Calculation of Manin's invariant for Del Pezzo surfaces

For and 8 we consider an action of the Weyl group of type on a unimodular lattice of rank . We give the tables of the first cohomology groups for all cyclic subgroups of the Weyl group with respect to this action. These are important in the arithmetic theory of Del Pezzo surfaces.

     

Universal torsors of Del Pezzo surfaces and homogeneous spaces

Let Cox(Sr) be the homogeneous coordinate ring of the blow-up Sr of P² in r general points, i.e., a smooth Del Pezzo surface of degree 9−r. We prove that for r∈{6,7}, Proj(Cox(Sr)) can be embedded into Gr/Pr, where Gr is an algebraic group with root system given by the primitive Picard lattice of Sr and Pr⊂Gr is a certain maximal parabolic subgroup.     

Cohomology of moduli spaces of Del Pezzo surfaces

We compute the rational Betti cohomology groups of the coarse moduli spaces of geometrically marked Del Pezzo surfaces of degree 3 and 4 as representations of the Weyl groups of the corresponding root systems. The proof uses a blend of methods from point counting over finite fields and techniques from arrangement complements.     

A boundedness result for toric log Del Pezzo surfaces

In this paper we give an upper bound for the Picard number of the rational surfaces which resolve minimally the singularities of toric log Del Pezzo surfaces of given index ℓ. This upper bound turns out to be a quadratic polynomial in the variable ℓ. Received: 18 June 2008     

Del Pezzo surfaces with nonrational singularities

Normal algebraic surfacesX with the property rk(Div(X)⊗ℚ/≡)=1, numerically ample canonical classes, and nonrational singularities are classified. It is proved, in particular, that any such surfaceX is a contraction of an exceptional section of a (possibly singular) relatively minimal ruled surface with a nonrational base. Moreover, f is uniquely determined by the surfaceX. Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 451–467, September, 1997. Translated by O. V. Sipacheva     

Density of rational points on del Pezzo surfaces of degree one

We state conditions under which the set S(k)$S(k)$ of k-rational points on a del Pezzo surface S of degree 1 over an infinite field k   of characteristic not equal to 2 or 3 is Zariski dense. For example, it suffices to require that the elliptic fibration S?P¹$S?{\mathbb{P}}^1$ induced by the anticanonical map has a nodal fiber over a k  -rational point of P¹${\mathbb{P}}^1$. It also suffices to require the existence of a point in S(k)$S(k)$ that does not lie on six exceptional curves of S   and that has order 3 on its fiber of the elliptic fibration. This allows us to show that within a parameter space for del Pezzo surfaces of degree 1 over R$\mathbb{R}$, the set of surfaces S   defined over Q$\mathbb{Q}$ for which the set S(Q)$S(\mathbb{Q})$ is Zariski dense, is dense with respect to the real analytic topology. We also include conditions that may be satisfied for every del Pezzo surface S and that can be verified with a finite computation for any del Pezzo surface S that does satisfy them.     

On the Arithmetic of Del Pezzo Surfaces of Degree 2

We study the arithmetic of certain del Pezzo surfaces of degree2. We produce examples of Brauer-Manin obstruction to the Hasseprinciple, coming from 2- and 4-torsion elements in the Brauergroup. 2000 Mathematics Subject Classification 14G05 (primary),12G05 (secondary).     

Del Pezzo foliations with log canonical singularities

In this paper, we classify del Pezzo foliations of rank at least 3 on projective manifolds and with log canonical singularities in the sense of McQuillan.     

Log del Pezzo surfaces with not small fractional indices

For a log del Pezzo surface S, the fractional index is the maximum of r with which can be written as r times some Cartier divisor. We classify all the log del Pezzo surfaces S with , after the technique of Nakayama.     

