The explicit factorization of the Cremona transformation which is an extension of the Nagata automorphism into elementary links

2005 ◽  
Vol 278 (7-8) ◽  
pp. 833-843
Author(s):  
Takashi Kishimoto
Keyword(s):  
Cremona Transformation

scholarly journals Special cubic Cremona transformations of ℙ6 and ℙ7

2019 ◽  
Vol 19 (2) ◽  
pp. 191-204 ◽  
Author(s):  
Giovanni Staglianò
Keyword(s):  
Cremona Transformation ◽  
Cremona Transformations ◽  
Base Locus ◽  
Famous Result

Abstract A famous result of Crauder and Katz (1989) concerns the classification of special Cremona transformations whose base locus has dimension at most two. They also proved that a special Cremona transformation with base locus of dimension three has to be one of the following: 1) a quinto-quintic transformation of ℙ5; 2) a cubo-quintic transformation of ℙ6; or 3) a quadro-quintic transformation of ℙ8. Special Cremona transformations as in Case 1) have been classified by Ein and Shepherd-Barron (1989), while in our previous work (2013), we classified special quadro-quintic Cremona transformations of ℙ8. Here we consider the problem of classifying special cubo-quintic Cremona transformations of ℙ6, concluding the classification of special Cremona transformations whose base locus has dimension three.

scholarly journals XIII.—The General Form of the Involutive 1-1 Quadric Transformation in a Plane

1905 ◽  
Vol 40 (2) ◽  
pp. 253-262
Author(s):  
Charles Tweedie
Keyword(s):  
Special Property ◽  
Cremona Transformation ◽  
Straight Line ◽  
Image Position

§ 1. In a communication read before the Society, 3rd December 1900, Dr Muir discusses the generalisation, for more than two pairs of variables, of the proposition that: IfthenIf we interpret (x, y) and (ξ, η) iis points in a plane, it is manifest that the transformation thereby obtained is a Cremona transformation. It has the special property of being reciprocal or involutive in character; i.e., if the point P is transformed into Q, then the repetition of the same transformation on Q transforms Q into P. Symbolically, if the transformation is denoted by T. T(P) = Q, and T(Q) = T2(P) = P; so that T2 = 1, and T = T−1. Moreover, if the locus of P (x, y) is a straight line, the locus of Q (ξ, η) is in general a conic.

The resolution of monoidal Cremona transformations of three-dimensional space

1938 ◽  
Vol 34 (1) ◽  
pp. 22-26
Author(s):  
D. W. Babbage
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Suitable Choice ◽  
Cremona Transformation ◽  
Cremona Transformations ◽  
Image Position ◽  
Three Dimensional Space

A Cremona transformation Tn, n′ between two three-dimensional spaces is said to be monoidal if the surfaces of order n in one space which form the homaloidal system corresponding to the planes of the second space have a fixed (n − 1)-ple point O. If the surfaces of order n′ forming the homaloidal system in the second space have a fixed (n′ − 1)-ple point O′, the transformation is said to be bimonoidal. A particularly simple bimonoidal transformation is that which transforms lines through O into lines through O′, and planes through O into planes through O′. Such a transformation we shall call an M-transformation. Its equations can, by suitable choice of coordinates, be expressed in the formwhere φn−1(x, y, z, w) = 0, φn(x, y, z, w) = 0 are monoids with vertex (0, 0, 0, 1).

DYNAMICS OF CREMONA MAPS FROM PHYSICAL MODELS

2001 ◽  
Vol 15 (24n25) ◽  
pp. 3279-3286
Author(s):  
W. SCHWALM ◽  
B. MORITZ ◽  
M. SCHWALM
Keyword(s):  
Dynamical Systems ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Physical Models ◽  
Invariant Curves ◽  
Chiral Potts Model ◽  
Cremona Transformation ◽  
Rational Mapping ◽  
Area Preserving Maps ◽  
The Matrix

A Cremona transformation X=f(x, y), Y=g(x, y) is a rational mapping (meaning that f and g are ratios of polynomials) with rational inverse x=F(X, Y), y=G(X, Y). Discrete dynamical systems defined by such transformations are well studied. They include symmetries of the Yang-Baxter equations and their generalizations. In this paper we comment on two types of dynamical systems based on Cremona transformations. The first is the P1 case of Bellon et al. which pertains to the inversion relation for the matrix of Boltzmann weights of the 4-state chiral Potts model. The resulting dynamical system decouples completely to one in a single variable. The sub case z=x corresponds to the symmetric Ashkin-Teller model. We solve this case explicitly giving orbits as closed formulas in the number n of iterations. The second type of system treated is an extension from the famous example due to McMillan of invariant curves of area preserving maps in two dimensions to the case of invariant curves and surfaces of three dimensional Cremona maps that preserve volume. The trace map of the renormalization of transmission through a Fibonacci chain, first introduced by Kohmoto, Kadanoff and Tang, is considered as an example of such a system.

scholarly journals On the Stereometric Generation of the De Jonquières Transformation

1918 ◽  
Vol 37 ◽  
pp. 48-58
Author(s):  
J. F. Tinto
Keyword(s):  
Three Dimensions ◽  
Plane Figure ◽  
Cremona Transformation ◽  
Quartic Surface ◽  
Cremona Transformations ◽  
Intrinsic Interest ◽  
Four Dimensions ◽  
Single Instance

In the geometry of the plane the logical interrelations of figures may often be rendered clearer by considering the plane to be a part of space of three dimensions. Thus, by taking the plane figure as part of a more extensive configuration in space of three dimensions, the elucidation of its properties, and in particular its relation with other figures, are often facilitated. Similarly, the figures of space of three dimensions can sometimes be treated more advantageously and compendiously by considering them as parts of figures in a space of four dimensions, and so on. As a single instance we may take Segre's elegant and powerful mode of treatment of the quartic surface which possesses a nodal conic. This surface he obtains as a projection in space of four dimensions of the quartic surface which constitutes the base of a pencil of quadratic varieties. In the following paper this mode of treatment has been applied to the interesting variety of the Cremona transformation in the plane known as the De Jonquieres transformation, a transformation which possesses some intrinsic interest in view of the fundamental rôle which it plays in the theory of Cremona Transformations. By the aid of a surface in space of three dimensions, a variety in space of four dimensions, etc., simple constructions are given for the De Jonquières transformation between two planes, between two spaces of three dimensions, etc., respectively.

Sign in / Sign up

Export Citation Format

Share Document