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 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.

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.

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:

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.

scholarly journals Del Pezzo surfaces over finite fields and their Frobenius traces

2018 ◽  
Vol 167 (01) ◽  
pp. 35-60 ◽  
Author(s):  
BARINDER BANWAIT ◽  
FRANCESC FITÉ ◽  
DANIEL LOUGHRAN
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Del Pezzo Surfaces ◽  
Cubic Surface ◽  
Inverse Galois Problem ◽  
Complete Answer ◽  
Cubic Surfaces ◽  
Analogous Question ◽  
Special Cases ◽  
Del Pezzo

AbstractLet S be a smooth cubic surface over a finite field $\mathbb{F}$q. It is known that #S($\mathbb{F}$q) = 1 + aq + q2 for some a ∈ {−2, −1, 0, 1, 2, 3, 4, 5, 7}. Serre has asked which values of a can arise for a given q. Building on special cases treated by Swinnerton–Dyer, we give a complete answer to this question. We also answer the analogous question for other del Pezzo surfaces, and consider the inverse Galois problem for del Pezzo surfaces over finite fields. Finally we give a corrected version of Manin's and Swinnerton–Dyer's tables on cubic surfaces over finite fields.

scholarly journals The nodal cubic surfaces and the surfaces from which they are derived by projection

1928 ◽  
Vol 119 (782) ◽  
pp. 213-248 ◽  
Keyword(s):  
Detailed Examination ◽  
Singular Surface ◽  
Ruled Surfaces ◽  
Singular Surfaces ◽  
Cubic Surfaces ◽  
The Subject ◽  
Geometrical Significance

The cubic surfaces have been classified according to the character of their singularities by Schlafli and by Cayley, who find that there are 21 types in addition to ruled surfaces. In their treatment of the matter each case is considered separately by algebraical methods, and there is a marked lack of any simple unifying principle, which it is the object of this paper to supply. A means whereby this can be done is suggested by the theorem that every surface is the projection of a non-singular surface in higher space. The considerations employed in the proof of this result are somewhat abstruse, and the purely geometrical significance is obscure, so that the more detailed examination of particular cases is of genuine interest. Accordingly, the subject of this paper is the generation of the various nodal cubic surfaces by the projection of non-singular surfaces, specifically the non-ruled surfaces of order n in space of n dimensions (denoted throughout by F n ); it will be shown that these arise by the projection of the same surface, F 9 .

scholarly journals Automorphisms of cubic surfaces in positive characteristic

2019 ◽  
Vol 83 (3) ◽  
pp. 15-92
Author(s):  
Игорь Владимирович Долгачев ◽  
Igor Vladimirovich Dolgachev ◽  
Alexander Duncan ◽  
Alexander Duncan
Keyword(s):  
Isomorphism Class ◽  
Normal Forms ◽  
Positive Characteristic ◽  
Automorphism Groups ◽  
Del Pezzo Surfaces ◽  
Closed Field ◽  
Intermediate Step ◽  
Cubic Surface ◽  
Cubic Surfaces ◽  
Del Pezzo

We classify all possible automorphism groups of smooth cubic surfaces over an algebraically closed field of arbitrary characteristic. As an intermediate step we also classify automorphism groups of quartic del Pezzo surfaces. We show that the moduli space of smooth cubic surfaces is rational in every characteristic, determine the dimensions of the strata admitting each possible isomorphism class of automorphism group, and find explicit normal forms in each case. Finally, we completely characterize when a smooth cubic surface in positive characteristic, together with a group action, can be lifted to characteristic zero. Bibliography: 29 titles.

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