Relative hyperplane line bundle
= Relative hyperplane line bundle
{title2=$L=S^*,\quad L|_{\mathbb P(E_x)}=\mathcal O(1)$}
Dualize the <relative tautological line bundle> to obtain the positive relative hyperplane <line bundle>. It restricts to $\mathcal O(1)$ on each projective fibre. This sign depends on the convention that projectivization parametrizes lines; fixing the convention avoids confusing it with the tautological line itself.