Bielliptic surfaces as covers of rational surfaces

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

Positive elimination in valued fields

Let K be an algebraically closed field with a valuation ring or a real closed field with a convex valuation ring . We show that the projection of a basic (see “Introduction”) subset of to K ⁿ is again basic.     

Fibred rational surfaces with extremal Mordell-Weil lattices

Slope inequalities are given for fibred rational surfaces according as the Clifford index of a general fibre. For fibred rational surfaces of Clifford index two, the Mordell-Weil lattices of maximal ranks are completely determined.Supported by The 21st Century COE Program named “Towards a new basic science: depth and synthesis”.     

Overdetermined elliptic problems in topological disks

We introduce a method, based on the Poincaré–Hopf index theorem, to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk. Applications to some well-known nonlinear elliptic PDEs are provided. Our result can be seen as the analogue of Hopf's uniqueness theorem for constant mean curvature spheres, but for the general analytic context of overdetermined elliptic problems.     

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.     

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.     

On the geometries of the rational unfoldings of X k

With a view to applications to self-regulating dynamical processes in biology, we determine the geometric structure of what we call isotangent curves, i.e. curves parametrized by the slopes of their points. They come up naturally as bifurcation curves of rational unfoldings of X ^k, and we classify them according to degree and number of cusps. They, as well as their isotangent involute curves, turn up in simulations of these processes.     

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     

The Klein correspondence of line congruences immersed in an elliptic spaceS 3 onto the hyperquadricP 4

This article is devoted to the investigation and the construction of the Klein correspondence of line congruences referred to a specialized moving frame in a 3-dimensional elliptic spaceS ₃ to the hyperquadricP ⁴ of the Klein 5-dimensional elliptic spaceS ₅. The Klein correspondence is given and characterized by Theorems 1, 2. The methods adapted here are based on Cartan's differential calculus [1], [6].     

On a question of Sean Keel

Let X be a smooth projective surface defined over , and let L be a line bundle over X such that for every complete curve Y contained in X. A question of Keel asks whether L is ample. If X is a P¹-bundle over a curve, we prove that this question has an affirmative answer.     

Construction of rational surfaces yielding good codes

In the present article, we consider Algebraic Geometry codes on some rational surfaces. The estimate of the minimum distance is translated into a point counting problem on plane curves. This problem is solved by applying the upper bound à la Weil of Aubry and Perret together with the bound of Homma and Kim for plane curves. The parameters of several codes from rational surfaces are computed. Among them, the codes defined by the evaluation of forms of degree 3 on an elliptic quadric are studied. As far as we know, such codes have never been treated before. Two other rational surfaces are studied and very good codes are found on them. In particular, a [57,12,34] code over F₇ and a [91,18,53] code over F₉ are discovered, these codes beat the best known codes up to now.     

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.     

On congruences related to central binomial coefficients

It is known that

     

On a class of rational cuspidal plane curves

We obtain new examples and the complete list of the rational cuspidal plane curvesC with at least three cusps, one of which has multiplicitydegC-2. It occurs that these curves are projectively rigid. We also discuss the general problem of projective rigidity of rational cuspidal plane curves.