Showing papers on "Plane curve published in 1976"

Elementary algebraic geometry

Abstract: I Examples of curves.- 1 Introduction.- 2 The topology of a few specific plane curves.- 3 Intersecting curves.- 4 Curves over ?.- II Plane curves.- 1 Projective spaces.- 2 Affine and projective varieties examples.- 3 Implicit mapping theorems.- 4 Some local structure of plane curves.- 5 Sphere coverings.- 6 The dimension theorem for plane curves.- 7 A Jacobian criterion for nonsingularity.- 8 Curves in ?2(?) are connected.- 9 Algebraic curves are orientable.- 10 The genus formula for nonsingular curves.- III Commutative ring theory and algebraic geometry.- 1 Introduction.- 2 Some basic lattice-theoretic properties of varieties and ideals.- 3 The Hilbert basis theorem.- 4 Some basic decomposition theorems on ideals and varieties.- 5 The Nullstellensatz: Statement and consequences.- 6 Proof of the Nullstellensatz.- 7 Quotient rings and subvarieties.- 8 Isomorphic coordinate rings and varieties.- 9 Induced lattice properties of coordinate ring surjections examples.- 10 Induced lattice properties of coordinate ring injections.- 11 Geometry of coordinate ring extensions.- IV Varieties of arbitrary dimension.- 1 Introduction.- 2 Dimension of arbitrary varieties.- 3 The dimension theorem.- 4 A Jacobian criterion for nonsingularity.- 5 Connectedness and orientability.- 6 Multiplicity.- 7 Bezout's theorem.- V Some elementary mathematics on curves.- 1 Introduction.- 2 Valuation rings.- 3 Local rings.- 4 A ring-theoretic characterization of nonsingularity.- 5 Ideal theory on a nonsingular curve.- 6 Some elementary function theory on a nonsingular curve.- 7 The Riemann-Roch theorem.- Notation index.

The Hyperelliptic Locus with Special Reference to Characteristic Two

Abstract: Let Jg0 be the coarse moduli space for curves of genus g(>2) defined over an algebraically closed field k, and denote by o~'.0 the subset of ~¢/0 corresponding to the hyperelliptic curves (the hyperelliptic locus). It is proved below that Yfg is closed and irreducible, and that any two points of ~ may be connected by a (non-complete) rational curve reside ~ . In the case k= C this has been proved by the Italian geometers, see e.g. [9]; more recently Rauch [8] has proved the closedness of ~ , and ArbareUo [1] has shown that ~g is irreducible and unirational. For g= 2 and char(k) arbitrary, the results follow immediately from Igusa's explicit description of ~2 =~/g2, [5]. For arbitrary g, but char (k) :t: 2, the irreducibility of "~¢'0 is a trivial consequence of Geyer's construction of a factorial affine moduli space for the hypereltiptic curves of genus g, [4]. Our proof of the irreducibility and the (uni-)rationatity question follow the classical (Italian) pattern with a slight modification inspired by Igusa's treatment for genus two: we exhibit a rational family of plane curves of degree g + 2, possessing only one singularity, which is ordinary of multiplicity g, such that every birational isomorphism class of hyperelliptic curves is represented in the family. Next, we blow-up these singularities simultaneously and acquire a smooth family with analogous properties. The universal property of ~'g now yields the desired results, but the closedness of ~ . The latter is achieved by a cohomological argument. We should like to express our thanks to F. Oort, who proposed the irreducibility question in characteristic two.

