Relative hyperplane line bundle ($L=S^{\ast },L{|}_{\mathbb{P}(E_x)}=\mathcal{O}(1)$)

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.