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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.”

The complete system of quadratic forms maps ${\mathbb{P}}^2$ to a smooth surface of degree four in ${\mathbb{P}}^5$. The six coordinates are $x_0^2,x_0{x}_1,x_0{x}_2,x_1^2,x_1{x}_2,x_2^2$. It occurs as a degree-two image of a K3 surface and, after taking a projective cone, in the plane-quintic exception to the K3 linear system dichotomies.