Cremona transformations and the conundrum of dimensionality and signature of macro-spacetime
The paper under review has a deep philosophical motivation: understanding why (apparently) our universe has four dimensions and a \((-,+,+,+)\) signature, and why the time arrow corresponds to the negatively defined direction. These facts do not seem to derive from the first principles of Physics (they are some ``observationally fixed'' parameters). The author constructs a theory of pencil-generated spacetimes. The spatial coordinates are identified with pencils of lines and the time dimension is identified with a pencil of conics, all arranged in the three-dimensional projective space. The separation of spatial and time-like coordinates is linked with the ``quadro-cubic'' Cremona transformations of the projective space. In the author's opinion, the algebro-geometric classifications of pencils suggest a plausible explanation for the fundamental physical and philosophical problems described above.
scientific article; zbMATH DE number 1755327