Non-conservative plane quartic curves in characteristic five

1990 ◽  
Vol 66 (1) ◽  
pp. 183-204 ◽  
Author(s):  
Karl Otto Stöhr
Keyword(s):  
Quartic Curves

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).

Compactifying the space of elliptic quartic curves

1992 ◽  
pp. 47-58 ◽  
Author(s):  
D. Avritzer ◽  
I. Vainsencher
Keyword(s):  
Quartic Curves

scholarly journals Generators and integral points on certain quartic curves

2019 ◽  
Vol 54 (2) ◽  
pp. 321-343
Author(s):  
Yasutsugu Fujita ◽  
◽  
Tadahisa Nara ◽  
Keyword(s):  
Integral Points ◽  
Quartic Curves

XI—The Identification of Klein's Quartic

1944 ◽  
Vol 62 (1) ◽  
pp. 83-91
Author(s):  
W. L. Edge
Keyword(s):  
Standard Form ◽  
Geometrical Interpretation ◽  
The Other ◽  
Veronese Surface ◽  
Quartic Polynomial ◽  
New Form ◽  
Quartic Curves ◽  
Quartic Polynomials

SummaryThere is a mode of specialising a quartic polynomial which causes a binary quartic to become equianharmonic and a ternary quartic to become a Klein quartic, admitting a group of 168 linear self-transformations. The six relations which must be satisfied by the coefficients of the ternary quartic were given by Coble forty years ago, but their true significance was never suspected and they have remained until now an isolated curiosity. In § 2 we give, in terms of a quadric and a Veronese surface, the geometrical interpretation of the six relations; we also give, in terms of the adjugate of a certain matrix, their algebraical interpretation. Both these interpretations make it abundantly clear that this set of relations specialising a ternary quartic has analogues for quartic polynomials in any number of variables, and point unmistakably to what these analogues are.That a ternary quartic is, when so specialised, a Klein quartic is proved in §§ 4–6. The proof bifurcates after (5.3); one branch leads instantly to the standard form of the Klein quartic while the other leads to another form which, on applying a known test, is found also to represent a Klein quartic. One or two properties of the curve follow from this new form of its equation. In §§ 8–10 some properties of a Veronese surface are established which are related to known properties of plane quartic curves; and these considerations lead to a discussion, in § 11, of certain hexads of points associated with a Klein curve.

scholarly journals Theta Surfaces

2020 ◽  
Author(s):  
Daniele Agostini ◽  
Türkü Özlüm Çelik ◽  
Julia Struwe ◽  
Bernd Sturmfels
Keyword(s):  
Algebraic Geometry ◽  
Theta Functions ◽  
Minkowski Sum ◽  
Zero Set ◽  
Analytic Surface ◽  
Numerical Algebraic Geometry ◽  
Abelian Integrals ◽  
Space Curves ◽  
Quartic Curves ◽  
Special Plane

Abstract A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that any analytic surface that is the Minkowski sum of two space curves in two different ways is a theta surface. The four space curves that generate such a double translation structure are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.

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