Composition of Points and the Mordell—Weil Problem for Cubic Surfaces

Affine cones over cubic surfaces are flexible in codimension one

Forum Mathematicum ◽

10.1515/forum-2020-0191 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Alexander Perepechko

Keyword(s):

Automorphism Group ◽

Open Subset ◽

Pezzo Surface ◽

Ample Divisor ◽

Transitive Action ◽

Del Pezzo Surface ◽

Codimension One ◽

Cubic Surfaces ◽

Special Automorphism ◽

Del Pezzo

AbstractLet Y be a smooth del Pezzo surface of degree 3 polarized by a very ample divisor that is not proportional to the anticanonical one. Then the affine cone over Y is flexible in codimension one. Equivalently, such a cone has an open subset with an infinitely transitive action of the special automorphism group on it.

Download Full-text

Cubic surfaces and their invariants: Some memories of Raymond Stora

Nuclear Physics B ◽

10.1016/j.nuclphysb.2016.05.031 ◽

2016 ◽

Vol 912 ◽

pp. 374-425 ◽

Cited By ~ 1

Author(s):

Michel Bauer

Keyword(s):

Cubic Surfaces

Download Full-text

Nodal cubic surfaces

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821050001163x ◽

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.

Download Full-text

VII. A memoir on cubic surfaces

Philosophical Transactions of the Royal Society of London ◽

10.1098/rstl.1869.0010 ◽

1869 ◽

Vol 159 ◽

pp. 231-326 ◽

Cited By ~ 24

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:

Download Full-text