= Solution
For a smooth projective <algebraic curve>, a <linear system of divisors> comes from a finite-dimensional space $V\subseteq H^0(C,\mathcal O(D))$. A basis of sections $s_0,\ldots,s_n$ defines $f(P)=[s_0(P):\cdots:s_n(P)]$, using any local trivialization; a common scalar change does not alter the projective point. A <base-point-free linear system> makes this defined everywhere, and $f^*\mathcal O_{\mathbb P^n}(1)\cong\mathcal O(D)$. Pulling back a hyperplane gives the zero divisor of a linear combination of the sections, an element of the original system.
A system with base divisor $B$ can instead define a map after removing $B$, but its hyperplane pullbacks have class $D-B$. Thus base-point-freeness is needed for the stated original-system property. For a <closed immersion>, the system must also separate distinct points and tangent directions; equivalently it is a <very ample linear system>. On a smooth curve the latter tangent condition is separation of the length-two subscheme at each point.
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