scholarly journals Space sextic curves with six bitangents, and some geometry of the diagonal cubic surface

1997 ◽  
Vol 40 (1) ◽  
pp. 85-97
Author(s):  
R. H. Dye
Keyword(s):  
Finite Number ◽  
Space Curve ◽  
Rational Space ◽  
Cubic Surface ◽  
General Space ◽  
Cubic Surfaces ◽  
Related Pair ◽  
Sextic Curve

A general space curve has only a finite number of quadrisecants, and it is rare for these to be bitangents. We show that there are irreducible rational space sextics whose six quadrisecants are all bitangents. All such sextics are projectively equivalent, and they lie by pairs on diagonal cubic surfaces. The bitangents of such a related pair are the halves of the distinguished double-six of the diagonal cubic surface. No space sextic curve has more than six bitangents, and the only other types with six bitangents are certain (4,2) curves on quadrics. In the course of the argument we see that space sextics with at least six quadrisecants are either (4,2) or (5,1) quadric curves with infinitely many, or are curves which each lie on a unique, and non-singular, cubic surface and have one half of a double-six for quadrisecants.

Nodal cubic surfaces

1979 ◽  
Vol 83 (3-4) ◽  
pp. 333-346
Author(s):  
W. L. Edge
Keyword(s):  
Singular Surface ◽  
Cubic Surface ◽  
Cubic Surfaces ◽  
Geometrical Relation ◽  
Elliptic Cone

SynopsisThe cubic surfaces in, save for the elliptic cone, are, whatever their singularities, projections of del Pezzo's non-singular surface F, of order 9 in. It is explained how, merely by specifying the geometrical relation of the vertex of projection to F, each cubic surface is obtainable “at a stroke”, without using spaces of intermediate dimensions.

scholarly journals VII. A memoir on cubic surfaces

1869 ◽  
Vol 159 ◽  
pp. 231-326 ◽  
Keyword(s):  
Singular Points ◽  
Third Order ◽  
Cubic Surface ◽  
The Third ◽  
Cubic Surfaces ◽  
Present Subject

The present Memoir is based upon, and is in a measure supplementary to that by Pro­fessor Schläfli, “On the Distribution of Surfaces of the Third Order into Species, in reference to the presence or absence of Singular Points, and the reality of their Lines,” Phil. Trans, vol. cliii. (1863) pp. 193—241. But the object of the Memoir is different. I disregard altogether the ultimate division depending on the reality of the lines, attend­ing only to the division into (twenty-two, or as I prefer to reckon it) twenty-three cases depending on the nature of the singularities. And I attend to the question very much on account of the light to be obtained in reference to the theory of Reciprocal Surfaces. The memoir referred to furnishes in fact a store of materials for this purpose, inasmuch as it gives (partially or completely developed) the equations in plane-coordinates of the several cases of cubic surfaces, or, what is the same thing, the equations in point-coor­dinates of the several surfaces (orders 12 to 3) reciprocal to these repectively. I found by examination of the several cases, that an extension was required of Dr. Salmon’s theory of Reciprocal Surfaces in order to make it applicable to the present subject ; and the preceding “Memoir on the Theory of Reciprocal Surfaces” was written in connexion with these investigations on Cubic Surfaces. The latter part of the Memoir is divided into sections headed thus:— “Section I = 12, equation (X, Y, Z, W ) 3 = 0” &c. referring to the several cases of the cubic surface; but the paragraphs are numbered continuously through the Memoir. The twenty-three Cases of Cubic Surfaces—Explanations and Table of Singularities . Article Nos. 1 to 13. 1. I designate as follows the twenty-three cases of cubic surfaces, adding to each of them its equation:

scholarly journals A proof of Aronhold's theorem upon quartic curves

1940 ◽  
Vol 6 (3) ◽  
pp. 190-191
Author(s):  
H. W. Richmond
Keyword(s):  
Finite Number ◽  
Simple Proof ◽  
Plane Curves ◽  
Cubic Surface ◽  
Quartic Curves ◽  
Order Four ◽  
The Given

It is to be expected that a finite number of plane curves of order four should have seven given lines as bitangents, because the number of conditions imposed is equal to the number of effective free constants in the equation of such a curve, viz., 14. Aronhold made the interesting discovery that one curve could be determined in which no three of the given lines have their six points of contact on a conic. The method, due to Geiser, of obtaining the bitangents as projections of the lines of a cubic surface leads to a simple proof of the existence of this quartic.

scholarly journals III. A memoir on cubic surfaces

1869 ◽  
Vol 17 ◽  
pp. 221-222
Keyword(s):  
Singular Points ◽  
Third Order ◽  
Cubic Surface ◽  
The Third ◽  
Cubic Surfaces ◽  
Present Subject

