The Coxeter Quotient of the Fundamental Group of a Galois Cover of 𝕋 × 𝕋

ABSTRACT

Let X be the surface 𝕋 × 𝕋, where 𝕋 is the complex torus. This article is the third in a series studying the fundamental group of the Galois cover of X with respect to a generic projection onto ??².

Van Kampen Theorem gives a presentation of the fundamental group of the complement of the branch curve, with 54 generators and more than 2000 relations. Here we introduce a certain natural quotient (obtained by identifying pairs of generators), prove it is a quotient of a Coxeter group related to the degeneration of X, and show that this quotient is virtually nilpotent.

Communicated by C. Pedrini.     

Constructing C3 shape preserving interpolating space curves

We present a method for constructing shape-preserving C ³ interpolants in R ³. The resulting curve is obtained by adding a polynomial perturbation of high degree to a curve which is shape-preserving but not sufficiently smooth. The degree of the perturbed curve is selected in order to maintain the shape properties of the basic curve.     

Cubic algebraic spline curves design

In this paper we propose a construction method of the planar cubic algebraic spline curve with endpoint interpolation conditions and a specific analysis of its properties. The piecewise cubic algebraic curve has G ² continuous contact with the control polygon at two endpoints and is G ² continuous between each segments of itself. The process of this method is simple and clear, and provides a new way of thinking to design implicit curves.     

Curve Shortening Flow in Arbitrary Dimensional Euclidian Space

In this paper, curve shortening flow in Euclidian space R^n（n≥3） is studied, and S. Altschuler＇s results about flow for space curves are generalized. We prove that the curve shortening flow converges to a straight line in infinite time if the initial curve is a ramp. We also prove the planar phenomenon when the curve shortening flow blows up.     

Normal forms for singularities of pedal curves produced by non-singular dual curve germs in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

For an n-dimensional spherical unit speed curve r and a given point P, we can define naturally the pedal curve of r relative to the pedal point P. When the dual curve germs are non-singular, singularity types of such pedal curves depend only on locations of pedal points. In this paper, we give a complete list of normal forms for singularities and locations of pedal points when the dual curve germs are non-singular. As an application of our list, we characterize C ^∞ left equivalence classes of pedal curve germs (I, s ₀) → S ⁿ produced by non-singular dual curve germ from the viewpoint of the relation between tangent space and tangent space.      

Modified Schwarz–Christoffel mappings using approximate curve factors

The Schwarz–Christoffel mapping from the upper half-plane to a polygonal region in the complex plane is an integral of a product with several factors, where each factor corresponds to a certain vertex in the polygon. Different modifications of the Schwarz–Christoffel mapping in which factors are replaced with the so-called curve factors to achieve polygons with rounded corners are known since long times. Among other requisites, the arguments of a curve factor and its correspondent scl factor must be equal outside some closed interval on the real axis.In this paper, the term approximate curve factor is defined such that many of the already known curve factors are included as special cases. Additionally, by alleviating the requisite on the argument from exact to asymptotic equality, new types of curve factors are introduced. While traditional curve factors have a C¹ regularity, C^∞ regular approximate curve factors can be constructed, resulting in smooth boundary curves when used in conformal mappings.Applications include modelling of wave scattering in waveguides. When using approximate curve factors in modified Schwarz–Christoffel mappings, numerical conformal mappings can be constructed that preserve two important properties in the waveguides. First, the direction of the boundary curve can be well controlled, especially towards infinity, where the application requires two straight parallel walls. Second, a smooth (C^∞) boundary curve can be achieved.     

Le relazioni irreduttibili fra gľinvarianti d’una forma qualunque Dľottavo ordine

Let C ⊂ P ³ be a smooth and connected projective curve of genus g ≥ 2. Here in characteristic ≠ 2 we prove that its generic plane section is in uniform position and that its numerical character is connected. Partially supported by MIUR and GNSAGA of INdAM (Italy)     

A Combinatorial Characterization of Hermitian Curves

A unital U with parameter q is a 2 – (q ³ + 1, q + 1, 1) design. If a point set U in PG(2, q ²) together with its (q + 1)-secants forms a unital, then U is called a Hermitian arc. Through each point p of a Hermitian arc H there is exactly one line L having with H only the point p in common; this line L is called the tangent of H at p. For any prime power q, the absolute points and nonabsolute lines of a unitary polarity of PG(2, q ²) form a unital that is called the classical unital. The points of a classical unital are the points of a Hermitian curve in PG(2, q ²).Let H be a Hermitian arc in the projective plane PG(2, q ²). If tangents of H at collinear points of H are concurrent, then H is a Hermitian curve. This result proves a well known conjecture on Hermitian arcs.     

The Most Symmetric Nonsingular Plane Curves of Degree n≤20, I*

If n is an odd prime less than 20, then the most symmetric nonsingular plane curves in P ² of degree n are projectively equivalent to the Fermat curve x ⁿ +y ⁿ +z ⁿ .     

