linear systems and maps Given a regular map $\varphi:C\to \mathbb P^n,P\mapsto \mathbb (f_0(P):f_1(P):\ldots:f_n(P))$, we can associate a linear system $|\varphi|$ in the following manner: let the divisor $D=-\min div(f_i)$ and $V$ be the vector space of every linear combination of the functions $f_i$. Thus we define $|\varphi|=\{div(g)+D\mid g\in V\}$.
My question: is it possible to do the converse?, i.e., given a linear system, can we find a map associate with this linear system?
Thanks!
 A: As Relapsarian mentioned in the comments, this is all discussed in Chapter II.7 of Hartshorne, but I'll try to strip away some of the language of schemes in my answer (although I assume the reader knows what a line bundle is, I provide in translation to the language of Fulton's book in the edit below).
Throughout, let $X$ be variety over a field $k$ and let $\mathcal{L}$ be a line bundle on $X$. Then, a linear system is a vector subspace $V \subset \Gamma(X,\mathcal{L})$. Take a basis of sections $s_1,\ldots,s_m$ of $V$, then consider the map $X \to \mathbb{P}^{m-1}$ given by
$$
x \mapsto [s_1(x) \colon \ldots \colon s_m(x) ],
$$
which is a map from $X$ to a projective space! But wait - what could go wrong here? Well, we could have that $s_1(x) = \ldots = s_m(x) = 0$ for some $x$, which is a problem because $[0 \colon \ldots \colon 0 ]$ is not a point of $\mathbb{P}^{m-1}$. 
Denote by $B = \{ x \in X \colon s_1(x) = \ldots = s_m(x) = 0 \}$ the set of basepoints, then the process above gives a regular map $X - B \to \mathbb{P}^{m-1}$. In the case where $B = \emptyset$ (e.g. if $X$ is a smooth projective variety and the line bundle $\mathcal{L}$ is generated by the global sections $V$), then we do indeed get a regular map $X \to \mathbb{P}^{m-1}$.
Edit: here is a translation between the language in Fulton's book and that of line bundles. I assume that $X$ is now a curve. Every divisor $D$ on $X$ determines a line bundle $\mathcal{O}(D)$ with the property that
$$
\Gamma(X,\mathcal{O}(D)) = \{ f \in K(X) \colon \textrm{div}(f) + D \geq 0 \} = L(D).
$$
Therefore, in my answer above, a linear system is a vector subspace $V \subset L(D)$. Notice that if we know the set of divisors $\{ D + \textrm{div}(f) \colon f \in V \}$, then we know $V$, and vice-versa (so this definition of a linear system is equivalent to that of user42912). 
