On 'spatially anisotropic' pencil-space-times associated with a quadro-cubic Cremona transformation

This paper deals with a particular type of quadro-cubic Cremona transformations and its associated fundamental pencil-space-time. The transformation is generated by a homaloidal web of quadrics which share, apart from a real line, three isolated real points of which one falls on the line in question. It is shown that the corresponding pencil-space-time exhibits the same dimensionality (4) and signature (3 + 1) as that characterizing a generic case, yet its spatial dimensions are not all equivalent; one of them is found to stand on a slightly different footing than the other two. A brief account of possible Riemannian, Cantorian space E-(infinity) and heterotic string space-time parallels of this intriguing spatial anisotropy is given. (C) 2001 Elsevier Science Ltd. All rights reserved.