Surfaces à points doubles isolés

We give a characterization of the generic projection on P ² of an algebraic surface of P ³ with a finite number of nodes. The construction of an algebraic surface of P ³ with a given number of nodes is thus equivalent to the construction of a plane curve with nodes and cusps in some special position. Received: November 9, 1996     

Offset approximation based on reparameterizing the path of a moving point along the base circle

This paper presents a novel algorithm for planar curve offsetting. The basic idea is to regard the locus relative to initial base circle, which is formed by moving the unit normal vectors of the base curve, as a unit circular arc first, then accurately to represent it as a rational curve, and finally to reparameterize it in a particular way to approximate the offset. Examples illustrated that the algorithm yields fewer curve segments and control points as well as C^1 continuity, and so has much significance in terms of saving computing time, reducing the data storage and smoothing curves entirely.     

Artin–Schreier curves and weights of two-dimensional cyclic codes

Let be the finite field with q elements of characteristic p, be the extension of degree m>1 and f(x) be a polynomial over . The maximum number of affine -rational points that a curve of the form y^q−y=f(x) can have is q^m+1. We determine a necessary and sufficient condition for such a curve to achieve this maximum number. Then we study the weights of two-dimensional (2-D) cyclic codes. For this, we give a trace representation of the codes starting with the zeros of the dual 2-D cyclic code. This leads to a relation between the weights of codewords and a family of Artin–Schreier curves. We give a lower bound on the minimum distance for a large class of 2-D cyclic codes. Then we look at some special classes that are not covered by our main result and obtain similar minimum distance bounds.     

Remarks on Normal Generation of Special Line Bundles on Algebraic Curves

Let L be a very ample line bundle with h ¹(L) ≥2 on a curve of genus g. We prove that L is normally generated if deg(L) ≥2g ? 1 ? 4h ¹(L) for large enough genus g.     

On the size of the Shafarevich--Tate group of elliptic curves over function fields

Let E be a nonconstant elliptic curve, over a global field K of positive, odd characterisitc. Assuming the finiteness of the Shafarevich-Tate group of E, we show that the order of theShafarevich-Tate group of E, is given by O ^{(N1/2+6 log(2)/ log(q))}, where N is the conductor of E,q isthe cardinality of the finite field of constants of K, and where the constant in the bound depends only on K. The method of proof is to workwith the geometric analog of the Birch-Swinnerton Dyer conjecture for thecorresponding elliptic surface over the finite field, as formulatedby Artin-Tate, and to examine the geometry of this elliptic surface.     

Blow-up rates for the general curve shortening flow

The blow-up rates of derivatives of the curvature function will be presented when the closed curves contract to a point in finite time under the general curve shortening flow. In particular, this generalizes a theorem of M.E. Gage and R.S. Hamilton about mean curvature flow in R².     

Translating Solutions of the Nonparametric Mean Curvature Flow with Nonzero Neumann Boundary Data in Product Manifold Mn × R*

In this paper, the authors can prove the existence of translating solutions to the nonparametric mean curvature flow with nonzero Neumann boundary data in a prescribed product manifold Mⁿ × R, where Mⁿ is an n-dimensional (n ≥ 2) complete Riemannian manifold with nonnegative Ricci curvature, and R is the Euclidean 1-space.     

On the dynamics of the singularly perturbed rational maps

In this paper, we study the dynamics of the family of rational maps fλ,（z） = zn - λ/zm, n ≥2, m ≥ 1,λ ∈ C. We construct an example of buried Sierpinski curve Julia set in this family. We also give an estimate of the location of bifurcation locus of fλ.     

The multiplication map for global sections of line bundles and rank 1 torsion free sheaves on curves

LetX be an integral projective curve andL ∃ Pic^a(X),M ∃ Pic^b (X) with h¹(X, L)= h¹(X, M) = 0 andL, M general. Here we study the rank of the multiplication map μ _L,M :H ⁰(X,L)⊗H ⁰(X,M)→H ⁰(X,L⊗M). We also study the same problem whenL andM are rank 1 torsion free sheaves onX. Most of our results are forX with only nodes as singularities.     

Zerlegung der Darboux-Drehung in Zwei Ebene Drehungen

The Darboux rotation for space curves in Euclidean space E³ is decomposed into two simultaneous rotations. The axes of these simultaneous rotations are joined by a simple mechanism. One of these axes is a parallel of the principal normal of the curve, the direction of the other is the direction of the Darboux vector of the curve. This decomposition of the Darboux rotation yields a necessary condition for the curve to be closed.

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Herrn Prof. Dr.Oswald Giering zum 60. Geburtstag gewidmet     

On fundamental groups related to the Hirzebruch surface <Emphasis Type="Italic">F</Emphasis><Subscript>1</Subscript>

Given a projective surface and a generic projection to the plane, the braid monodromy factorization (and thus, the braid monodromy type) of the complement of its branch curve is one of the most important topological invariants, stable on deformations. From this factorization, one can compute the fundamental group of the complement of the branch curve, either in ℂ² or in ℂℙ². In this article, we show that these groups, for the Hirzebruch surface F _1,(a,b), are almost-solvable. That is, they are an extension of a solvable group, which strengthen the conjecture on degeneratable surfaces. This work was supported by the Emmy Noether Institute Fellowship (by the Minerva Foundation of Germany) and Israel Science Foundation (Grant No. 8008/02-3)