On non-vanishing of cohomologies of generalized Raynaud polarized surfaces

We consider a family of slightly extended version of Raynaud’s surfaces X over the field of positive characteristic with Mumford-Szpiro type polarizations Z, which have Kodaira non-vanishing H¹(X,Z⁻ⁿ)≠0 for all 1≤n≤N with some N≥1. The surfaces are at least normal but smooth under a special condition. We also give a fairly large family of non-Mumford-Szpiro type polarizations Z_a,b with Kodaira non-vanishing.     

Floor diagrams relative to a conic,and GW–W invariants of Del Pezzo surfaces

We enumerate, via floor diagrams, complex and real curves in CP²$\mathbb{C}{P}^2$ blown up in n points on a conic. As an application, we deduce Gromov–Witten and Welschinger invariants of Del Pezzo surfaces. These results are mainly obtained using Li's degeneration formula and its real counterpart.     

Torsion cohomology classes and algebraic cycles on complex projective manifolds

Atiyah and Hirzebruch gave examples of even degree torsion classes in the singular cohomology of a smooth complex projective manifold, which are not Poincaré dual to an algebraic cycle. We notice that the order of these classes must be small compared to the dimension of the manifold. However, building upon a construction of Kollár, one can provide such examples with arbitrary high prime order, the dimension being fixed. This method also provides examples of torsion algebraic cycles, which are non trivial in the Griffiths group, and lie in a arbitrary high level of the H. Saito filtration on Chow groups.     

Motivic integral of K3 surfaces over a non-archimedean field

We prove a formula expressing the motivic integral (Loeser and Sebag, 2003) [34] of a K3 surface over C((t)) with semi-stable reduction in terms of the associated limit mixed Hodge structure. Secondly, for every smooth variety over a complete discrete valuation field we define an analogue of the monodromy pairing, constructed by Grothendieck in the case of abelian varieties, and prove that our monodromy pairing is a birational invariant of the variety. Finally, we propose a conjectural formula for the motivic integral of maximally degenerate K3 surfaces over an arbitrary complete discrete valuation field and prove this conjecture for Kummer K3 surfaces.     

Deformations of equivelar Stanley–Reisner abelian surfaces

The versal deformation of Stanley–Reisner schemes associated to equivelar triangulations of the torus is studied. The deformation space is defined by binomials and there is a toric smoothing component which I describe in terms of cones and lattices. Connections to moduli of abelian surfaces are considered. The case of the Möbius torus is especially nice and leads to a projective Calabi–Yau 3-fold with Euler number 6.     

Katoʼs residue homomorphisms and reciprocity laws on arithmetic surfaces

We explicitly study Kato?s residue homomorphisms in Milnor K-theory, which are closely related to Contou-Carrère symbols. As applications we establish several reciprocity laws for certain locally defined maps on K-groups that are associated to arithmetic surfaces.     

Remarks on abelian surfaces in nonsingular toric Fano 4-folds

In this paper, we investigate whether the 124 nonsingular toric Fano 4-folds admit totally nondegenerate embeddings from abelian surfaces or not. In consequence, we determine the possibilities of these embeddings, except for the remaining 18 nonsingular toric Fano 4-folds. Received: 12 July 2002     

Halves of a real Enriques surface

The real partE _ℝ of a real Enriques surfaceE admits a natural decomposition in two halves,E _ℝ=E _ℝ ⁽¹⁾ ∪E _ℝ ⁽²⁾ , each half being a union of components ofE _ℝ. We classify the triads (E _ℝ;E _ℝ ⁽¹⁾ ,E _ℝ ⁽²⁾ ) up to homeomorphism. Most results extend to surfaces of more general nature than Enriques surfaces. We use and study in details the properties of Kalinin's filtration in the homology of the fixed point set of an involution, which is a convenient tool not widely known in topology of real algebraic varieties.     

Abelian surfaces in projective toric 4-folds

We show fundamental properties on embedding of abelian surfaces into projective toric 4-folds, and study the case of the toric Del Pezzo 4-fold from the viewpoint of the moduli space of abelian surfaces with polarization of type (1, 5). Received: 31 January 2005; revised: 15 April 2005     

Local Euler obstructions of toric varieties

We use Matsui and Takeuchi's formula for toric A-discriminants to give algorithms for computing local Euler obstructions and dual degrees of toric surfaces and 3-folds. In particular, we consider weighted projective spaces. As an application we give counterexamples to a conjecture by Matsui and Takeuchi. As another application we recover the well-known fact that the only defective normal toric surfaces are cones.     