A geometric inequality for plane curves with restricted curvature

Abstract: A geometric proof is given that a closed plane curve of length L and curvature bounded by K can be contained inside a circle of radius L/4 (iT 2)/2K. Let K be a positive constant and let En be n-dimensional Euclidean space. A continuously differentiable curve X in En parametrized by arc length s is called a K-curve if and only if IIX'(sl) X'(S2)11 < Klsl S21 for all sl and 52 The purpose of this note is to give a geometric proof of an inequality obtained previously by calculus of variations methods [4]: namely, that if X is a closed K-curve in E2 of length L, then X lies in a circle of radius R where (1) R < L/4-('7-2)/2K. Since the components of X' are functions of bounded variation, X"(s) and k(s) = IIX"(s)jj exist for almost all s and, when k(s) exists, k(s) < K. Thus, Kcurves are a generalization of C2 curves with curvature bounded by K. They share many of the geometrical properties of the latter but are to be preferred in several respects. Dubins [3] showed that among K-curves with prescribed initial and terminal points and prescribed initial and terminal tangent vectors there exists a K-curve of minimal length. We show below (Proposition 3) that the convex envelope of a closed K-curve in E2 is also a closed K-curve. Both of these properties fail if K-curves are replaced by C2 K-curves. In fact, Proposition 3 fails if K-curves are replaced by piecewise C2 K-curves. To prove inequality (1) we first generalize a theorem of Blaschke [1, p. 116] to the case of convex K-curves, then apply a geometrical construction to show that (1) holds for convex K-curves, and then extend (1) to all K-curves by Proposition 3. An alternative geometric proof of (1) may be obtained by combining results of Dubins [3, Proposition 1 and Theorem 1] with Theorem 2 of Johnson [4]. (We are indebted to the referee for alerting us to Dubins' interesting work and to the existence of the alternative proof.) For information on related problems, consult [2]-[4]. Let C be a closed convex K-curve in E2 with arc length parameter s. (The arc length parameter of a closed K-curve is understood to assume all real values by periodic extension. Also, a closed K-curve in E2 is convex if and only Received by the editors June 11, 1975 and, in revised form, September 22, 1975. AMS (MOS) subject classifications (1970). Primary 53A05, 52A40.

A method for plotting curves defined by implicit equations

Abstract: A square chain of points is found that approximates a plane curve of the form F (x,y) =0. In case F (x,y) =0 represents a conic section, the method requires no multiplication or division, and so is well adapted to hardware or small computer software implementation. The algorithm may also be used to find approximate solutions of differential equations of the form F1dx + F2dy = 0.

Pseudospline interpolation for space curves

Abstract: A method for interpolating a curve through points in space is described. It is the direct analogue of Fowler-Wilson or pseudospline interpolation for plane curves in that local coordinate systems, cubic polynomials of suitable parameters, and mildly nonlinear equations are used to obtain a continuous interpolating curve with continuous tangent and curvature vectors.

A method for plotting curves defined by implicit equations

Abstract: A square chain of points is found that approximates a plane curve of the form F (x,y) =0. In case F (x,y) =0 represents a conic section, the method requires no multiplication or division, and so is...

Die algebraischen Konoide mit ebener Striktionslinie

Abstract: Among the algebraic ruled surfaces a theorem ofH. Brauner characterizes certain helicoids of projective screw motions by the property to possess continuous sets of plane shadow curves. Making use of this theorem all algebraic conoids with a plane striction curve are determined in euclidean 3-space and some properties of these curves are discussed.

Hypersurfaces of order two

Abstract: A hypersurface Sn 1 of order two in the real projective n-space is met by every straight line in maximally two points; cf. [ 1, p. 3911. We develop a synthetic theory of these hypersurfaces inductively, basing it upon a concept of differentiability. We define the index and the degree of degeneracy of an Sn 1 and classify the Sn in terms of these two quantities. Our main results are (i) the reduction of the theory of the Sn-Ito the nondegenerate case; (ii) the Theoremn (A.5.1 1) that a nondegenerate Sn-1 of positive index must be a quadric and (iii) a comparison of our theory with Marchaud's discussion of "linearly connected" sets; cf. [31. Preface. The theory of plane curves is of major importance in our study of the hypersurfaces of order two. This theory is the first step in our induction and the means by which we define tangents. The introduction of these curves follows the approach of P. Scherk in [5] and R. Park in [4]. A. Marchaud introduced in [2] the "surfaces of order two" in the real projective three-space. Our theory is based on that paper. We compare our hypersurfaces with the quadrics by direct construction (see Appendix) and also by showing that these hypersurfaces are identical with the common boundaries of certain pairs of linearly connected sets; cf. [3]. This paper has developed out my doctoral thesis of the same title, written at the University of Toronto, under the supervision of Professor Peter Scherk. My thanks are due to him for his continued help in the preparation of this paper. I would also like to thank Professor 0. Haupt for bringing to my attention the theory of linearly connected sets. 1. Curves in p2. The theory of hypersurfaces is based on that of curves in the real projective plane. In this chapter, we give a precise definition of the curves of order two and introduce tangents. 1.