Rational Surfaces with Many Nodes

We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain n b ₂ –2 disjoint smooth rational curves with self-intersection –2, where b ₂ is the second Betti number. In the last section this is applied to the study of minimal complex surfaces of general type with p _g = 0 and K² = 8, 9 which admit an automorphism of order 2.     

Imposing Singular Points and Many Nodes to Curves on Surfaces

 Let S be a smooth projective surface. Here we study the conditions imposed to curves of a fixed very ample linear system by a general union of types of singularities τ when most of connected components of τ are ordinary double points. This problem is related to the existence of “good” families of curves on S with prescribed singularities, most of them being nodes, and to the regularity of their Hilbert scheme. Received 6 July 2000; in revised form 16 June 2001     

Imposing Singular Points and Many Nodes to Curves on Surfaces

 Let S be a smooth projective surface. Here we study the conditions imposed to curves of a fixed very ample linear system by a general union of types of singularities τ when most of connected components of τ are ordinary double points. This problem is related to the existence of “good” families of curves on S with prescribed singularities, most of them being nodes, and to the regularity of their Hilbert scheme.     

有理曲线和曲面的分片多项式逼近

本文利用摄动的思想,以摄动有理曲线（曲面）的系数的无穷模作为优化目标,给出了用多项式曲线（曲面）逼近有理曲线（曲面）的一种新方法．同以前的各种方法相比,该方法不仅收敛而且具有更快的收敛速度,并且可以与细分技术相结合,得到有理曲线与曲面的整体光滑、分片多项式的逼近．     

Explicit Bounds for Rational Curves on Kahler Surfaces of Non-Negative Kodaira dimension

1991 Mathematics Subject Classification: 14J10, 32J15.     

On the Parameterization of Rational Ringed Surfaces and Rational Canal Surfaces

Ringed surfaces and canal surfaces are surfaces that contain a one-parameter family of circles. Ringed surfaces can be described by a radius function, a directrix curve and vector field along the directrix curve, which specifies the normals of the planes that contain the circles. In particular, the class of ringed surfaces includes canal surfaces, which can be obtained as the envelopes of a one-parameter family of spheres. Consequently, canal surfaces can be described by a spine curve and a radius function. We present parameterization algorithms for rational ringed surfaces and rational canal surfaces. It is shown that these algorithms may generate any rational parameterization of a ringed (or canal) surface with the property that one family of parameter lines consists of circles. These algorithms are used to obtain rational parameterizations for Darboux cyclides and to construct blends between pairs of canal surfaces and pairs of ringed surfaces.     

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 Linear Systems of Curves on Rational Scrolls

In this paper we prove a conjecture on the dimension of linear systems, with base points of multiplicity 2 and 3, on an Hirzebruck surface.     

Simple Curves on Surfaces

We study simple closed geodesics on a hyperbolic surface of genus g with b geodesic boundary components and c cusps. We show that the number of such geodesics of length at most L is of order L^6g+2b+2c–6. This answers a long-standing open question.     

On Rational Cuspidal Projective Plane Curves

In 2002, L. Nicolaescu and the fourth author formulated a verygeneral conjecture which relates the geometric genus of a Gorensteinsurface singularity with rational homology sphere link withthe Seiberg--Witten invariant (or one of its candidates) ofthe link. Recently, the last three authors found some counterexamplesusing superisolated singularities. The theory of superisolatedhypersurface singularities with rational homology sphere linkis equivalent with the theory of rational cuspidal projectiveplane curves. In the case when the corresponding curve has onlyone singular point one knows no counterexample. In fact, inthis case the above Seiberg--Witten conjecture led us to a veryinteresting and deep set of ‘compatibility properties’of these curves (generalising the Seiberg--Witten invariantconjecture, but sitting deeply in algebraic geometry) whichseems to generalise some other famous conjectures and propertiesas well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yauinequalities). Namely, we provide a set of ‘compatibilityconditions’ which conjecturally is satisfied by a localembedded topological type of a germ of plane curve singularityand an integer d if and only if the germ can be realized asthe unique singular point of a rational unicuspidal projectiveplane curve of degree d. The conjectured compatibility propertieshave a weaker version too, valid for any rational cuspidal curvewith more than one singular point. The goal of the present articleis to formulate these conjectured properties, and to verifythem in all the situations when the logarithmic Kodaira dimensionof the complement of the corresponding plane curves is strictlyless than 2. 2000 Mathematics Subject Classification 14B05,14J17, 32S25, 57M27, 57R57 (primary), 14E15, 32S45, 57M25 (secondary).     

On Rational Surfaces Ruled by Conics

Abstract

We study projective rational surfaces ruled by conics, describing their singularities and special fibres. In particular, if Sis smooth, we give a “canonical” procedure to determine a minimal model among the geometrically ruled surfaces birational to S.     

