Quadro-quartic Cremona transformations and four-dimensional pencil-space-times with the reverse signature

This paper is concerned with a couple of quadro-quartic Cremona transformations and their associated fundamental pencil-space-times. Both space-times are found to have four dimensions. Their signature is, however, reverse to what we observe, i.e. they feature three time dimensions and a single space coordinate only. The global structure of the three temporal dimensions is briefly examined. In a generic case, all the three are seen to be structurally equivalent. In a particular case, they exhibit an intriguing (2+1) split-up. Strikingly, this is found to completely parallel the quadro-cubic case, except for the swapped roles of space and time.