= Projectivization by quotients
{title2=$\mathbb P(E)_x=\{E_x\twoheadrightarrow\ell:\dim\ell=1\}$}
This convention for a <projective bundle> parametrizes one-dimensional quotients of the fibres of a <vector bundle> $E$. Its universal quotient is $\pi^*E\twoheadrightarrow\mathcal O_{\mathbb P(E)}(1)$, and $\pi_*\mathcal O(1)=E$. Tensoring $E$ by a <line bundle> on the base leaves the projective bundle unchanged but tensors its universal quotient by the pullback of that line bundle. The alternative convention of <holomorphic projectivization by lines> replaces $E$ by its dual and reverses the tautological sign.
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