scholarly journals On degrees and genera of curves on smooth quartic surfaces in P3

1984 ◽  
Vol 96 ◽  
pp. 127-132 ◽  
Author(s):  
Shigefumi Mori
Keyword(s):  
Singular Curve ◽  
Quartic Surface ◽  
Quartic Surfaces

Our result is motivated by the results [GP] of Gruson and Peskin on characterization of the pair of degree d and genus g of a non-singular curve in P3. In the last step, they construct the required curve C on a singular quartic surface when Here we consider curves on smooth quartic surfaces.

scholarly journals A type of periodicity of certain quartic surfaces

1942 ◽  
Vol 7 (1) ◽  
pp. 73-80 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Present Note ◽  
Quartic Surface ◽  
Periodic Property ◽  
A Chain ◽  
Quartic Surfaces ◽  
Do So

The following pages have been written in consequence of reading some paragraphs by Reye, in which he obtains, from a quartic surface, a chain of contravariant quartic envelopes and of covariant quartic loci. This chain is, in general, unending; but Reye at once foresaw the possibility of the quartic surface being such that the chain would be periodic. The only example which he gave of periodicity being realised was that in which the quartic surface was a repeated quadric. It is reasonable to suppose that, had he been able to do so, he would have chosen some surface which had the periodic property without being degenerate; in the present note two such surfaces are signalised.

scholarly journals Matrices of integers associated with self-transformations of surfaces

1948 ◽  
Vol 193 (1032) ◽  
pp. 25-43
Keyword(s):  
Rational Curve ◽  
Pell Equation ◽  
Quartic Surface ◽  
Twisted Cubic ◽  
Cubic Curve ◽  
The Matrix ◽  
Quartic Surfaces ◽  
Infinite Sequences ◽  
Square Matrix

Let c = ( c 1 , . . . , c r ) be a set of curves forming a minimum base on a surface, which, under a self-transformation, T , of the surface, transforms into a set T c expressible by the equivalences T c = Tc, where T is a square matrix of integers. Further, let the numbers of common points of pairs of the curves, c i , c j , be written as a symmetrical square matrix Г. Then the matrix T satisfies the equation TГT' = Г. The significance of solutions of this equation for a given matrix Г is discussed, and the following special surfaces are investigated: §§4-7. Surfaces, in particular quartic surfaces, wìth only two base curves. Self-transformations of these depend on the solutions of the Pell equation u 2 - kv 2 = 1 (or 4). §8. The quartic surface specialized only by being made to contain a twisted cubic curve. This surface has an involutory transformation determined by chords of the cubic, and has only one other rational curve on it, namely, the transform of the cubic. The appropriate Pell equation is u 2 - 17 v 2 = 4. §9. The quartic surface specialized only by being made to contain a line and a rational curve of order m to which the line is ( m - 1)⋅secant (for m = 1 the surface is made to contain two skew lines). The surface has two infinite sequences of self-transformations, expressible in terms of two transformations R and S , namely, a sequence of involutory transformations R S n , and a sequence of non-involutory transformations S n .

scholarly journals On a set of quartic surfaces in space of four dimensions; and a certain involutory transformation

1926 ◽  
Vol 110 (756) ◽  
pp. 700-708
Keyword(s):  
Present Note ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Quartic Surface ◽  
Four Dimensions ◽  
Cubic Surfaces ◽  
Mutual Relations ◽  
Quartic Surfaces ◽  
Three Dimensional Space

There exists in space of four dimensions an interesting figure of 15 lines and 15 points, first considered by Stéphanos (‘Compt. Rendus,’ vol. 93, 1881), though suggested very clearly by Cremona’s discussion of cubic surfaces in three-dimensional space. In connection with the figure of 15 lines there arises a quartic surface, the intersection of two quadrics, which is of importance as giving rise by projection to the Cyclides, as Segre has shown in detail (‘Math. Ann.,’ vol. 24, 1884). The symmetry of the figure suggests, howrever, the consideration of 15 such quartic surfaces; and it is natural to inquire as to the mutual relations of these surfaces, in particular as to their intersections. In general, two surfaces in space of four dimensions meet in a finite number of points. It appears that in this case any two of these 15 surfaces have a curve in common; it is the purpose of the present note to determine the complete intersection of any two of these 15 surfaces. Similar results may be obtained for a system of cubic surfaces in three dimensions, corresponding to those here given for this system of quartic surfaces in four dimensions, since the surfaces have one point in common, which may be taken as the centre of a projection.

