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     

Del Pezzo surfaces with log terminal singularities

We prove that there are no del Pezzo surfaces with five log terminal singularities and the Picard number 1. In the course of the proof, we make use of fibrations with general fiber ?¹.     

Exceptional linear systems on curves on Del Pezzo surfaces

Work performed during the author's stay at UCLA, partially supported by Dottorato di Ricerca funds of the Universities of Genova, Milano, Pavia and Torino (1988–1989) and a C.N.R. scholarship (1989–1990)     

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.     

Del Pezzo surfaces of degree 1 and Jacobians

We construct absolutely simple jacobians of nonhyperelliptic genus 4 curves, using Del Pezzo surfaces of degree 1.     

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.

     

Lines on Del Pezzo surfaces and transfinite heterotic string space-time

It is pointed out that the hierarchy of fractal dimensions characterizing transfinite heterotic string space-times bears a striking resemblance to the sequence of the number of lines lying on Del Pezzo surfaces.     

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     

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.     

On birational morphisms between pencils of Del Pezzo surfaces

Let and be two Del Pezzo fibrations of degrees , respectively. Assume that and differ by a flop. Then we prove that and give a short list of values of other basic numerical invariants of and .

     

Subcanonical, gorenstein and complete intersection curves on Del Pezzo surfaces

In this note we classify subcanonical, Gorenstein and complete intersection smooth connected curves lying on del Pezzo surfaces, by showing their classes in Picard groups of the surfaces.

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Sunto  In questa nota si classificano le curve liscie connesse, che sono sottocanoniche, Gorenstein o intersezioni complete, tracciate sulle superfici di del Pezzo, esibendone le classi nei gruppi di Picard delle superfici stesse.

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To Mario Fiorentini     

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.     

Global smoothings of degenerate Del Pezzo surfaces with normal crossings

Fujita classified one-parameter degenerations of Del Pezzo manifolds with smooth total spaces, which includes the complete classification of semi-stable degenerations of Del Pezzo surfaces. We prove the converse, namely, for a given semi-stable Del Pezzo surface of each type in the list of Fujita, there exists a smoothing of it with a smooth total space.     

Rational points of bounded height on Del Pezzo surfaces of degree six

LetK be a number field. Denote byV ₃ a split Del Pezzo surface of degree six overK and by ω its canonical divisor. Denote byW ₃ the open complement of the exceptional lines inV ₃. LetN _{W s}(−ω, X) be the number ofK-rational points onW ₃ whose anticanonical heightH _−ω is bounded byX. Manin has conjectured that asymptoticallyN _{W 3}(−ω, X) tends tocX(logX)³, wherec is a constant depending only on the number field and on the normalization of the height. Our goal is to prove the following theorem: For each number fieldK there exists a constantc _K such thatN _{W 3}(−ω, X)≤c_KX(logX)^3+2r , wherer is the rank of the group of units ofO _K. The constantc _K is far from being optimal. However, ifK is a purely imaginary quadratic field, this proves an upper bound with a correct power of logX. The proof of Manin's conjecture for arbitrary number fields and a precise treatment of the constants would require a more sophisticated setting, like the one used by [Peyre] to prove Manin's conjecture and to compute the correct asymptotic constant (in some normalization) in the caseK=ℚ. Up to now the best result for arbitraryK goes back, as far as we know, to [Manin-Tschinkel], who gives an upper boundN _{W 3}(−ω,X)≤cX^l+ε. The author would like to express his gratitude to Daniel Coray and Per Salberger for their generous and indispensable support.     

On the birational rigidity of some pencils of Del Pezzo surfaces

In this note, we discuss birational properties of some three-dimensional Del Pezzo fibrations of degree two. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 62, Algebraic Geometry-10, 1999.     

Welschinger invariants of toric Del Pezzo surfaces with nonstandard real structures

The Welschinger invariants of real rational algebraic surfaces are natural analogs of the Gromov-Witten invariants, and they estimate from below the number of real rational curves passing through prescribed configurations of points. We establish a tropical formula for the Welschinger invariants of four toric Del Pezzo surfaces equipped with a nonstandard real structure. Such a formula for real toric Del Pezzo surfaces with a standard real structure (i.e., naturally compatible with the toric structure) was established by Mikhalkin and the author. As a consequence we prove that for any real ample divisor D on a surface Σ under consideration, through any generic configuration of c ₁(Σ)D − 1 generic real points, there passes a real rational curve belonging to the linear system |D|. To Vladimir Igorevich Arnold on the occasion of his 70th birthday     

Stabilité du fibré tangent des surfaces de Del Pezzo

Sans résumé     

Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface X _k obtained by blowing up ℂℙ² at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration W _k :M _k →ℂ with k+3 singular fibers, equipped with a suitable symplectic form. Moreover, we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of X _k , and give an explicit correspondence between the deformation parameters for X _k and the cohomology class [B+iω]∈H ²(M _k ,ℂ).     

Del Pezzo surface fibrations obtained by blow-up of a smooth curve in a projective manifold

This Note is devoted to the study of the Fano manifolds X obtained by blow-up along a smooth curve C in a complex projective manifold Y. By the Mori theory, we can ensure the existence of an extremal contraction $\phi :X\to Z$ different from the blow-up $\pi :X\to Y$. Here we give the complete classification of the corresponding pairs $(Y,C)$ in the case where φ is a fiber type contraction of relative dimension 2, i.e. the general fibers of φ are del Pezzo surfaces. In Tsukioka (Thesis, Nancy University 1, 2005), the relative dimension 1 case is also considered. To cite this article: T. Tsukioka, C. R. Acad. Sci. Paris, Ser. I 340 (2005).