A rationality problem of some Cremona transformation
Resource
Proceedings of the Japan academy series A-Mathematical Sciences 84 (8): 133-137
Journal
Proceedings of the Japan Academy Series A: Mathematical Sciences
Journal Volume
84
Journal Issue
8
Pages
133-137
Date Issued
2008
Author(s)
Abstract
In this note we give a new approach to the rationality problem of some Cremona transformation. Let k be any field, k(x, y) be the rational function field of two variables over k. Let σ be a k-automorphism of k(x, y) be defined by σ(x) =-x(3x - 9y - y2)3/(27x + 2x 2 + 9xy + 2xy2 - y3)2, σ(y) = -(3x + y2(3x - 9y - y2)/27x + 2x2 + 9xy + 2xy2 - y3. Theorem. The fixed field k(x, y)〈σ〉 is rational (= purely transcendental) over k. Embodied in a new proof of the above theorem are several general guidelines for solving the rationality problem of Cremona transformations, which may be applied elsewhere. ©2008 The Japan Academy.
Type
journal article
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