Showing papers on "Plane curve published in 1997"

Complex Analysis and Geometry

Abstract: On the Limits of Manifolds with nef Canonical Bundles, M. Andreatta and T. Peternell On the Stability of the Restriction of TPn to Projective Curves, E. Ballico and B. Russo Theorie des (a,b)-Modules II. Extensions, D. Barlet Moduli of Reflexive K3 Surfaces, C. Bartocci, U. Bruzzo, and D. Hernandez Ruiperez New Examples of Domains with Non-Injective Proper Holomorphic Self-Maps, F. Berteloot and J. J. Loeb Q-Convexivity. A Survey, M. Coltoiu Commuting maps and Families of Hyperbolic Automorphisms, C. de Fabritiis An Alternative Proof of a Theorem of Boas-Straube-Yu, K. Diederich and G. Herbort Large Polynomial Hulls with No Analytic Structure, J. Duval and N. Levenberg Canonical Connections for Almost-Hypercomplex Structures, P. Gauduchon The Tangent Bundle of P2 Restricted to Plane Curves, G. Hein Quotients with Respects to Holomorphic Actions of Reductive Groups, P. Heinzner and L. Migliorini Adjunction Theory on Terminal Varieties, M. Mella Runge Theorem in Higher Dimensions, V. Vajaitu Only Countably Many Simply-Connected Lie Groups Admit Lattices, J. Winkelmann

Real deformations and complex topology of plane curve singularities

Abstract: This is the paper as published. The topology of a complex plane curve singularity with real branches is deduced from any real deformation having delta crossings. An example of the computation of the global geometric monodromy of a polynomial mapping is added.

The Arnold-Gelfand mathematical seminars : [geometry and singularity theory]

Abstract: Discriminants and Local Invariants of Planar Fronts.- Crofton Densities, Symplectic Geometry and Hilbert's Fourth Problem.- Projective Convex Curves.- Topological Classification of Real Trigonometric Polynomials and Cyclic Serpents Polyhedron.- Singularities of Short Linear Waves on the Plane.- New Generalizations of Poincare's Geometric Theorem.- Explicit Formulas for Arnold's Generic Curve Invariants.- Nonlinear Integrable Equations and Nonlinear Fourier Transform.- Elliptic Solutions of the Yang-Baxter Equation and Modular Hypergeometric Functions.- Combinatorics of Hypergeometric Functions Associated with Positive Roots.- Local Invariants of Mappings of Surfaces into Three-Space.- Theorem on Six Vertices of a Plane Curve Via Sturm Theory.- The Arf-Invariant and the Arnold Invariants of Plane Curves.- Produit cyclique d'espaces et operations de Steenrod.- Characteristic Classes of Singularity Theory.- Value of Generalized Hypergeometric Function at Unity.- Harish-Chandra Decomposition for Zonal Spherical Function of type An.- Positive Paths in the Linear Symplectic Group.- Invariants of Submanifolds in Euclidean Space.- On Combinatorics and Topology of Pairwise Intersections of Schubert Cells in SLn/B.

Rational cuspidal plane curves of type (d, d-3)

Abstract: In the previous paper [E-print alg-geom/9507004] we classified the rational cuspidal plane curves C with a cusp of multiplicity deg C - 2. In particular, we showed that any such curve can be transformed into a line by Cremona transformations. Here we do the same for the rational cuspidal plane curves C with a cusp of multiplicity deg C - 3.

Introducing projective billiards

Abstract: We introduce and study a new class of dynamical systems, the projective billiards, associated with a smooth closed convex plane curve equipped with a smooth field of transverse directions. Projective billiards include the usual billiards along with the dual, or outer, billiards.

