= Sections of a projective bundle
{title2=$s:X\to\mathbb P(E)$}
A complex line subbundle $L\subset E$ defines a <section of a fiber bundle> $s:X\to\mathbb P(E)$ selecting $L_x$ at each point. Pullback of the <tautological bundle> along $s$ is $L$, so if $t$ is its <Euler class>, then $s^*t=e(L)$. For a decomposition $E=\bigoplus_iL_i$, the resulting sections and the open sets where coordinate projection is nonzero identify the tautological line with $\pi^*L_i$. These identifications support the projective-bundle factorization via a <vanishing cup product from an open cover>.
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