Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 21 / 3

For these linear systems of divisors, take a complex K3 surface $X$ and a nef big divisor $D$, so $D^2>0$. The exceptional terminology refers to the complete system $|D|$, rather than to an arbitrary subsystem.

A monogonal linear system on a K3 surface has the formwhere $|E|$ is an elliptic pencil on a K3 surface and $\Gamma$ is its section. Its intersection with an elliptic fibre is one, and its fixed part is $\Gamma$. For a concrete existence example, take the smooth Fermat quartic surface $X\subseteq {\mathbb{P}}^3$ and two skew lines on it. Such lines can be written aswith ${\zeta }^4=({\zeta }^{\prime }{)}^4={\eta }^4=({\eta }^{\prime }{)}^4=-1$, $\zeta \ne {\zeta }^{\prime }$ and $\eta \ne {\eta }^{\prime }$. The equations show that they are disjoint, and the four partial derivatives show that the quartic is smooth. The adjunction formula gives $K_X=0$, and the hypersurface exact sequence of sheaves $0\to {\mathcal{O}}_{{\mathbb{P}}^3}(-4)\to {\mathcal{O}}_{{\mathbb{P}}^3}\to {\mathcal{O}}_X\to 0$, together with the intermediate cohomology vanishing of these twists, gives $H^1(X,{\mathcal{O}}_X)=0$. Thus it is a K3 surface, and each line has square $-2$. If $H$ is the hyperplane class, the planes through ${\Gamma }_0$ cut out ${\Gamma }_0$ plus a residual cubic pencil $|E|$, $E=H-{\Gamma }_0$. This pencil has no base point off ${\Gamma }_0$. On ${\Gamma }_0$, its two local residual equations are the two normal derivatives of the quartic equation; they cannot vanish simultaneously because $X$ is smooth. Thus it is basepoint-free, withThe degree-one map of $\Gamma$ to the pencil base makes it a section and forces connected fibres. Bertini smoothness theorem and adjunction formula then make a general fibre a smooth genus-one curve.

Now choose $D=mE+\Gamma$. For every irreducible curve other than $\Gamma$, both $E$ and the effective curve $\Gamma$ have nonnegative intersection with it. Also $D\Gamma =m-2\ge 0$, so $D$ is nef, and $D^2=2m-2>0$. On a general elliptic fibre, the restriction of ${\mathcal{O}}_X(D)$ is the degree-one bundle of the section point. Its unique section vanishes at that point. Every global section of ${\mathcal{O}}_X(D)$ therefore vanishes along a dense subset of $\Gamma$, hence along $\Gamma$ itself. Removing this fixed divisor gives exactly $|mE|$. This verifies the monogonal example directly; its general divisor is a section plus $m$ fibres.

A hyperelliptic linear system on a K3 surface is a basepoint-free nef big system whose morphism is generically two-to-one. Equivalently its general smooth curve is hyperelliptic. Let $\pi :X\to {\mathbb{P}}^2$ be a double cover branched over a smooth sextic, and put $D={\pi }^{\ast }\mathcal{O}(1)$. The double-cover construction givesThus $X$ is a K3 surface. Further,since the second summand contributes $H^0(\mathcal{O}(-2))=0$. The map given by $|D|$ is precisely the double cover. A general member is the double cover of a line branched at six points, so Riemann-Hurwitz formula gives genus two. This is the hyperelliptic example.

A trigonal linear system on a K3 surface is a basepoint-free nonhyperelliptic nef big system whose general smooth curve has gonality three. For an example, take the same smooth quartic with a line ${\Gamma }_0$ and the hyperplane system $|H|$. Its general member is a smooth plane quartic of genus three; adjunction formula identifies its plane embedding with its canonical map, so it is not hyperelliptic. The elliptic pencil $E=H-{\Gamma }_0$ hasIts restriction to a general hyperplane curve is a basepoint-free degree-three pencil. The curve is neither rational nor hyperelliptic, so its gonality is exactly three. This verifies the trigonal example.

Thus the three examples exhibit respectively a fixed section with elliptic fibres, a double-plane map, and an embedded quartic with trigonal hyperplane sections. The last example also illustrates why low-genus trigonal sections should not be confused with a blanket numerical criterion for every higher-genus model.

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.”