ $k$-birationally equivalent to a projective space? We know that the unramified cohomology groups of degree 1 and 2 of $\mathcal{T}$ and $\mathcal{T}^c$ are reduced to their constant part. For the analogue of the third cohomology groups, we give a sufficient condition using the Galois structure of the geometrical Picard group of $X$. This enables us to show that $H^{3}_{nr}(\mathcal{T}^{c},\mathbb{Q}/\mathbb{Z}(2))/H^3(k,\mathbb{Q}/\mathbb{Z}(2))$ vanishes if $X$ is a generalised Ch\^atelet surface and that this group is reduced to its $2$-primary part if $X$ is a del Pezzo surface of degree at least 2.

Geometry of moduli space of sheaves on del Pezzo surfaces via Maruyama transform

In this paper, we prove that the moduli space of stable pairs (or sheaves) on del Pezzo surface S is isomorphic to the Maruyama type transformation of a projective bundle of a quiver representation space.

G-birational superrigidity of Del Pezzo surfaces of degree 2 and 3

Any minimal Del Pezzo G-surface S of degree smaller than 3 is G-birationally rigid. We classify those which are G-birationally superrigid, and for those which fail to be so, we describe the equations of a set of generators for the infinite group mathrm{Bir}^G(S) of G-birational automorphisms.

Del Pezzo surfaces, rigid line configurations and Hirzebruch–Kummer coverings

We prove the equisingular rigidity of the singular Hirzebruch–Kummer coverings X(n, $$\mathcal {L}$$ ) of the projective plane branched on line configurations $$\mathcal {L}$$ , satisfying some technical condition. In the case, $$\mathcal {L}=$$ the complete quadrangle, we give explicit equations of the Hirzebruch–Kummer covering $$S_n(=$$ the minimal desingularisation of $$X (n, \mathcal {L}))$$ in a product of four Fermat curves of degree n. Since $$S_n$$ is the $$(\mathbb {Z}/n)^5$$ covering of the Del Pezzo surface $$Y_5$$ of degree 5 branched on the 10 lines, these equations are derived from explicit equations of the image of $$Y_5$$ in $$(\mathbb {P}^1)^4$$ .We describe more generally determinantal equations for all Del Pezzo surfaces of degree $$9-k \le 6$$ as subvarieties of the k-fold product of the projective line.

On plane quartics with a Galois invariant Cayley octad

We describe a construction of plane quartics with prescribed Galois operation on the 28 bitangents, in the particular case of a Galois invariant Cayley octad. As an application, we solve the inverse Galois problem for degree two del Pezzo surfaces in the corresponding particular case.

Reflective modular forms and applications

The reflective modular forms of orthogonal type are fundamental automorphic objects generalizing the classical Dedekind eta-function. This article describes two methods for constructing such modular forms in terms of Jacobi forms: automorphic products and Jacobi lifting. In particular, it is proved that the first non-zero Fourier–Jacobi coefficient of the Borcherds modular form (the generating function of the so-called Fake Monster Lie Algebra) in any of the 23 one-dimensional cusps coincides with the Kac–Weyl denominator function of the affine algebra of the root system of the corresponding Niemeier lattice. This article gives a new simple construction of the automorphic discriminant of the moduli space of Enriques surfaces as a Jacobi lifting of the product of eight theta-functions and considers three towers of reflective modular forms. One of them, the tower of , gives a solution to a problem of Yoshikawa (2009) concerning the construction of Lorentzian Kac–Moody algebras from the automorphic discriminants connected with del Pezzo surfaces and analytic torsions of Calabi–Yau manifolds. The article also formulates some conditions on sublattices, making it possible to produce families of `daughter' reflective forms from a fixed modular form. As a result, nearly 100 such functions are constructed here. Bibliography: 77 titles.

The blow-up of $$\mathbb {P}^4$$ P 4 at 8 points and its Fano model, via vector bundles on a del Pezzo surface

Building on the work of Mukai, we explore the birational geometry of the moduli spaces $$M_{S,L}$$ of semistable rank two torsion-free sheaves, with $$c_1=-K_S$$ and $$c_2=2$$ , on a polarized degree one del Pezzo surface (S, L); this is related to the birational geometry of the blow-up X of $$\mathbb {P}^4$$ in 8 points. Our analysis is explicit and is obtained by looking at the variation of stability conditions. Then we provide a careful investigation of the blow-up X and of the moduli space $$Y=M_{S,-K_S}$$ , which is a remarkable family of smooth Fano fourfolds. In particular we describe the relevant cones of divisors of Y, the group of automorphisms, and the base loci of the anticanonical and bianticanonical linear systems.

Lorentzian Lattices and E-Polytopes

We consider certain E n -type root lattices embedded within the standard Lorentzian lattice Z n + 1 ( 3 ≤ n ≤ 8 ) and study their discrete geometry from the point of view of del Pezzo surface geometry. The lattice Z n + 1 decomposes as a disjoint union of affine hyperplanes which satisfy a certain periodicity. We introduce the notions of line vectors, rational conic vectors, and rational cubics vectors and their relations to E-polytopes. We also discuss the relation between these special vectors and the combinatorics of the Gosset polytopes of type ( n − 4 ) 21 .

Non-vanishing heterotic superpotentials on elliptic fibrations

We present models of heterotic compactification on Calabi-Yau threefolds and compute the non-perturbative superpotential for vector bundle moduli. The key feature of these models is that the threefolds, which are elliptically fibered over del Pezzo surfaces, have homology classes with a unique holomorphic, isolated genus-zero curve. Using the spectral cover construction, we present vector bundles for which we can explicitly calculate the Pfaffians associated with string instantons on these curves. These are shown to be non-zero, thus leading to a non-vanishing superpotential in the 4D effective action. We discuss, in detail, why such compactifications avoid the Beasley-Witten residue theorem.