scholarly journals The geometrical construction of Maschke's quartic surfaces

1945 ◽  
Vol 7 (2) ◽  
pp. 93-103 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Finite Order ◽  
Geometrical Construction ◽  
Remarkable Property ◽  
Quartic Surface ◽  
Quartic Form ◽  
Quartic Surfaces

There is, in the second (Cambridge, 1911) edition of Burnside's Theory of Groups of Finite Order, an example on p. 371 which must have aroused the curiosity of many mathematicians; a quartic surface, invariant for a group of 24.5! collineations, appears without any indication of its provenance or any explanation of its remarkable property. The example teases, whether because Burnside, if he obtained the result from elsewhere, gives no reference, or because, if the result is original with him, it is difficult to conjecture the process by which he arrived at it. But the quartic form which, when equated to zero, gives the surface, appears, together with associated forms, in a paper by Maschke1, and it is fitting therefore to call both form and surface by his name.

scholarly journals Conics on a Maschke Surface

1946 ◽  
Vol 7 (3) ◽  
pp. 153-161
Author(s):  
W. L. Edge
Keyword(s):  
Quartic Surface ◽  
Linear Complexes ◽  
Image Position ◽  
Quartic Surfaces

The six quaternary quartic formswere first obtained by Maschke; it has recently been explained that the quartic surfaces obtained by equating these forms to zero are important constituents of Klein's famous configuration derived from six linear complexes that are mutually in involution. The quartic surface Φi = 0 will be denoted, for each of the six suffixes i, by Mi.

On the quartic surface

1944 ◽  
Vol 40 (2) ◽  
pp. 121-145 ◽  
Author(s):  
B. Segre
Keyword(s):  
Geometric Characterization ◽  
Discontinuous Group ◽  
Quartic Surface ◽  
The Real ◽  
Birational Transformations ◽  
Projective Transformations ◽  
The Third ◽  
Real Domain ◽  
Different Types

Summary1. The projective transformations of F into itself … 1212. The flecnodal curve, and the lines of F … 1223. A geometric characterization of F, and the six different types of F in the real domain … 1234. The τ-points and τ-planes, and a notation for the lines of F … 1245. The incidence conditions for the lines of F … 1256. The tetrads of the first kind … 1267. The tetrads of the second kind … 1268. The pairs of lines of F … 1279. The tetrads of the third kind … 12910. The 16-tangent quadrics of F … 13011. The conics of the first kind … 13112. The conics of the second kind … 13213. No other irreducible conics lie on F … 13314. The tangent planes of F of multiplicity greater than 3 … 13615. On twisted curves, especially cubics and quartics, lying on F … 13816. The T-transformations … 13917. Construction of an infinite discontinuous group of birational transformations of F into itself … 14218. Deduction of an infinity of unicursal curves lying on F … 143

Configurations defined by six lines in space of three dimensions

1933 ◽  
Vol 29 (1) ◽  
pp. 52-68 ◽  
Author(s):  
J. A. Todd
Keyword(s):  
Linear System ◽  
Classical Problem ◽  
Three Dimensions ◽  
Quartic Surface ◽  
Lines In Space ◽  
Pass Through ◽  
Quartic Surfaces

The investigations which follow were originally suggested by the now classical problem of Cayley, the determination of the condition that seven lines in space, of which no two intersect, should lie on a quartic surface. This problem suggests the consideration of the linear system of quartic surfaces which pass through six given lines, and this, essentially, is the basis of all that follows.

scholarly journals Diagonal quartic surfaces and transcendental elements of the Brauer group

2010 ◽  
Vol 9 (4) ◽  
pp. 769-798 ◽  
Author(s):  
Evis Ieronymou
Keyword(s):  
Function Field ◽  
Sufficient Conditions ◽  
Complex Numbers ◽  
Weak Approximation ◽  
Brauer Group ◽  
Quartic Surface ◽  
Simple Algebras ◽  
Central Simple ◽  
Quartic Surfaces ◽  
Torsion Part

