= Solution
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 form
$$
D=mE+\Gamma,\qquad m\ge2,\quad
E^2=0,\quad E\Gamma=1,\quad\Gamma^2=-2,
\qquad |D|=\Gamma+|mE|,
$$
where $|E|$ 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 as
$$
\Gamma_0:\ x_0=\zeta x_1,\ x_2=\eta x_3,\qquad
\Gamma:\ x_0=\zeta' x_1,\ x_2=\eta' x_3,
$$
with $\zeta^4=(\zeta')^4=\eta^4=(\eta')^4=-1$, $\zeta\ne\zeta'$ and $\eta\ne\eta'$. 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> $0\to\mathcal O_{\mathbb P^3}(-4)\to\mathcal O_{\mathbb P^3}\to\mathcal O_X\to0$, 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>, with
$$
E^2=4-2-2=0,\qquad E\Gamma=1.
$$
The 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 $D\Gamma=m-2\ge0$, 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 $D=\pi^*\mathcal O(1)$. The double-cover construction gives
$$
\pi_*\mathcal O_X=\mathcal O\oplus\mathcal O(-3),\qquad
K_X=\pi^*(K_{\mathbb P^2}+3H)=0,\qquad
H^1(X,\mathcal O_X)=0.
$$
Thus $X$ is a <K3 surface>. Further,
$$
D^2=2,\qquad
H^0(X,D)=H^0(\mathbb P^2,\mathcal O(1)),
$$
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 $E=H-\Gamma_0$ has
$$
H\cdot E=4-1=3.
$$
Its 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 \b[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.
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