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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.

Let ${\mathcal{A}}^{p,q}(X)$ be the space of smooth complex differential forms of type $(p,q)$. The Dolbeault operator satisfies ${\bar{\partial }}^2=0$, and the Dolbeault cohomology isFor $q=0$ the denominator is zero; forms outside the dimension range are zero.

A sequence of sheaves is an exact sequence of sheaves precisely when its sequence of stalks at every point is exact. Here this means the first map is injective on stalks, the last is surjective on stalks, and the image equals the kernel at the middle stalk. In particular, a section of the last sheaf need only have local lifts; surjectivity on all global sections is not required.

The associated long exact sequence in sheaf cohomology of Čech cohomology, understood in the direct limit over open covers, isThe degree-zero groups are global sections. The connecting map is obtained by locally lifting a Čech cocycle to the middle sheaf and taking its Čech coboundary, which takes values in the first sheaf. Different lifts change it by a Čech coboundary.

The holomorphic de Rham complex is exact locally, which is the criterion for an exact sequence of sheaves. Work in a small star-shaped holomorphic coordinate neighbourhood in ${\mathbb{C}}^2$, centred at zero. A holomorphic function has $\partial f=0$ exactly when it is constant on this neighbourhood, giving exactness at ${\mathcal{O}}_M$ and the injection of the constant sheaf.

For exactness at ${\Omega }_M^1$, let $\alpha =a_1(z)d{z}_1+a_2(z)d{z}_2$ be holomorphic and $\partial$-closed. Then $\partial a_1/\partial z_2=\partial a_2/\partial z_1$. Define the holomorphic functionDifferentiation under the integral and the closedness identity giveThus $\partial F=\alpha$.

For exactness at ${\Omega }_M^2$, any holomorphic differential form of degree two is $\beta =b(z)d{z}_1\wedge d{z}_2$ and is automatically $\partial$-closed by dimension. DefineIts derivative isThese local primitives prove the holomorphic Poincaré lemma in the degrees needed here. Exactness on every stalk givesThis is sheaf exactness; the local primitives need not glue to global ones.