An explicit construction of ruled surfaces

The main goal of this paper is to give a general algorithm to compute, via computer-algebra systems, an explicit set of generators of the ideals of the projective embeddings of ruled surfaces, i.e. projectivizations of rank two vector bundles over curves, such that the fibers are embedded as smooth rational curves.There are two different applications of our algorithm. Firstly, given a very ample linear system on an abstract ruled surface, our algorithm allows computing the ideal of the embedded surface, all the syzygies, and all the algebraic invariants which are computable from its ideal as, for instance, the k-regularity. Secondly, it is possible to prove the existence of new embeddings of ruled surfaces.The method can be implemented over any computer-algebra system able to deal with commutative algebra and Gröbner-basis computations. An implementation of our algorithms for the computer-algebra system Macaulay2 (cf. [Daniel R. Grayson, Michael E. Stillman, Macaulay 2, a software system for research in algebraic geometry, 1993. Available at http://www.math.uiuc.edu/Macaulay2/]) and explicit examples are enclosed.     

Seshadri constants on rational surfaces with anticanonical pencils

We study a Seshadri constant at a general point on a rational surface whose anticanonical linear system contains a pencil. First, we describe a Seshadri constant of an ample line bundle on such a rational surface explicitly by the numerical data of the ample line bundle. Second, we classify log del Pezzo surfaces which are special in terms of the Seshadri constants of the anticanonical divisors when the anticanonical degree is between 4 and 9.     

Intersection theory over quantum ruled surfaces

In this paper, we will extend several results on intersection theory over commutative ruled surfaces to quantum ruled surfaces. Typically, we define the fiber of a closed point, the quasi-section, and the quasi-canonical divisor on a quantum rules surface, and study how these “curves” on a quantum ruled surface intersect with each other.     

Higher Embeddings of General Blowups of Ruled Surfaces

 In this note we give a criterion for a line bundle on a general blowup of a ruled surface to be k-very ample. (Received 29 November 2000; in revised form 2 April 2001)     

Simple immersions of wonderful varieties

Let G be a semisimple connected linear algebraic group over , and X a wonderful G-variety. We study the possibility of realizing X as a closed subvariety of the projective space of a simple G-module. We describe the wonderful varieties having this property as well as the linear systems giving rise to such immersions. We also prove that any ample line bundle on a wonderful variety is very ample. Research supported by European Research Training Network LIEGRITS (MRTN-CT 2003- 505078), in contract with CNRS DR17, No 2.     

Higher order bad loci

Zero-schemes on smooth complex projective varieties, forcing all elements of ample and free linear systems to be reducible, are studied. Relationships among the minimal length of such zero-schemes, the positivity of the line bundle associated with the linear system, and the dimension of the variety are established. Bad linear spaces are also investigated.     

Orbits of points on certain K3 surfaces

In this paper we show that, for a K3 surface within a certain class of surfaces and over a number field, the orbit of a point under the group of automorphisms is either finite or its exponent of growth is exactly the Hausdorff dimension of a fractal associated to the ample cone. In particular, the exponent depends on the geometry of the surface and not its arithmetic. For surfaces in this class, the exponent is 0.6527±0.0012.     

Embeddings of general blowups of abelian surfaces

Blowups of algebraic surfaces polarized by tensor powers of ample line bundles were studied by Coppens [8]. In this note we complete the picture in the case of abelian surfaces studying blowups of surfaces polarized by primitive line bundles. Received: 16 November 1999 / Revised version: 13 June 2000     

On the base point freeness of adjoint bundles¶on normal surfaces

We introduce some invariants of singularities which represent the anti-freeness of the adjoint linear systems. The invariants indicate that if either the singularities or the boundaries are worse then the adjoint linear systems are much global generative. Using these invariants, we prove effective global generation of adjoint linear systems on normal log surfaces. Received: 14 May 1999/ Revised version: 2 August 1999     

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     

Normally generated line bundles on general curves

Let L be a very ample line bundle of degree d on a general curve X of genus g≥2. Here we prove that if then L is globally generated, i.e. L embeds X as a projectively normal curve in PH⁰(L).     

