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     

Brill-Noether theory on Hirzebruch surfaces

In this paper, we investigate higher rank Brill-Noether problems for stable vector bundles on Hirzebruch surfaces. Using suitable non-splitting extensions, we deal with the non-emptiness. Results concerning the emptiness follow as a consequence of a generalization of Clifford’s theorem for line bundles on curves to vector bundles on surfaces.     

Rank 2 totally arithmetically Cohen–Macaulay vector bundles on Hirzebruch surfaces

Here we study the totally arithmetically Cohen–Macaulay rank 2 vector bundles on any Hirzebruch surface F _e . E. Ballico was partially supported by MIUR and GNSAGA of INdAM (Italy).     

Bubble tree compactification of moduli spaces of vector bundles on surfaces

We announce some results on compactifying moduli spaces of rank 2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so-called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Kollár. As an example, the compactification of the space of stable rank 2 vector bundles with Chern classes c ₁ = 0, c ₁ = 2 on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers.     

k—Very Ample Line Bundles on Del Pezzo Surfaces

A k-very ample line bundle L on a Del Pezzo Surface is numerically characterized, improving the results of Biancofiore— Ceresa in [7].     

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     

Exceptional bundles on nodal Enriques surfaces

In this paper we prove that an Enriques surfaceX has a smooth rational curve if and only if there exists an exceptional bundleE _t of rank 2 withc ₂ (E _t )=t for any integer t onX. We describe all exceptional bundles of rank 2 on Enriques surfaces and show that they are all stable with respect to any ample divisor.     

Holomorphic vector bundles on non-algebraic surfacesFibrés vectoriels holomorphes sur les surfaces non algébriques

The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is, in general, still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class VII. Our methods, which are based on Donaldson theory and deformation theory, can be used to solve the existence problem of holomorphic vector bundles on further classes of non-algebraic surfaces. To cite this article: A. Teleman, M. Toma, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 383–388.     

Applications of toric systems on surfaces

In this note we apply the techniques of the toric systems introduced by Hille–Perling to several problems on smooth projective surfaces: We showed that the existence of full exceptional collection of line bundles implies the rationality for small Picard rank surfaces; we proved equivalences of several notions of cyclic strong exceptional collection of line bundles; we also proposed a partial solution to a conjecture on exceptional sheaves on weak del Pezzo surfaces.     

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.     

Rank-two vector bundles on Hirzebruch surfaces

We survey some parts of the vast literature on vector bundles on Hirzebruch surfaces, focusing on the rank-two case.     

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).     

Jumping conics on a smooth quadric in $${\mathbb {P}_3}$$

We investigate the jumping conics of stable vector bundles E of rank 2 on a smooth quadric surface Q with the first Chern class c₁ = O_Q(-1,-1){c_1= \mathcal{O}_Q(-1,-1)} with respect to the ample line bundle O_Q(1,1){\mathcal {O}_Q(1,1)} . We show that the set of jumping conics of E is a hypersurface of degree c ₂(E) − 1 in \mathbb P₃^*{\mathbb {P}_3^{*}} . Using these hypersurfaces, we describe moduli spaces of stable vector bundles of rank 2 on Q in the cases of lower c ₂(E).     

DERIVED LENGTHS OF SOLVABLE GROUPS SATISFYING THE ONE-PRIME HYPOTHESIS. III

Abstract

The purpose of this paper is to extend the result in Park (Park, J. (2001). Birational maps of del Pezzo fibrations. J. Reine Angew. Math. 538:213–221) in the case of del Pezzo fibrations of degree 1. To this end we investigate the anticanonical linear systems of del Pezzo surfaces of degree 1. We then classify all possible effective anticanonical divisors on Gorenstein del Pezzo surfaces of degree 1 with canonical singularities.     

A theorem on the adjoint system for vector bundles

In this paper, we study the classification theory of uniruled varieties by means of the adjoint system for vector bundles on the varieties. We prove that ifE is an ample vector bundle on a smooth projective varietyX with rank(E)=dimX-2, thenK _X +C ₁ (E) is numerically effective except in a few cases. In all of the exceptional cases,X is a uniruled variety. As consequences, we generalized a result of Fujita [Fu3] and Ionescu [Io] and improve upon a theorem of Wiśniewski [Wi1].     

Globally generated vector bundles on with

One classifies the globally generated vector bundles on with the first Chern class c₁ = 3. The case c₁ = 1 is very easy, the case c₁ = 2 was done in [42], the case c₁ = 3, rank =2 was settled in [21] and the case c₁ ≤ 5, rank = 2 in [10]. Our work is based on Serre's theorem relating vector bundles of rank = 2 with codimension 2 lci subschemes and its generalization for higher ranks, considered firstly by Vogelaar in [48].     

Welschinger invariants of real Del Pezzo surfaces of degree ≥ 3

We give a recursive formula for purely real Welschinger invariants of real Del Pezzo surfaces of degree K ² ≥ 3, where in the case of surfaces of degree 3 with two real components we introduce a certain modification of Welschinger invariants and enumerate exclusively the curves traced on the non-orientable component. As an application, we prove the positivity of the invariants under consideration and their logarithmic asymptotic equivalence, as well as congruence modulo 4, to genus zero Gromov–Witten invariants.     

Curves on Generic Surfaces of High Degree Through a Complete Intersection in P 3

We consider general surfaces, S, of high degree containing a given complete intersection space curve, Y. We study integral curves in the subgroup of Pic(S) generated by Y and the plane section. We determine the cohomological invariants of these curves and classify the subcanonical ones. Then using these subcanonical curves we produce stable rank two vector bundles on P ³.     

On cubic surfaces with a rational line

We report on our project to construct non-singular cubic surfaces over \mathbbQ{\mathbb{Q}} with a rational line. Our method is to start with degree 4 Del Pezzo surfaces in diagonal form. For these, we develop an explicit version of Galois descent.     

Equations of log del Pezzo surfaces of index ≤ 2

Del Pezzo surfaces over with log terminal singularities of index ≤ 2 were classified by Alekseev and Nikulin. In this paper, for each of these surfaces, we find an appropriate morphism to projective space. These morphisms enable us to describe equations of natural embeddings of log del Pezzo surfaces of index ≤ 2 in some weighted projective space. The results obtained give a completion of similar results of Du Val, Hidaka, and Watanabe, describing del Pezzo surfaces of index 1. The work was done during the authors’ stay at the University of Liverpool supported by the Marie Curie program in Autumn 2004.