Clifford Algebra, Spin Representation, and Rational Parameterization of Curves and Surfaces

The Pythagorean hodograph (PH) curves are characterized by certain Pythagorean n-tuple identities in the polynomial ring, involving the derivatives of the curve coordinate functions. Such curves have many advantageous properties in computer aided geometric design. Thus far, PH curves have been studied in 2- or 3-dimensional Euclidean and Minkowski spaces. The characterization of PH curves in each of these contexts gives rise to different combinations of polynomials that satisfy further complicated identities. We present a novel approach to the Pythagorean hodograph curves, based on Clifford algebra methods, that unifies all known incarnations of PH curves into a single coherent framework. Furthermore, we discuss certain differential or algebraic geometric perspectives that arise from this new approach.     

Enumerating Rational Curves: The Rational Fibration Method

A new method to approach enumerative questions about rational curves on algebraic varieties is described. The idea is to reduce the counting problems to computations on the Néron-Severi group of a ruled surface. Applications include a short proof of Kontsevich's formula for plane curves and the solution of the analogous problem for the Hirzebruch surface F₃.     

Maximal Subsheaves of Torsion-Free Sheaves on Nodal Curves

Let Y be a reduced irreducible projective curve of arithmeticgenus g 2 with at most ordinary nodes as singularities. Fora subsheaf F of rank r', degree d' of a torsion-free sheaf Eof rank r, degree d, let s(E,F) = r'd-rd'. Define s_r'(E) = mins(E,F), where the minimum is taken over all subsheaves of Eof rank r'. For a fixed r', s_r' defines a stratification ofthe moduli space U(r,d) of stable torsion-free sheaves of rankr, degree d by locally closed subsets U_r',s. We study the nonemptinessand dimensions of the strata. We show that the general elementin each nonempty stratum is a vector bundle and it has onlyfinitely many (respectively unique) maximal subsheaves of rankr' for s r'(r-r')(g – 1) (respectively s < r'(r-r')(g– 1)). We prove that the tensor product of two generalstable vector bundles on an irreducible nodal curve Y is nonspecial.     

Rational Curves on Complex Manifolds

We give an overview of the study of rational curves on a complex manifold. Starting from Mori’s celebrated theorem on the extremal rays of the cone of curves, we describe some of its applications, and some of the tools used to produce rational curves in Algebraic Geometry and in Kähler Geometry.     

The Exchange Variations of Offset Curves and Surfaces

Results in Mathematics - This paper proposes a novel definition of generalized and standard offset curves and surfaces that was inspired by the needs in many engineering design practices. We also...     

Inequalities for the Curvature of Curves and Surfaces

In this paper we bound the difference between the total mean curvatures of two closed surfaces in in terms of their total absolute curvatures and the Frechet distance between the volumes they enclose. The proof relies on a combination of methods from algebraic topology and integral geometry. We also bound the difference between the lengths of two curves using the same methods.     

On Some Classical Methods to Improve Algebraic Curves and Surfaces

First, a modern presentation of the theory of the Halphen transform is given. This method associates to a plane projective curve C, once a general conic has been chosen, another birationally equivalent plane curve, whose singularities are simpler than those of C. Repeating, a curve is obtained whose only singularities are nodes. Next, it is studied how to apply this process to a family of plane curves. With this technique it is possible to transform a given family (with irreducible general member) into one where, generically, the curves are nodal. Finally, it is studied a similar process, called the Halphen–Picard transformation, for surfaces in three-space. By suitably reiterating this procedure, a surface can be transformed into a birationally equivalent one (in the same projective space), such that the sections with planes in a general pencil are, generically, nodal curves.     

Necessary Conditions for the Generic Global Rigidity of Frameworks on Surfaces

A result due in its various parts to Hendrickson, Connelly, and Jackson and Jordán, provides a purely combinatorial characterisation of global rigidity for generic bar-joint frameworks in \({{\mathbb {R}}}^2\) . The analogous conditions are known to be insufficient to characterise generic global rigidity in higher dimensions. Recently Laman-type characterisations of rigidity have been obtained for generic frameworks in \({\mathbb {R}}^3\) when the vertices are constrained to lie on various surfaces, such as the cylinder and the cone. In this paper we obtain analogues of Hendrickson’s necessary conditions for the global rigidity of generic frameworks on the cylinder, cone and ellipsoid.     

The Symmetry Principle and Nondegenerate Families of Curves on Abstract Surfaces

Siberian Mathematical Journal - Consider a&nbsp;Euclidean space domain whose volume element is induced by some weight function, while the arclength element of a&nbsp;curve at...