Cubic surfaces violating the Hasse principle are Zariski dense in the moduli scheme

We construct new examples of cubic surfaces, for which the Hasse principle fails. Thereby we show that, over every number field, the counterexamples to the Hasse principle are Zariski dense in the moduli scheme of non-singular cubic surfaces.     

Rational places in extensions and sequences of function fields of Kummer type

In this paper we prove results on the number of rational places in extensions of Kummer type over finite fields and give sufficient conditions for non-trivial lower bounds on the number of rational places at each step of sequences of function fields over a finite field, that we call (a, b)-sequences. In the case of a prime field, we apply these results to the study of rational places in certain sequences of function fields of Kummer type.     

Bielliptic surfaces as covers of rational surfaces

We present a construction of the bielliptic surfaces as covers of certain rational elliptic surfaces.     

The Eilenberg-Watts theorem over schemes

We study obstructions to a direct limit preserving right exact functor F between categories of quasi-coherent sheaves on schemes being isomorphic to tensoring with a bimodule. When the domain scheme is affine, or if F is exact, all obstructions vanish and we recover the Eilenberg-Watts Theorem. This result is crucial to the proof that the noncommutative Hirzebruch surfaces constructed in C. Ingalls, D. Patrick (2002) [6] are noncommutative P¹-bundles in the sense of M. Van den Bergh [10].     

Curves and 0-cycles on projective surfaces

LetC be a smooth curve with ag _n ¹ , i.e. a linear system of dimension 1 and degreen, lying on a smooth projective surfaceS. Let :S P ^N be the map associated to the line bundleK _S +[C] and letD be a general divisor of the given linear systemg _n ¹ . LetV be the linear space spanned by the image ofD through . We study the case in whichn:=dimV=1 and in general we discuss the case in whichn is small. The starting point is an analysis of the adjunction map using Bogomolov-Reider-Serrano techniques; several results from curve theory are also needed.     

Effectivity of Brauer–Manin obstructions on surfaces

We study Brauer–Manin obstructions to the Hasse principle and to weak approximation on algebraic surfaces over number fields. A technique for constructing Azumaya algebra representatives of Brauer group elements is given, and this is applied to the computation of obstructions.     

Classification of incidence scrolls (I)

The aim of this paper is to obtain a classification of scrolls of genus 0 and 1, which are defined by a one-dimensional family of lines meeting a certain set of linear spaces in p ⁿ . These ruled surfaces will be called incidence scrolls and such a set will be the base of the incidence scroll. Unless otherwise stated, we assume that the base spaces are in general position. Received: 1 December 2000     

The g-periodic subvarieties for an automorphism g of positive entropy on a compact Kähler manifold

For a compact Kähler manifold X and a strongly primitive automorphism g of positive entropy, it is shown that X has at most ρ(X) of g-periodic prime divisors. When X is a projective threefold, every prime divisor containing infinitely many g-periodic curves, is shown to be g-periodic (a result in the spirit of the Dynamic Manin-Mumford conjecture as in Zhang (2006) [17]).     

Local structure theorems for smooth maps of formal schemes

We continue our study on infinitesimal lifting properties of maps between locally noetherian formal schemes started in [L. Alonso Tarrío, A. Jeremías López, M. Pérez Rodríguez, Infinitesimal lifting and Jacobi criterion for smoothness on formal schemes, Comm. Alg. 35 (2007) 1341-1367]. In this paper, we focus on some properties which arise specifically in the formal context. In this vein, we make a detailed study of the relationship between the infinitesimal lifting properties of a morphism of formal schemes and those of the corresponding maps of usual schemes associated to the directed systems that define the corresponding formal schemes. Among our main results, we obtain the characterization of completion morphisms as pseudo-closed immersions that are flat. Also, the local structure of smooth and étale morphisms between locally noetherian formal schemes is described: the former factors locally as a completion morphism followed by a smooth adic morphism and the latter as a completion morphism followed by an étale adic morphism.