Nef cone volumes and discriminants of abelian surfaces

The nef cone volume appeared first in work of Peyre in a number-theoretic context on Fano varieties, and was then studied by Derenthal and co-authors in a series of papers on del Pezzo surfaces. The idea was subsequently extended to also measure the Zariski chambers of del Pezzo surfaces. We start in this paper to explore the possibility to use this attractive concept to effectively measure the size of the nef cone on algebraic surfaces in general. This provides an interesting way of measuring in how big a space an ample line bundle can be moved without destroying its positivity. We give here complete results for simple abelian surfaces that admit a principal polarization and for products of elliptic curves.     

Counting curves on rational surfaces

In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher-genus Gromov–Witten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret higher-genus Gromov–Witten invariants of certain K-nef surfaces, and then apply this to a degeneration of a cubic surface. Received: 30 June 1999 / Revised version: 1 January 2000     

Projective normality of abelian surfaces¶given by primitive line bundles

In this paper, we study projective normality of abelian surfaces, with embeddings given by ample line bundles of type (1,d). We show that if d≥ 7, the generic abelian surface is projectively normal. Received: 12 June 1998     

Elliptic curve configurations on Fano surfaces

The elliptic curves on a surface of general type constitute an obstruction for the cotangent sheaf to be ample. In this paper, we give the classification of the configurations of the elliptic curves on the Fano surface of a smooth cubic threefold. That means that we give the number of such curves, their intersections and a plane model. This classification is linked to the classification of the automorphism groups of theses surfaces.     

A note on non-reduced Picard schemes

The Picard scheme of a smooth curve and a smooth complex variety is reduced. In this note we discuss which classes of surfaces in terms of the Enriques-Kodaira classification can have non-reduced Picard schemes and whether there are restrictions on the characteristic of the ground field. It turns out that non-reduced Picard schemes are uncommon in Kodaira dimension κ≤0, that this phenomenon can be bounded for κ=2 (general type) and that it is as bad as can be for κ=1.     

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.     

On the finite generation of the effective monoid of rational surfaces

We give a numerical criterion for ensuring the finite generation of the effective monoid of the surfaces obtained by a blowing-up of the projective plane at the supports of zero dimensional subschemes assuming that these are contained in a degenerate cubic. Furthermore, this criterion also ensures the regularity of any numerically effective divisor on these surfaces. Thus the dimension of any complete linear system is computed. On the other hand, in particular and among these surfaces, we obtain ringed rational surfaces with very large Picard numbers and with only finitely many integral curves of strictly negative self-intersection. These negative integral curves except two (−1)-curves are all contained in the support of an anticanonical divisor. Thus almost all the geometry of such surfaces is concentrated in the anticanonical class.     

Zur Theorie verallgemeinerter torsaler Strahlflächen

In this paper we will investigate (k+1)-dimensional generalized ruled surfaces generated by a one-parameter family ofk-dimensional linear subspaces of then-dimensional Euclidean spaceE _n . Some results which are well-known for developable surfaces are proved for generalized ruled surfaces: Generalized developable surfaces are locally either cyclinders, cones or tangent surfaces. Each regular surface on a generalized ruled surface is locally Euclidean if and only if is developable. Each locally Euclidean hypersurface is a generalized developable hypersurface. Furthermore, the hypersurfaces with vanishing Gaussian curvature and the locally Euclidean hypersurfaces on generalized rule hypersurfaces will be characterized.     

Topologically unique maximal elementary abelian group actions on compact oriented surfaces

We determine all finite maximal elementary abelian group actions on compact oriented surfaces of genus σ≥2 which are unique up to topological equivalence. For certain special classes of such actions, we determine group extensions which also define unique actions. In addition, we explore in detail one of the families of such surfaces considered as compact Riemann surfaces and tackle the classical problem of constructing defining equations.