The present Memoir is based upon, and is in a measure supplementary to that by Professor Schläfli, “On the Distribution of Surfaces of the Third Order into Species, in reference to the presence or absence of Singular Points, and the reality of their Lines,” Phil. Trans, vol. cliii. (1863) pp. 193–241. But the object of the Memoir is different. I disregard altogether the ultimate division depending on the reality of the lines, attending only to the division into (twenty-two, or as I prefer to reckon it) twenty-three cases depending on the nature of the singularities. And I attend to the question very much on account of the light to be obtained in reference to the theory of Reciprocal Surfaces. The memoir referred to furnish.es in fact a store of materials for this purpose, inasmuch as it gives (partially or completely developed^the equations in plane-coordinates of the several cases of cubic surfaces; or, what is the same thing, the equations in point-coordinates of the several surfaces (orders 12 to 3) reciprocal to these respectively. I found by examination of the several cases, that an extension was required of Dr. Salmon’s theory of Reciprocal Surfaces in order to make it applicable to the present subject; and the preceding “Memoir on the Theory of Reciprocal Surfaces” was written in connexion with these investigations on Cubic Surfaces. The latter part of the Memoir is divided into sections headed thus:—“Section 1 = 12, equation (X, Y, Z, W) 3 = 0” &c. referring to the several cases of the cubic surface; but the paragraphs are numbered continuously through the Memoir. The principal results are included in the following Table of singularities. The heading of each column shows the number and character of the case referred to, viz. C denotes a conic node, B a biplanar node, and U a uniplanar node; these being further distinguished by subscript numbers, showing the reduction thereby caused in the class of the surface: thus XIII=12—B 3 —2 C 2 indicates that the case XIII is a cubic surface, the class whereof is 12—7, = 5, the reduction arising from a biplanar node, B 4 , reducing the class by 3, and from 2 conic nodes, C 2 , each reducing the class by 2.

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.

Cubic surfaces over finite fields

2010 ◽  
Vol 149 (3) ◽  
pp. 385-388 ◽  
Author(s):  
PETER SWINNERTON–DYER
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Cubic Surface ◽  
Cubic Surfaces

Let V be a nonsingular cubic surface defined over the finite field Fq. It is well known that the number of points on V satisfies #V(Fq) = q2 + nq + 1 where −2 ≤ n ≤ 7 and that n = 6 is impossible; see for example [1], Table 1. Serre has asked if these bounds are best possible for each q. In this paper I shall show that this is so, with three exceptions:

The number of contact primes of the canonical curve of genus p

1930 ◽  
Vol 26 (4) ◽  
pp. 453-457
Author(s):  
W. G. Welchman
Keyword(s):  
General Formula ◽  
Theta Functions ◽  
Cubic Surface ◽  
General Curve ◽  
Projective Model ◽  
Quartic Curve ◽  
Canonical Curve ◽  
Riemann Theta Functions ◽  
Sextic Curve ◽  
Special Case

1. It is known, from the theory of the Riemann theta-functions, that the canonical series of a general curve of genus p has 2P−1 (2P − 1) sets which consist of p − 1. points each counted twice. Taking as projective model of the curve the canonical curve of order 2p − 2 in space of p− 1 dimensions, whose canonical series is given by the intersection of primes, we have the number of contact primes of the curve. The 28 bitangents of a plane quartic curve, the canonical curve of genus 3, have been studied in detail since the days of Plücker. The number, 120, of tritangent planes of the sextic curve of intersection of a quadric and a cubic surface, the canonical curve of genus 4, has been obtained directly by correspondence arguments by Enriques. Enriques also remarks that the general formula 2P−1 (2p −1) is a special case of the formula of de Jonquières, which was proved, by correspondence methods, by Torelli§.

scholarly journals Cohomology of the universal smooth cubic surface

2020 ◽  
Author(s):  
Ronno Das
Keyword(s):  
Finite Field ◽  
Cubic Surface ◽  
Cubic Surfaces ◽  
Universal Family ◽  
Rational Cohomology

Abstract We compute the rational cohomology of the universal family of smooth cubic surfaces using Vassiliev’s method of simplicial resolution. Modulo embedding, the universal family has cohomology isomorphic to that of $\mathbb{P}^2$. A consequence of our theorem is that over the finite field $\mathbb{F}_q$, away from finitely many characteristics, the average number of points on a smooth cubic surface is q2+q + 1.

scholarly journals The double cover of cubic surfaces branched along their Hessian

2014 ◽  
Vol 25 (08) ◽  
pp. 1450075 ◽  
Author(s):  
Atsushi Ikeda
Keyword(s):  
Hodge Structure ◽  
Double Cover ◽  
Cubic Surface ◽  
Cubic Surfaces ◽  
Variations Of Hodge Structure

We establish a relation between the Hodge structure of the double cover of a nonsingular cubic surface branched along its Hessian and the Hodge structure of the triple cover of P3 branched along the cubic surface. Then we introduce a method to study the infinitesimal variations of Hodge structure of the double cover of the cubic surface. Using these results, we compute the Néron–Severi lattices for the double cover of a generic cubic surface and the Fermat cubic surface.

scholarly journals Automorphisms of cubic surfaces without points

2020 ◽  
Vol 31 (11) ◽  
pp. 2050083
Author(s):  
Constantin Shramov
Keyword(s):  
Automorphism Group ◽  
Finite Groups ◽  
Characteristic Zero ◽  
Automorphism Groups ◽  
Sharp Bound ◽  
Cubic Surface ◽  
Birational Transformations ◽  
Cubic Surfaces ◽  
Birational Automorphism

We classify finite groups acting by birational transformations of a nontrivial Severi–Brauer surface over a field of characteristic zero that are not conjugate to subgroups of the automorphism group. Also, we show that the automorphism group of a smooth cubic surface over a field [Formula: see text] of characteristic zero that has no [Formula: see text]-points is abelian, and find a sharp bound for the Jordan constants of birational automorphism groups of such cubic surfaces.

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