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The rank-two bundle of fibre spans is globally generated and splits as $\mathcal{O}(a_1)\oplus \mathcal{O}(a_2)$ with $a_1+a_2=g-2$. On its rational normal scroll, $L{|}_C$ is the degree-three pencil and $M{|}_C=K_C$. The adjunction formula then forces the displayed divisor class. Writing $a=a_2-a_1$ and $M=B+a_{2}L$ on the Hirzebruch surface gives $C\in |3B+(a+a_2+2)L|$.

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.

Use projectivization by quotients, and setThe fibre over $t$ parametrizes one-dimensional quotients of $E_t$, so it is ${\mathbb{P}}^{n-1}$. The standard line bundles areThus $L$ is the fiber class of a rational normal scroll of the ruling and $M$ is the universal quotient line bundle of ${\pi }^{\ast }E$. In additive divisor notation we also write $L,M$ for their first Chern classes. For $n\ge 2$ these generate the Picard group; for $n=1$, $F={\mathbb{P}}^1$ and $M=a_{1}L$. This fixes the sign convention for the rational normal scroll.

Put $s={\sum }_i{a}_i$. The projective bundle formula for Chow groups givesHere the first relation comes from the one-dimensional base, and the second is the projective-bundle relation with $c_1(E)=sL$ and higher Chern classes zero on the base. Integration over a fibre gives $L{M}^{n-1}=1$. Therefore all top-degree mixed intersection numbers areIn particular, for integers $u,v$,For the canonical divisor of a smooth variety, the relative Euler sequencehas determinant ${\mathcal{O}}_F(nM-sL)$. Combining its dual determinant with $K_{{\mathbb{P}}^1}=-2L$ gives

A common twist does not change the abstract projective bundle. Choose an integer $b$ such that $d_i=a_i+b\ge 1$ for every $i$. The complete linear system of divisors $|M+bL|$ hasIts map to ${\mathbb{P}}^N$, where $N=s+nb+n-1$, has coordinate blocksHere $[s_0:t_0]$ is a point of the base and the ${\lambda }_i$ are quotient coordinates in a local trivialization of $E(b)$. These expressions have the correct transition factors and define the global map. Any nonzero block recovers the base point through its degree-$d_i$ rational normal curve. Once that point is known, the blocks recover the projective fibre coordinates. On the open where the first coordinate of a nonzero block is nonzero, the ratio of the next coordinate to the first recovers $t_0/s_0$; the analogous last-coordinate chart recovers $s_0/t_0$. The remaining coordinate ratios then recover the fibre coordinates regularly. Thus the map identifies $F$ with its closed image and is a projective embedding, not merely a pointwise injection.

The hyperplane class on this image is $M+bL$, soConsequently every such natural embedding hasThe tautological choice $b=0$ is an embedding when all $a_i>0$. If the $a_i$ are only nonnegative, $|M|$ is basepoint-free but can contract the projective subbundle arising from zero summands; one must distinguish that scroll image from the smooth abstract scroll.

Take $D$ nef and big on a complex K3 surface, and put $g=1+D^2/2$. The K3 linear system dichotomies are successive alternatives, so each later statement assumes that the exceptional cases in the earlier ones have been excluded.

The first dichotomy separates basepoint-free systems from monogonal linear systems on a K3 surface. In the exceptional case,Otherwise $|D|$ is basepoint-free. There are no isolated base points left after excluding the fixed-section case; this is the first dichotomy.

For a basepoint-free system, the second dichotomy separates hyperelliptic linear systems on a K3 surface from birational maps to normal images. The degree-two cases are characterized byTheir images are surfaces of minimal degree: the plane, a rational normal scroll, or the Veronese surface. In the birational case, the map is an embedding away from the $(-2)$-curves $\Gamma$ with $D\Gamma =0$, which it contracts to Du Val singularities. Thus “birational” does not assert an embedding of the original surface. These are the degree-two and birational projective-model alternatives.

In the third dichotomy, the birational image is either defined scheme-theoretically by quadrics or has an exceptional cubic requirement. The exceptional threefold containing it is ${\mathbb{P}}^3$ for $g=3$, a quadric hypersurface in ${\mathbb{P}}^4$ for $g=4$, a rational normal threefold scroll in the higher-genus trigonal case, or the cone over the Veronese surface in the plane-quintic case. For $g\ge 5$, trigonal sections come from an elliptic pencil with $DE=3$; the plane-quintic exception hasAll other birational cases have homogeneous ideal generated by quadrics. For $g\ge 4$, the exceptional ideals are generated by quadrics and cubics. For $g=3$, the image is a quartic hypersurface in ${\mathbb{P}}^3$ and its ideal needs its degree-four defining equation; that case must not be included in the cubic-generation assertion. The geometric alternatives, including the low-genus exceptions, are the plane-quintic exception to the third dichotomy; the higher-genus trigonal conclusion is also stated in the projective-model theorem.

The point of the final exception is that a plane quintic is not a trigonal curve. Its canonical model is the degree-two Veronese embedding of the plane curve, and its quadrics retain an ambient Veronese surface. Accordingly, one must include the cone-over-Veronese case instead of calling every failure of quadratic generation “trigonal.”