Topics in singularity theory : V.I. Arnold's 60th anniversary collection

Abstract: On the geometry of caustics by J. W. Bruce and V. M. Zakalyukin Lagrangian embeddings and Lagrangian cobordism by Yu. V. Chekanov Polynomial invariants of Legendrian links and plane fronts by S. Chmutov and V. Goryunov Solutions of the elliptic qKZB equations and Bethe ansatz I by G. Felder, V. Tarasov, and A. Varchenko Singular deformations of Lie algebras. Example: Deformations of the Lie algebra $L_1$ by A. Fialowski and D. Fuchs On imaginary plane curves and spin quotients of complex surfaces by complex conjugation by S. Finashin and E. Shustin Stationary phase integrals, quantum toda lattices, flag manifolds and the mirror conjecture by A. Givental On a problem of B. Teissier by S. M. Gusein-Zade Embedding theorems for local maps, slow-fast systems and bifurcation from Morse-Smale to Smale-Williams by Yu. Ilyashenko Topological invariants of fiber singularities by M. E. Kazaryan Informal complexification and Poisson structures on moduli spaces by B. A. Khesin Consistent partitions of polytopes and polynomial measures by A. Khovanskii On primitive elements in the bialgebra of chord diagrams by S. K. Lando Spaces of meromorphic functions on Riemann surfaces by S. M. Natanzon Hamiltonian loops and Arnold's principle by L. Polterovich Singularities in the presence of symmetries by I. Scherbak Discrete versions of the four-vertex theorem by V. D. Sedykh Excitation of elliptic normal modes of invariant tori in Hamiltonian systems by M. B. Sevryuk Ramified coverings of $S^2$ with one degenerate branching point and enumeration of edge-ordered graphs by B. Shapiro, M. Shapiro, and A. Vainshtein On zeros of the Schwarzian derivative by S. Tabachnikov Stratified Picard-Lefschetz theory with twisted coefficients by V. A. Vassiliev.

La formule de cauchy sur la longueur d'une courbe

Abstract: plan. ABSTRACT. We give a general proof of the Cauchy formula about the length of a plane curve. The formula is given in two ways: as the integral of the variation of orthogonal projections of the curve, and as a double integ ral of the number of intersections of the curve with an arbitrary line of the plan e.

Theorem on six vertices of a plane curve via Sturm theory

Abstract: We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the “projective version” of the well known four vertices theorem for a curve in the Euclidean plane.) We obtain this classical fact as a corollary of some general Sturm-type theorems.

Fundamental groups of complements to singular plane curves

Abstract: The classical hyperplane section theorem of Zariski about the fundamental groups of the complements to hypersurfaces in the complex projective space is generalized to the weighted homogeneous case. Using this result, we study how the fundamental group of the complement to a projective plane curve changes under covering of the projective plane by another projective plane. We also give an example of plane curves whose complements have nonabelian and finite fundamental groups. Few examples of such curves have been known. By the weighted Zariski's hyperplane section theorem, we can determine the group structure of these finite nonabelian groups. 0. Introduction. In the first part of this paper, we formulate and prove a weighted homogeneous version of Zariski's hyperplane section theorem on the fundamental groups of the complements to hypersurfaces in a complex projective space, and discuss some direct applications to the fundamental groups of com plements to projective plane curves. In the second part, we investigate a certain singular plane curve C(q9 k)9 which is a generalization of Zariski's three cuspidal quartics, and calculate the fundamental group ^(P2 \ C(q9k)) using the main result of the first part. This group turns out to be finite and nonabelian. Let jci, ... 9xn be variables with weights

Counting curves of any genus on rational ruled surfaces

Abstract: In this paper we study the geometry of the Severi varieties parametrizing curves on the rational ruled surface $\fn$. We compute the number of such curves through the appropriate number of fixed general points on $\fn$, and the number of such curves which are irreducible. These numbers are known as Severi degrees; they are the degrees of unions of components of the Hilbert scheme. As (i) $\fn$ can be deformed to $\eff_{n+2}$, (ii) the Gromov-Witten invariants are deformation-invariant, and (iii) the Gromov-Witten invariants of $\eff_0$ and $\eff_1$ are enumerative, Theorem \ref{irecursion} computes the genus $g$ Gromov-Witten invariants of all $\fn$. (The genus 0 case is well-known.) The arguments are given in sufficient generality to also count plane curves in the style of L. Caporaso and J. Harris and to lay the groundwork for computing higher genus Gromov-Witten invariants of blow-ups of the plane at up to five points (in a future paper).

