Weierstrass points of order three on smooth quartic curves

2019 ◽  
Vol 18 (01) ◽  
pp. 1950020 ◽  
Author(s):  
Alwaleed Kamel ◽  
Waleed Khaled Elshareef
Keyword(s):  
Projective Plane ◽  
Weierstrass Points ◽  
Smooth Projective Plane ◽  
Quartic Curve ◽  
The Subject ◽  
Quartic Curves

In this paper, we study the [Formula: see text]-Weierstrass points on smooth projective plane quartic curves and investigate their geometry. Moreover, we use a technique to determine in a very precise way the distribution of such points on any smooth projective plane quartic curve. We also give a variety of examples that illustrate and enrich the subject.

XII.—Some Remarks occasioned by the Geometry of the Veronese Surface

1942 ◽  
Vol 61 (2) ◽  
pp. 140-159
Author(s):  
W. L. Edge
Keyword(s):  
Subject Matter ◽  
Algebraic Representation ◽  
Fresh Properties ◽  
Veronese Surface ◽  
Quartic Curve ◽  
Mise En Scene ◽  
Dual Character ◽  
The Subject ◽  
Quartic Curves

The subject-matter of these pages may be briefly summarised as follows: the geometry of the Veronese surface, with an algebraic representation of it that does justice to its self-dual character; the relations of the secant planes of the surface to quadrics which either contain the surface or are outpolar to it; and the derivation of an invariant and two contravariants of a ternary quartic in the light of the (1, 1) correspondence between the quartic curves in a plane and the quadrics outpolar to a Veronese surface. There is no suggestion of discovering fresh properties of the surface, though possibly the results in § 12 § 13 may be new; but the geometrical considerations lead naturally to some algebraical results which it seems worth while to have on record, such as, for example, the identity 8.2 and the remarks concerning the rank of the determinant which appears there, and the form found in § 13 for the harmonic envelope of a plane quartic curve. These algebraical results lie very close to properties of the surface; so close in fact that one might say that the Veronese surface is the proper mise en scène for them.

scholarly journals A plane quartic curve with twelve undulations

1945 ◽  
Vol 35 ◽  
pp. 10-13 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Homogeneous Coordinates ◽  
Quartic Curve ◽  
Base Points ◽  
Octahedral Group ◽  
Quartic Form ◽  
Image Position ◽  
Quartic Curves ◽  
Set Of Points

The pencil of quartic curveswhere x, y, z are homogeneous coordinates in a plane, was encountered by Ciani [Palermo Rendiconli, Vol. 13, 1899] in his search for plane quartic curves that were invariant under harmonic inversions. If x, y, z undergo any permutation the ternary quartic form on the left of (1) is not altered; nor is it altered if any, or all, of x, y, z be multiplied by −1. There thus arises an octahedral group G of ternary collineations for which every curve of the pencil is invariant.Since (1) may also be writtenthe four linesare, as Ciani pointed out, bitangents, at their intersections with the conic C whose equation is x2 + y2 + z2 = 0, to every quartic of the pencil. The 16 base points of the pencil are thus all accounted for—they consist of these eight contacts counted twice—and this set of points must of course be invariant under G. Indeed the 4! collineations of G are precisely those which give rise to the different permutations of the four lines (2), a collineation in a plane being determined when any four non-concurrent lines and the four lines which are to correspond to them are given. The quadrilateral formed by the lines (2) will be called q.

On Five Lines, in Space of Four Dimensions, which lie upon a, Quadric Threefold and the Normal Quartic Curves of which they are Chords

1924 ◽  
Vol 22 (2) ◽  
pp. 189-199
Author(s):  
F. Bath
Keyword(s):  
Three Dimensions ◽  
Apparent Contradiction ◽  
Four Dimensions ◽  
Quartic Curve ◽  
Lines In Space ◽  
Quadric Threefold ◽  
Quartic Curves

The connexion between the conditions for five lines of S4(i) to lie upon a quadric threefold,and (ii) to be chords of a normal quartic curve,leads to an apparent contradiction. This difficulty is explained in the first paragraph below and, subsequently, two investigations are given of which the first uses, mainly, properties of space of three dimensions.

Transformations depending on sets of associated points

1952 ◽  
Vol 48 (3) ◽  
pp. 383-391
Author(s):  
T. G. Room
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Three Dimensional ◽  
Cubic Curves ◽  
Birational Transformations ◽  
Quadric Surfaces ◽  
Base Points ◽  
Dimensional Projective Space ◽  
Quartic Curves

This paper falls into three sections: (1) a system of birational transformations of the projective plane determined by plane cubic curves of a pencil (with nine associated base points), (2) some one-many transformations determined by the pencil, and (3) a system of birational transformations of three-dimensional projective space determined by the elliptic quartic curves through eight associated points (base of a net of quadric surfaces).

scholarly journals Cubic Arcs in the Projective Plane Over a Finite Field of Order Twenty Three

2019 ◽  
Vol 54 (6) ◽  
Author(s):  
Najm A.M. Al-Seraji ◽  
Asraa A. Monshed
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Coding Theory ◽  
Finite Projective Plane ◽  
Cubic Curves ◽  
The Subject

In this research we are interested in finding all the different cubic curves over a finite projective plane of order twenty-three, learning which of them is complete or not, constructing the stabilizer groups of the cubics in, studying the properties of these groups, and, finally, introducing the relation between the subject of coding theory and the projective plane of order twenty three.

