= Solution
Work over $\mathbb C$, and let $C$ be a <smooth projective curve> of <genus> $g\ge3$ and gonality three. A <trigonal curve> has a <basepoint-free> degree-three pencil, giving a finite morphism $f:C\to\mathbb P^1$ of degree three. It is not <hyperelliptic>, so its <canonical map> is an embedding. Let $D_t=f^{-1}(t)$, including its scheme multiplicities. The complete pencil has $h^0(D_t)=2$, and <Riemann-Roch theorem> gives
$$
h^0(K_C-D_t)=g-2.
$$
Thus the quotient of canonical sections evaluated on $D_t$ has dimension two, and $D_t$ spans a line in $\mathbb P^{g-1}$. This argument includes ramified fibres, rather than treating only three distinct points.
These two-dimensional quotients vary in a globally generated rank-two bundle $E$ on $\mathbb P^1$. More explicitly, the image of
$$
H^0(C,K_C)\otimes\mathcal O_{\mathbb P^1}\longrightarrow f_*K_C
$$
has constant rank two by the preceding dimension count, so it is a subbundle. The equality of the rank at every fibre justifies the bundle construction. On <global sections> it contains every canonical section, and therefore $H^0(\mathbb P^1,E)=H^0(C,K_C)$. The <Birkhoff–Grothendieck theorem> gives
$$
E=\mathcal O(a_1)\oplus\mathcal O(a_2),
\qquad 0\le a_1\le a_2.
$$
Global generation forces nonnegative splitting degrees. Since $h^0(E)=a_1+a_2+2=g$,
$$
\boxed{a_1+a_2=g-2.}
$$
The natural map from $f^*E$ to $K_C$ is a <surjection> at every point, since the canonical system has no base points. By <projectivization by quotients> it gives a map $C\to F=\mathbb P(E)$ over $\mathbb P^1$, with $M|_C=K_C$. Its composition with the tautological map of $F$ is the canonical embedding of $C$, so $C\to F$ is itself a closed embedding. Each fibre of the ruling maps to the spanning line of $D_t$; their union is the <canonical scroll of a trigonal curve>, and the ruling restricts to the given $g^1_3$.
Put $a=a_2-a_1$. Tensoring $E$ by $\mathcal O(-a_2)$ identifies the abstract ruled surface with $\mathbb F_a=\mathbb P(\mathcal O(-a)\oplus\mathcal O)$. The section coming from the quotient onto the smaller summand has self-intersection $-a$ and will be denoted by $B$. In this normalized presentation its tautological class is $B$; restoring the twist gives
$$
\boxed{M=B+a_2L.}
$$
This class gives the <scroll> map. It is an embedding if $a_1>0$. When $a_1=0$, it contracts $B$; thus the printed embedding description needs this positivity qualification. For example, in <genus> three the ruled model is $\mathbb F_1$ and its image is $\mathbb P^2$; in <genus> four the type $(0,2)$ has a quadric-cone image. In both cases the smooth curve and its canonical image are still described by the construction above.
As a divisor on $F$, the curve meets each fibre in degree three, so $C=3M+tL$ for an integer $t$. The <scroll> formula gives
$$
K_F=-2M+(g-4)L.
$$
By <adjunction formula> and $M|_C=K_C$,
$$
(K_F+C-M)|_C=(g-4+t)L|_C=0.
$$
Taking degree on $C$ uses $\deg(L|_C)=3$ and gives $t=4-g$. Consequently
$$
C\in|3M+(4-g)L|
=|3B+(3a_2+4-g)L|
=|3B+(a+a_2+2)L|.
$$
Writing $d=a+a_2+2$, this is
$$
\boxed{C\in|3B+dL|,\qquad
d=\frac{g+3a+2}{2}.}
$$
A divisor $rB+dL$ with $r\ge0$ on $\mathbb F_a$ is <basepoint-free> when $d\ge ra$. Indeed, its pushforward along the ruling splits into
$$
\bigoplus_{j=0}^r\mathcal O_{\mathbb P^1}(d-ja);
$$
if each degree is nonnegative, sections generate these coefficient functions, and the fibre <monomials> generate at every point. In particular $3B+dL$ is <basepoint-free> when $d\ge3a$. The <Bertini smoothness theorem> then makes its general member nonsingular, including the equality case, where the general member avoids $B$.
Conversely, if $d<3a$, then $(3B+dL)\cdot B=d-3a<0$, so every effective member contains $B$. Because the canonical <scroll> has $a_1\ge0$,
$$
d=2a+a_1+2>2a.
$$
The residual system $|2B+dL|$ is <basepoint-free>. Its general member $R$ does not contain $B$ and intersects it in
$$
R\cdot B=d-2a=a_1+2>0.
$$
Thus a general divisor in the original system is $B+R$ with intersecting components and is singular at an intersection point. The fixed component occurs exactly once for a general member, since $R$ is not forced to contain it. This proves both directions, with the canonical-scroll hypotheses retained:
$$
\boxed{|3B+(a+a_2+2)L|\text{ has a nonsingular general member}
\iff d\ge3a\iff 3a\le g+2.}
$$
This is also the bound on the <Maroni invariant>. Merely observing a fixed component would not by itself prove singularity; the positive intersection with its residual divisor is essential.
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