= Solution
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,
$$
D=mE+\Gamma,\quad m\ge2,\quad
E^2=0,\quad E\Gamma=1,\quad\Gamma^2=-2,
\qquad |D|=\Gamma+|mE|.
$$
Otherwise $|D|$ is <basepoint-free>. There are no isolated base points left after excluding the fixed-section case; this is the https://arxiv.org/pdf/alg-geom/9602006[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 by
$$
D^2=2,\qquad\text{or}\qquad
E^2=0,\ E\text{ an elliptic pencil},\ DE=2,
\qquad\text{or}\qquad D=2B,\ B^2=2,\ |B|\text{ basepoint-free}.
$$
Their 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 https://www.math.uni-bonn.de/people/huybrech/K3Global.pdf[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 $g\ge5$, trigonal sections come from an elliptic pencil with $DE=3$; the plane-quintic exception has
$$
g=6,\qquad D=2B+\Gamma,\qquad
B^2=2,\ B\Gamma=1,\ \Gamma^2=-2,\quad |B|\text{ basepoint-free}.
$$
All other birational cases have <homogeneous ideal> generated by quadrics. For $g\ge4$, 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 https://arxiv.org/pdf/alg-geom/9602006[plane-quintic exception to the third dichotomy]; the higher-genus trigonal conclusion is also stated in https://web.math.ucsb.edu/~drm/conferences/ITP99/morrison/cortona.pdf[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.”
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