Rational normal octavic surfaces with a double line, in space of five dimensions

1933 ◽  
Vol 29 (1) ◽  
pp. 95-102 ◽  
Author(s):  
D. W. Babbage
Keyword(s):  
Rational Surface ◽  
Simple Rule ◽  
Plane Curves ◽  
Double Line ◽  
Five Dimensions ◽  
Multiple Line ◽  
Image Position

The following paper arises from a remark in a recent paper by Professor Baker. In that paper he gives a simple rule, under which a rational surface has a multiple line, expressed in terms of the system of plane curves which represent the prime sections of the surface. The rule is that, if one system of representing curves is given by an equation of the formthe surface being given, in space (x0, x1,…, xr), by the equationsthen the surface contains the linecorresponding to the curve φ = 0; and if the curve φ = 0 has genus q, this line is of multiplicity q + 1.

XVIII.—The Invariant Theory of Forms in Six Variables relating to the Line Complex

1927 ◽  
Vol 46 ◽  
pp. 210-222 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Invariant Theory ◽  
Line Geometry ◽  
Five Dimensions ◽  
Ordinary Space ◽  
Homogeneous Coordinates ◽  
Plücker Coordinates ◽  
Algebraic Theories ◽  
Straight Line ◽  
Image Position ◽  
Theory Of Forms

It is well known that the Plücker coordinates of a straight line in ordinary space satisfy a quadratic identitywhich may also be considered as the equation of a point-quadric in five dimensions, if the six coordinates Pij are treated as six homogeneous coordinates of a point. Projective properties of line geometry may therefore be treated as projective properties of point geometry in five dimensions. This suggests that certain algebraic theories of quaternary forms (corresponding to the geometry of ordinary space) can best be treated as algebraic theories of senary forms: that is, forms in six homogeneous variables.

A Note on the Multiplicity of Cohen-Macaulay Algebras with Pure Resolutions

1985 ◽  
Vol 37 (6) ◽  
pp. 1149-1162 ◽  
Author(s):  
Craig Huneke ◽  
Matthew Miller
Keyword(s):  
Tangent Cone ◽  
Betti Numbers ◽  
Rational Surface ◽  
Minimal Resolution ◽  
Homogeneous Ideal ◽  
Elliptic Singularity ◽  
Rational Surface Singularity ◽  
Generic Matrix ◽  
Image Position ◽  
Coordinate Rings

Let R = k[X1, …, Xn] with k a field, and let I ⊂ R be a homogeneous ideal. The algebra R/I is said to have a pure resolution if its homogeneous minimal resolution has the formSome of the known examples of pure resolutions include the coordinate rings of: the tangent cone of a minimally elliptic singularity or a rational surface singularity [15], a variety defined by generic maximal Pfaffians [2], a variety defined by maximal minors of a generic matrix [3], a variety defined by the submaximal minors of a generic square matrix [6], and certain of the Segre-Veronese varieties [1].If I is in addition Cohen-Macaulay, then Herzog and Kühl have shown that the betti numbers bi are completely determined by the twists di.

Fast Algorithms of Line Outage Distribution Factors with Cascading Failures

2014 ◽  
Vol 1070-1072 ◽  
pp. 843-848
Author(s):  
Xiao Long Fan ◽  
Jin Quan Zhao
Keyword(s):  
Fast Algorithms ◽  
Computationally Efficient ◽  
Double Line ◽  
Cascading Outages ◽  
Security Applications ◽  
Multiple Line ◽  
Recursive Theory ◽  
Efficient Expression ◽  
Line Outages

The prominent role of cascading outages in recent blackouts has created a need in security applications for evaluating line outage distribution factors (LODFs) under the multiple-line outages. Two fast algorithms of LODFs with multiple-line outages are proposed in this paper. In the first method, the double-line outage LODFs are expressed in terms of single-line outage LODFs, and it can be extended to N-k (k≥2) contingencies without any complex matrix operation through the recursive theory. In the second method, a computationally efficient expression of LODFs based on power transfer distribution factors (PTDFs) in pre-contingency network is presented. Numerical simulations are carried out on IEEE 14 and 118-bus test systems. The results show that both methods can effectively improve the computation speed of multiple-line outage LODFs.

III.—Double Binary Forms

1924 ◽  
Vol 43 ◽  
pp. 43-50 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Integral Function ◽  
Quarterly Journal ◽  
Plane Curves ◽  
Binary Forms ◽  
Cartesian Coordinates ◽  
German School ◽  
Symbolic Methods ◽  
Image Position ◽  
Rational Integral

In the Quarterly Journal, No. 162, 1910, Professor A. R. Forsyth considered some of the problems arising from a homographic transformation of plane curves whose equations could be written in the formwhere F is a rational integral function of z and z′, and where z = x + iy, z′ = x − iy determine the rectangular Cartesian coordinates of the plane. It was suggested that the theory could be developed algebraically by using the symbolic methods of the German school which proved so powerful in furthering the theory of binary forms.

