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.

On decidable varieties of Heyting algebras

1992 ◽  
Vol 57 (3) ◽  
pp. 988-991 ◽  
Author(s):  
Devdatt P. Dubhashi
Keyword(s):  
Order Theory ◽  
Boolean Algebras ◽  
Semantic Interpretation ◽  
Structure Theory ◽  
Heyting Algebras ◽  
Quantitative Classification ◽  
First Order ◽  
First Order Theory ◽  
Famous Result

In this paper we present a new proof of a decidability result for the firstorder theories of certain subvarieties of Heyting algebras. By a famous result of Grzegorczyk, the full first-order theory of Heyting algebras is undecidable. In contrast, the first-order theory of Boolean algebras and of many interesting subvarieties of Boolean algebras is decidable by a result of Tarski [8]. In fact, Kozen [6] gives a comprehensive quantitative classification of the complexities of the first-order theories of various subclasses of Boolean algebras (including the full variety).This stark contrast may be reconciled from the standpoint of universal algebra as arising out of the byplay between structure and decidability: A good structure theory entails positive decidability results. Boolean algebras have a well-developed structure theory [5], while the corresponding theory for Heyting algebras is quite meagre. Viewed in this way, we may hope to obtain decidability results if we focus attention on subclasses of Heyting algebras with good structural properties.K. Idziak and P. M. Idziak [4] have considered an interesting subvariety of Heyting algebras, , which is the variety generated by all linearly-ordered Heyting algebras. This variety is shown to be the largest subvariety of Heyting algebras with a decidable theory of its finite members. However their proof is rather indirect, proceeding via semantic interpretation into the monadic second order theory of trees. The latter is a powerful theory—it interprets many other theories—but is computationally highly infeasible. In fact, by a celebrated theorem of Rabin, its complexity is not bounded by any elementary recursive function. Consequently, the proof of [4], besides being indirect, also gives no information on the quantitative computational complexity of the theory of .Here we pursue the theme of structure and decidability. We isolate the indecomposable algebras in and use this to prove a theorem on the structure of if -algebras. This theorem relates the -algebras structurally to Boolean algebras. This enables us to bootstrap the known decidability results for Boolean algebras to the variety if .

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).

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.

The XJC-correspondence

2016 ◽  
Vol 2016 (716) ◽  
Author(s):  
Luc Pirio ◽  
Francesco Russo
Keyword(s):  
Cremona Transformations

AbstractFor anyThese three equivalences form what we call theWe also provide some applications to the classification of particular types of varieties in the class defined above and of quadro-quadric Cremona transformations.

scholarly journals On planar Cremona maps of prime order

2004 ◽  
Vol 174 ◽  
pp. 1-28 ◽  
Author(s):  
Tommaso de Fernex
Keyword(s):  
Prime Order ◽  
Higher Dimensions ◽  
Conjugacy Classes ◽  
Geometric Constructions ◽  
Cremona Transformations ◽  
Rational Surfaces ◽  
Mori Theory ◽  
First Results

AbstractThis paper contains a new proof of the classification of prime order elements of Bir(ℙ2) up to conjugation. The first results on this topic can be traced back to classic works by Bertini and Kantor, among others. The innovation introduced by this paper consists of explicit geometric constructions of these Cremona transformations and the parameterization of their conjugacy classes. The methods employed here are inspired to [4], and rely on the reduction of the problem to classifying prime order automorphisms of rational surfaces. This classification is completed by combining equivariant Mori theory to the analysis of the action on anticanonical rings, which leads to characterize the cases that occur by explicit equations (see [28] for a different approach). Analogous constructions in higher dimensions are also discussed.

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.

QUADRO-QUADRIC CREMONA MAPS AND VARIETIES 3-CONNECTED BY CUBICS: SEMI-SIMPLE PART AND RADICAL

2013 ◽  
Vol 24 (13) ◽  
pp. 1350105 ◽  
Author(s):  
LUC PIRIO ◽  
FRANCESCO RUSSO
Keyword(s):  
Jordan Algebras ◽  
Cremona Transformation ◽  
Cremona Transformations ◽  
Simple Part ◽  
Rationally Connected ◽  
Structure Theorems ◽  
Twisted Cubics

Via the XJC-correspondence proved in [L. Pirio and F. Russo, Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and rank 3 Jordan algebras, submitted] we provide some structure theorems for quadro-quadric Cremona transformations and for extremal varieties 3-covered by twisted cubics by reinterpreting for these objects the algebraic results on the solvability of the radical of Jordan algebras. In this way, we can define the semi-simple part and the radical part of a quadro-quadric Cremona transformation, respectively of an extremal variety 3-covered by twisted cubics, and then describe how general objects are constructed from the semi-simple ones, which are completely classified modulo certain equivalences, via suitable null radical extensions.

Some Quartic Primals and Associated Cremona Transformations of Four-Dimensional Space

1936 ◽  
Vol 32 (1) ◽  
pp. 12-22 ◽  
Author(s):  
D. W. Babbage
Keyword(s):  
Dimensional Space ◽  
The Other ◽  
Subsequent Paper ◽  
Cremona Transformation ◽  
Cremona Transformations

The object of this paper is to draw attention to a series of five types of rational quartic primal in [4]. Two of these are already known. The greater part of this paper deals with one of the other three types of primal and with a new symmetrical quarto-quartic Cremona transformation of [4] determined by a homaloidal system of primals of this type passing through a certain surface of order nine. It is hoped in a subsequent paper to continue the investigation into the remaining two types of primal.

Some effectivity questions for plane Cremona transformations in the context of symmetric key cryptography

2021 ◽  
Vol 64 (1) ◽  
pp. 1-28
Author(s):  
N. I. Shepherd-Barron
Keyword(s):  
Lower Bound ◽  
Hyperbolic Plane ◽  
Cremona Transformation ◽  
Cremona Transformations ◽  
Henon Maps ◽  
Hénon Maps ◽  
Symmetric Key ◽  
Symmetric Key Cryptography ◽  
Feistel Ciphers ◽  
Plane Cremona Transformations

An effective lower bound on the entropy of some explicit quadratic plane Cremona transformations is given. The motivation is that such transformations (Hénon maps, or Feistel ciphers) are used in symmetric key cryptography. Moreover, a hyperbolic plane Cremona transformation g is rigid, in the sense of [5], and under further explicit conditions some power of g is tight.

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), with S a rational surface and ℒ a linear system on S. We give a classification theorem for such pairs, and we determine, for each irreducible plane curve B, its Cremona minimal models, that is, those plane curves which are equivalent to B via a Cremona transformation and have minimal degree under this condition.

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