Projectivization by quotients ($\mathbb{P}(E{)}_x=\{E_x\twoheadrightarrow \ell :\dim \ell =1\}$)

This convention for a projective bundle parametrizes one-dimensional quotients of the fibres of a vector bundle $E$. Its universal quotient is ${\pi }^{\ast }E\twoheadrightarrow {\mathcal{O}}_{\mathbb{P}(E)}(1)$, and ${\pi }_{\ast }\mathcal{O}(1)=E$. 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.

On the projectivization by quotients, the fibre over a one-dimensional quotient of $E_x$ is that quotient line itself. These lines glue to the universal quotient. Its restriction to a projective fibre has degree one. It is the dual of the relative tautological line after identifying quotient projectivization of $E$ with line projectivization of $E^{\ast }$.

For this smooth projective bundle, put $L={\pi }^{\ast }{\mathcal{O}}_{{\mathbb{P}}^1}(1)$ and $M={\mathcal{O}}_{\mathbb{F}}(1)$, using projectivization by quotients. If every $a_i>0$, the complete system $|M|$ embeds it as a rational normal scroll. If every $a_i\ge 0$, the same system defines a scroll image which may be singular. Allowing arbitrary integers gives an abstract scroll; a common twist makes an embedding available. Its intersection numbers are $L^2=0$, $L{M}^{n-1}=1$, $M^n={\sum }_i{a}_i$, and its canonical divisor is $-nM+({\sum }_i{a}_i-2)L$.

The divisor class of a fibre of the ruling of a rational normal scroll has square zero. Its complete pencil recovers the projection to ${\mathbb{P}}^1$, and its intersection with $M^{n-1}$ is one.

Two rank-$n$ abstract rational normal scrolls are isomorphic exactly when their splitting degrees agree as multisets after a common integer shift. For $n\ge 3$, the primitive nef class $L$ of the ruling is intrinsically identified by $L^2=0$: restriction to a projective fibre forces every other divisor class with square zero to have zero coefficient of $M$. An isomorphism therefore preserves the ruling, and its effect on $M$ is a twist by $cL$. Pushforward gives an isomorphism of the defining bundles up to this twist; uniqueness in the Birkhoff–Grothendieck theorem recovers their splitting degrees. For surfaces the unique nef square-zero ray identifies the ruling except on ${\mathbb{P}}^1\times {\mathbb{P}}^1$, where exchanging the two rulings changes no normalized splitting type.