Explicit formulas for Arnold’s generic curve invariants

Abstract: We review the explicit formulas for Arnold’s generic curve invariants due to Viro, Shumakovich and Polyak and add some remarks concerning the invariants of spherical curves and curves immersed in arbitrary orientable surfaces.

Generalized Bezout's theorem and its applications in coding theory

Abstract: This paper presents a generalized Bezout theorem which can be used to determine a tighter lower bound of the number of distinct points of intersection of two or more plane curves. A new approach to determine a lower bound on the minimum distance for algebraic-geometric codes defined from a class of plane curves is introduced, based on the generalized Bezout theorem. Examples of more efficient linear codes are constructed using the generalized Bezout theorem and the new approach. For d=4, the linear codes constructed by the new construction are better than or equal to the known linear codes. For d/spl ges/5, these new codes are better than the known AG codes defined from whole spaces. The Klein codes [22, 16, 5] and [22, 15, 6] over GF(2/sup 3/), and the improved Hermitian code [64, 56, 6] over GF(2/sup 4/) are also constructed.

On the Bennequin Invariant and the Geometry of Wave Fronts

Abstract: The theory of Arnold's invariants of plane curves and wave fronts is applied to the study of the geometry of wave fronts in the standard 2-sphere, in the Euclidean plane and in the hyperbolic plane. Some enumerative formulae similar to the Plucker formulae in algebraic geometry are given in order to compute the generalized Bennequin invariant J + in terms of the geometry of the front. It is shown that in fact every coefficient of the polynomial invariant of Aicardi can be computed in this way. In the case of affine wave fronts, some formulae previously announced by S.L. Tabachnikov are proved. This geometric point of view leads to a generalization to generic wave fronts of a result shown by Viro for smooth plane curves. As another application, the Fabricius-Bjerre and Weiner formulae for smooth plane and spherical curves are generalized to wave fronts.

Contact Cohomology of the Projective Plane

Abstract: We construct an associative ring which is a deformation of the quantum cohomology ring of the projective plane. Just as the quantum cohomology encodes the incidence characteristic numbers of rational plane curves, the contact cohomology encodes the tangency characteristic numbers.

On Campedelli branch loci

Abstract: We construct new degree ten plane curves having six [3, 3] points that do not belong to a conic and degree ten plane curves with five [3, 3] points and a quadruple point, having the six singularities that again do not lie on a conic. In the second family we find an irriducible curve.

Some remarks on complete simply connected minimal surfaces meeting the planes x3 = Constant Transversally

Abstract: The helicoid and plane are the only known complete embedded minimal surfaces inR 3 that are simply connected We prove the helicoid and plane are the only surfaces of this type that have bounded curvature and meet each plane x3 = constant in (at most) one smooth connected curve

A Short Proof of a Gauss Problem

Abstract: The traversal of a self crossing closed plane curve, with points of multiplicity at most two, defines a double occurrence sequence.

Data Reduction of Piecewise Linear Curves

Abstract: We present and study two new algorithms for data reduction or simplification of piecewise linear plane curves. Given a curve P and a tolerance e ≥ 0, both methods determine a new curve Q, with few vertices, which is at most e in Hausdorff distance from P. The methods differ from most existing methods in that they do not require a vertex in Q to be a vertex in P. Several examples are given where we show that the methods presented here compare favorably to other methods found in the literature. We also show how the vertices of a curve can be reordered so that the first, say n, vertices of the reordered sequence form an approximation to the curve itself.

Rational surfaces in ℙ4 containing a plane curve

Abstract: The families of smooth rational surfaces in ℙ4 have been classified in degree ⩽ 10. All known rational surfaces in ℙ4 can be represented as blow-ups of the plane P2. The fine classification of these surfaces consists of giving explicit open and closed conditions which determine the configurations of points corresponding to all surfaces in a given family. Using a restriction argument originally due independently to Alexander and Bauer we achieve the fine classification in two cases, namely non-special rational surfaces of degree 9 and special rational surfaces of degree 8. The first case completes the fine classification of all non-special rational surfaces. In the second case we obtain a description of the moduli space as the quotient of a rational variety by the symmetric group S5. We also discuss in how far this method can be used to study other rational surfaces in ℙ4.