Classification of Algebraic Surfaces with Sectional Genus less than or Equal to Six. II: Ruled Surfaces with dim ϕKx⊗L(x) = l

1986 ◽  
Vol 38 (5) ◽  
pp. 1110-1121 ◽  
Author(s):  
Elvira Laura Livorni
Keyword(s):  
Line Bundle ◽  
Hyperplane Section ◽  
Ample Line Bundle ◽  
Rational Surface ◽  
Algebraic Surfaces ◽  
Sectional Genus ◽  
Rational Surfaces ◽  
Image Position ◽  
Smooth Hyperplane Section

Let L be a very ample line bundle on a smooth, connected, projective, ruled not rational surface X. We have considered the problem of classifying biholomorphically smooth, connected, projected, ruled, non rational surfaces X with smooth hyperplane section C such that the genus g = g(C) is less than or equal to six and dim where is the map associated to . L. Roth in [10] had given a birational classification of such surfaces. If g = 0 or 1 then X has been classified, see [8].If g = 2 ≠ hl,0(X) by [12, Lemma (2.2.2) ] it follows that X is a rational surface. Thus we can assume g ≦ 3.Since X is ruled, h2,0(X) = 0 andsee [4] and [12, p. 390].

On the fundamental group of the complement of certain singular plane curves

1987 ◽  
Vol 102 (3) ◽  
pp. 453-457 ◽  
Author(s):  
András Némethi
Keyword(s):  
Projective Space ◽  
Fundamental Group ◽  
Algebraic Curve ◽  
Plane Curves ◽  
Image Position

Let C be a complex algebraic curve in the projective space ℙ2. The purpose of this paper is to calculate the fundamental group G of the complement of C in the case when C = X ∩ H1 ∩ … ∩ Hn−2, whereand Hi are generic hyperplanes (i = 1, … n − 2).

scholarly journals The Brauer group of cubic surfaces

1993 ◽  
Vol 113 (3) ◽  
pp. 449-460 ◽  
Author(s):  
Sir Peter Swinnerton-Dyer
Keyword(s):  
Finite Group ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Rational Surface ◽  
Hasse Principle ◽  
Weak Approximation ◽  
Brauer Group ◽  
Cubic Surfaces ◽  
Image Position

1. Let V be a non-singular rational surface defined over an algebraic number field k. There is a standard conjecture that the only obstructions to the Hasse principle and to weak approximation on V are the Brauer–Manin obstructions. A prerequisite for calculating these is a knowledge of the Brauer group of V; indeed there is one such obstruction, which may however be trivial, corresponding to each element of Br V/Br k. Because k is an algebraic number field, the natural injectionis an isomorphism; so the first step in calculating the Brauer–Manin obstruction is to calculate the finite group H1 (k), Pic .

On certain nets of plane curves

1924 ◽  
Vol 22 (1) ◽  
pp. 1-10 ◽  
Author(s):  
F. P. White
Keyword(s):  
Fixed Points ◽  
Plane Curves ◽  
Cubic Form ◽  
Straight Line ◽  
Pass Through ◽  
Image Position ◽  
Quartic Curves

1. The plane quartic curves which pass through twelve fixed points g, of which no three lie on a straight line, no six on a conic and no ten on a cubic, form a net of quartics represented by the equation

scholarly journals Systems of Plane Curves whose Orthogonals form a Similar System

1895 ◽  
Vol 20 ◽  
pp. 497-498
Author(s):  
Tait
Keyword(s):  
Perfect Fluid ◽  
Plane Curves ◽  
Similar System ◽  
Parallel Lines ◽  
Infinite Mass ◽  
Image Position

While tracing the lines of motion and the meridian sections of their orthogonal surfaces for an infinite mass of perfect fluid disturbed by a moving sphere :—the question occurred to me “When are such systems similar?” In the problem alluded to, the equations of the curves are, respectively,It was at once obvious that any sets of curves such asare orthogonals. But they form similar systems only whenHence the only sets of similar orthogonal curves, having equations of the above form, are (a) groups of parallel lines and (b) their electric images (circles touching each other at one point).

scholarly journals Birational classification of curves on rational surfaces

2010 ◽  
Vol 199 ◽  
pp. 43-93
Author(s):  
Alberto Calabri ◽  
Ciro Ciliberto
Keyword(s):  
Linear System ◽  
Plane Curve ◽  
Rational Surface ◽  
Minimal Degree ◽  
Classification Theorem ◽  
Plane Curves ◽  
Cremona Transformation ◽  
Rational Surfaces

AbstractIn this paper we consider the birational classification of pairs (S, ℒ), withSa rational surface andℒa linear system onS. We give a classification theorem for such pairs, and we determine, for each irreducible plane curveB, itsCremona minimalmodels, that is, those plane curves which are equivalent toBvia a Cremona transformation and have minimal degree under this condition.

Sign in / Sign up

Export Citation Format

Share Document