Degeneration of torsors over families of del Pezzo surfaces

Let S be a split family of del Pezzo surfaces over a discrete valuation ring such that the general fiber is smooth and the special fiber has $\mathbf{ADE}$-singularities. Let G be the reductive group given by the root system of these singularities. We construct a G-torsor over S whose restriction to the generic fiber is the extension of structure group of the universal torsor. This extends a construction of Friedman and Morgan for individual singular del Pezzo surfaces. In case of very good residue characteristic, this torsor is unique and infinitesimally rigid.     

The Tate conjecture for powers of ordinary cubic fourfolds over finite fields

Recently, Levin proved the Tate conjecture for ordinary cubic fourfolds over finite fields. In this paper, we prove the Tate conjecture for self-products of ordinary cubic fourfolds. Our proof is based on properties of so-called polynomials of K3-type introduced by the author about 12 years ago.     

Permutation polynomials from trace functions over finite fields

In this paper, we propose several classes of permutation polynomials based on trace functions over finite fields of characteristic 2. The main result of this paper is obtained by determining the number of solutions of certain equations over finite fields.     

Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces

Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., D₄,E₆,E₇,E₈). To such a diagram one can attach a group G whose generators correspond to the legs of the affinization, have orders equal to the leg lengths plus 1, and the product of the generators is 1. The group G is then a 2-dimensional crystallographic group: G=Z??Z², where ? is 2, 3, 4, and 6, respectively. In this paper, we define a flat deformation H(t,q) of the group algebra C[G] of this group, by replacing the relations saying that the generators have prescribed orders by their deformations, saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots (which are deformation parameters). The algebra H(t,q) for D₄ is the Cherednik algebra of type C^∨C₁, which was studied by Noumi, Sahi, and Stokman, and controls Askey-Wilson polynomials. We prove that H(t,q) is the universal deformation of the twisted group algebra of G, and that this deformation is compatible with certain filtrations on C[G]. We also show that if q is a root of unity, then for generic t the algebra H(t,q) is an Azumaya algebra, and its center is the function algebra on an affine del Pezzo surface. For generic q, the spherical subalgebra eH(t,q)e provides a quantization of such surfaces. We also discuss connections of H(t,q) with preprojective algebras and Painlevé VI.     

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.     

Log Canonical Thresholds of Del Pezzo Surfaces

We study global log canonical thresholds of del Pezzo surfaces. All varieties are assumed to be defined over . Received: May 2007 Revision: October 2007 Accepted: April 2008     

Del Pezzo surfaces of degree 6 over an arbitrary field

We give a characterization of all del Pezzo surfaces of degree 6 over an arbitrary field F. A surface is determined by a pair of separable algebras. These algebras are used to compute the Quillen K-theory of the surface. As a consequence, we obtain an index reduction formula for the function field of the surface.     

Reconstruction of singularities on orbifold del Pezzo surfaces from their Hilbert series

Abstract

The Hilbert series of a polarized algebraic variety (X, D) is a powerful invariant that, while it captures some features of the geometry of (X, D) precisely, often cannot recover much information about its singular locus. This work explores the extent to which the Hilbert series of an orbifold del Pezzo surface fails to pin down its singular locus, which provides nonexistence results describing when there are no orbifold del Pezzo surfaces with a given Hilbert series, supplies bounds on the number of singularities on such surfaces, and has applications to the combinatorics of lattice polytopes in the toric case.     

On the resolvent of the Laplacian on functions for degenerating surfaces of finite geometry

We consider families (Yn) of degenerating hyperbolic surfaces. The surfaces are geometrically finite of fixed topological type. Let Zn be the Selberg Zeta function of Yn, and let zn be the contribution of the pinched geodesics to Zn. Extending a result of Wolpert's, we prove that Zn(s)/zn(s) converges to the Zeta function of the limit surface if Re(s)>1/2. The technique is an examination of resolvent of the Laplacian, which is composed from that for elementary surfaces via meromorphic Fredholm theory. The resolvent ⁻¹(Δn−t) is shown to converge for all t∉[1/4,∞). We also use this property to define approximate Eisenstein functions and scattering matrices.