An algorithm for a lifted Massey triple product of a smooth projective plane curve

2020 ◽  
Vol 30 (08) ◽  
pp. 1651-1669
Author(s):  
Younggi Lee ◽  
Jeehoon Park ◽  
Junyeong Park ◽  
Jaehyun Yim
Keyword(s):  
Projective Plane ◽  
Homogeneous Polynomial ◽  
Plane Curve ◽  
Main Idea ◽  
Triple Product ◽  
Cup Product ◽  
Smooth Projective Plane ◽  
Defining System ◽  
Polynomial Formula ◽  
Massey Triple Product

We provide an explicit algorithm to compute a lifted Massey triple product relative to a defining system for a smooth projective plane curve [Formula: see text] defined by a homogeneous polynomial [Formula: see text] over a field. The main idea is to use the description (due to Carlson and Griffiths) of the cup product for [Formula: see text] in terms of the multiplications inside the Jacobian ring of [Formula: see text] and the Cech–deRham complex of [Formula: see text]. Our algorithm gives a criterion whether a lifted Massey triple product vanishes or not in [Formula: see text] under a particular nontrivial defining system of the Massey triple product and thus can be viewed as a generalization of the vanishing criterion of the cup product in [Formula: see text] of Carlson and Griffiths. Based on our algorithm, we provide explicit numerical examples by running the computer program.

scholarly journals The Tritangent Circles of a Circular Quartic Curve

1932 ◽  
Vol 3 (1) ◽  
pp. 46-52
Author(s):  
H. W. Richmond
Keyword(s):  
Fixed Points ◽  
Space Curve ◽  
Cubic Surface ◽  
Quartic Curve ◽  
Family Of Curves ◽  
The Family ◽  
The Subject ◽  
Simple Notation ◽  
Pass Through

§1. In a recent paper with this title Prof. W. P. Milne has discussed the properties of the conics which pass through two fixed points of a plane quartic curve and touch the curve at three other points. In dealing with a numerous family of curves such as this it is very desirable to have a scheme of marks or labels to distinguish the different members of the family; Hesse's notation for the double tangents of a C4 illustrates this. By using another line of approach to the subject, by projecting the curve of intersection of a quadric and a cubic surface from a point at which (under exceptional circumstances) the surfaces touch, I find that a fairly simple notation for the 64 conics, in harmony with that for the bitangents, can be obtained. This paper, let it be said, from start to finish is no more than an adaptation of results known for the sextic space-curve referred to; it will be sufficient therefore to state results with short explanations.

Quartic curves in characteristic 2

1995 ◽  
Vol 117 (3) ◽  
pp. 393-414 ◽  
Author(s):  
C. T. C. Wall
Keyword(s):  
Finite Deformation ◽  
Normal Forms ◽  
Positive Characteristic ◽  
Surface Singularities ◽  
The Subject ◽  
Characteristic 2 ◽  
Simple Singularities ◽  
Module Type ◽  
Quartic Curves ◽  
Article 5

Simple singularities in positive characteristicSimple singularities in positive characteristic have been discussed by many authors, and the article [5] in particular establishes the subject on a firm footing. In it a simple, or ‘ADE’ singularity is defined by a list of normal forms and it is shown that the following conditions on a singularity are equivalent: (i) it is simple, (ii) it has finite deformation type, (iii) it has finite Cohen-Macaulay module type. Moreover, the normal forms for surface singularities coincide with the earlier list of Artin [1] and those for curves with the list of [9]: in those papers further characterizations were obtained.

scholarly journals Hypersurfaces with linear type singular loci

2019 ◽  
Vol 19 (09) ◽  
pp. 2050169
Author(s):  
Amir Behzad Farrahy ◽  
Abbas Nasrollah Nejad
Keyword(s):  
Projective Plane ◽  
Plane Curve ◽  
Isolated Singularity ◽  
Linear Type ◽  
Jacobian Ideal ◽  
Quartic Curve ◽  
Singular Loci ◽  
Necessary And Sufficient Criteria ◽  
Simple Singularities ◽  
Necessary And Sufficient

In this paper, necessary and sufficient criteria for the Jacobian ideal of a reduced hypersurface with isolated singularity to be of linear type are presented. We prove that the gradient ideal of a reduced projective plane curve with simple singularities ([Formula: see text]) is of linear type. We show that any reduced projective quartic curve is of gradient linear type.

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