Projective bundle
= Projective bundle
= Projective-space bundle
{synonym}
The projective bundle of a real or complex <vector bundle> $E\to X$ has fiber the space of one-dimensional subspaces of $E_x$. It carries a <tautological bundle> $\lambda\subset p^*E$; in the real case this is the <projectivization of a real vector bundle>.
The same construction applies to an algebraic <vector bundle> over an <algebraic variety>: local trivializations glue products with <projective space>. In the lines convention its universal <tautological bundle> is a <line bundle>; the alternative <projectivization by quotients> convention parametrizes one-dimensional quotient spaces. Both conventions give dimension $\dim X+\operatorname{rank}(E)-1$ and a <proper morphism> to the base.