ABSTRACTWe exhibit central simple algebras over the function field of a diagonal quartic surface over the complex numbers that represent the 2-torsion part of its Brauer group. We investigate whether the 2-primary part of the Brauer group of a diagonal quartic surface over a number field is algebraic and give sufficient conditions for this to be the case. In the last section we give an obstruction to weak approximation due to a transcendental class on a specific diagonal quartic surface, an obstruction which cannot be explained by the algebraic Brauer group which in this case is just the constant algebras.

scholarly journals Real quartic surfaces containing16skew lines

2004 ◽  
Vol 2004 (44) ◽  
pp. 2331-2345
Author(s):  
Isidro Nieto
Keyword(s):  
Three Dimensional ◽  
Real Surface ◽  
Two Dimensional ◽  
Viewing Angle ◽  
Complex Surfaces ◽  
Quartic Surface ◽  
The Real ◽  
Visual Properties ◽  
Quartic Surfaces

It is well known that there is an open three-dimensional subvarietyMsof the Grassmannian of lines inℙ3which parametrizes smooth irreducible complex surfaces of degree 4 which are Heisenberg invariant, and each quartic contains 32 lines but only 16 skew lines, being determined by its configuration of lines, are called adouble 16. We consider here the problem of visualizing in a computer the real Heisenberg invariant quartic surface and the real double 16. We construct a family of pointsl∈Msparametrized by a two-dimensional semialgebraic variety such that under a change of coordinates oflinto its Plüecker, coordinates transform into the real coordinates for a lineLinℙ3, which is then used to construct a program in Maple 7. The program allows us to draw the quartic surface and the set of transversal lines toL. Additionally, we include a table of a group of examples. For each test example we specify a parameter, the viewing angle of the image, compilation time, and other visual properties of the real surface and its real double 16. We include at the end of the paper an example showing the surface containing the double 16.

Surface From Circle Rotation

2017 ◽  
Vol 5 (1) ◽  
pp. 32-35 ◽  
Author(s):  
Гирш ◽  
A. Girsh
Keyword(s):  
Real World ◽  
Rotation Axis ◽  
Geometric Constructions ◽  
Quartic Surface ◽  
The Real ◽  
Circle Plane ◽  
Large Numbers ◽  
Circle Rotation ◽  
Arbitrary Axis ◽  
Quartic Surfaces

Descriptive geometry, as the elementary one, studies the real world by its abstractions. But Euclid’s geometry of the real world is conjugated to pseudo-Euclidean geometry, and they make a conjugated pair. As a consequence, each real figure is conjugated with some imaginary pattern. This paper apart from some science facts demonstrates the presence of imaginary patterns in geometric constructions, where the imaginary patterns manifest themselves as singularities or as geometrically imaginary points (GIP) in “Real — Imaginary” conjugate pairs. The study is conducted, as a rule, from simple to complex, from particulars to generals. Rotation of a circle around an arbitrary axis generates, in the general case, a quartic surface. Among the quartic surfaces are a circular torus and a sphere as a special case of the torus. The torus is obtained from the circle rotation around an axis lying in the circle plane. If the axis does not intersect the generating circle, then the surface is called an open torus; when the axis intersects the generating circle, then the surface is called a closed torus; when the rotation axis passes through the center of the generating circle, then the surface is a sphere. The open torus is associated with a bagel, and the closed one — with an apple. The torus is a perfect example for the application of two well-known Guldin’s formulas. Next, the imaginary torus support is considered in this paper, at the end of which the sphere and its imaginary sup - port are considered. Imaginary patterns lead to the complex numbers, in regards to which grieved the great J. Steiner, calling them "hieroglyphs of analysis". But imaginary patterns exist apart from analysis formulas — they are the part of geometry. J.V. Poncelet was the first who understood the imaginary points in 1812, being in Russian captivity in Saratov and, what is important, without analysis formulas at all. Computational geometry often shows quantities, large numbers of real figures, because it takes into account the imaginary images too.

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