= Isomorphism classification of abstract rational scrolls
{title2=$\{a_i\}_{\mathrm{mult}}=\{b_i+c\}_{\mathrm{mult}}$}
Two rank-$n$ abstract <rational normal scrolls> are <isomorphic> exactly when their splitting degrees agree as multisets after a common integer shift. For $n\ge3$, 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.
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