A rationality problem of some Cremona transformation

AbstractWe consider the problem of whether the norm one torus defined by a finite separable field extensionK/kis stably (or retract) rational overk. This has already been solved for the case whereK/kis a Galois extension. In this paper, we solve the problem for the case whereK/kis a non-Galois extension such that the Galois group of the Galois closure ofK/kis nilpotent or metacyclic.

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

Mathematische Nachrichten ◽

10.1002/mana.200310276 ◽

2005 ◽

Vol 278 (7-8) ◽

pp. 833-843

Author(s):

Takashi Kishimoto

Keyword(s):

Cremona Transformation

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Special cubic Cremona transformations of ℙ6 and ℙ7

Advances in Geometry ◽

10.1515/advgeom-2018-0001 ◽

2019 ◽

Vol 19 (2) ◽

pp. 191-204 ◽

Cited By ~ 1

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.

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

Transactions of the Royal Society of Edinburgh ◽

10.1017/s0080456800034311 ◽

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.

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Kurt Baier on Reason and Morality

Dialogue ◽

10.1017/s0012217300017686 ◽

1997 ◽

Vol 36 (4) ◽

pp. 813-818

Author(s):

Ishtiyaque Haji

Keyword(s):

Moral Order ◽

Necessary Condition ◽

Moral Conduct ◽

Rationality Problem ◽

Do So

The Rational and the Moral Order, a work of sweeping scope and depth, opens with three problems: the Rationality Problem, briefly, is that the following set is inconsistent, although each of its elements seems true: our conduct cannot be rationally justified unless it promotes our own good; moral conduct is rationally justified; but morality often requires that we do things that do not promote our own good. The Motivation Problem distills to this: can something be a reason for someone to do something without its actually motivating him to do so (the so-called “externalist” position), or is being a motivator a necessary condition of being a reason for that person (the “internalist position”)? Finally, the Sanction Problem notes that, although it seems plausible and generally accepted that immorality should be sanctioned, it seems neither plausible, nor is it generally accepted, that irrationality should be. Why this asymmetry? I restrict my attention, in what follows, to aspects of Baier's fascinating discussion on the Rationality Problem and the Motivation Problem.

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