= Solution
Fix a point $[L]\in\operatorname{Pic}^d(C)$ and put $V=H^0(C,L)$. The set of effective <Cartier divisors> in its <divisor class> is the <complete linear system of a divisor>, but identifying points alone would not prove the requested scheme assertion. We identify its functor on arbitrary parameter <schemes>, including nonreduced ones.
Let $T$ be a $k$-scheme and let $\mathcal D\subset C\times T$ be a relative effective degree-$d$ <Cartier divisor> whose <Abel map of an algebraic curve> is the constant morphism $[L]$. The definition of the <Picard functor of a curve> says
$$
\mathcal O(\mathcal D)\cong\operatorname{pr}_C^*L\otimes\pi^*M
$$
for a <line bundle> $M$ on $T$. Its canonical section is nonzero on every geometric fibre. Since the constant family has direct image $V\otimes_k\mathcal O_T$, this section determines a line subbundle
$$
M^{-1}\hookrightarrow V\otimes_k\mathcal O_T.
$$
It is a subbundle, rather than merely an injective sheaf map: locally some coordinate is nonzero on each residue field, hence a unit on a neighbourhood, so the quotient is locally free. Such a subbundle is exactly a $T$-point of the <projective space> $\mathbb P_{\mathrm{lines}}(V)$.
Conversely, its tautological line subbundle evaluates to a section of $\operatorname{pr}_C^*L\otimes\pi^*M$, nonzero on every geometric fibre. On the smooth integral fibres its zero locus is an effective <Cartier divisor>. The fibrewise regularity criterion gives a relative <Cartier divisor> flat over $T$, of degree $d$, and its Picard class is constant $[L]$. These constructions are inverse and commute with arbitrary <base change>. Thus they identify functors, and therefore the scheme fibre itself:
$$
\boxed{a_d^{-1}([L])\cong\mathbb P_{\mathrm{lines}}(H^0(C,L))\cong\mathbb P^{h^0(C,L)-1}_k.}
$$
If $h^0(C,L)=0$, the fibre is empty, which is smooth as well. The argument remains valid over a point's residue field, so every scheme fibre, not just each closed geometric fibre, is smooth. Fibre dimensions can jump when $h^0(C,L)$ jumps; this does not assert that the entire <Abel map of an algebraic curve> is a <smooth morphism>.
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