Remark 53.3.6 (Classical definition). Let $X$ be a smooth projective curve over an algebraically closed field $k$. We say two effective Cartier divisors $D, D' \subset X$ are linearly equivalent if and only if $\mathcal{O}_ X(D) \cong \mathcal{O}_ X(D')$ as $\mathcal{O}_ X$-modules. Since $\mathop{\mathrm{Pic}}\nolimits (X) = \text{Cl}(X)$ (Divisors, Lemma 31.27.7) we see that $D$ and $D'$ are linearly equivalent if and only if the Weil divisors associated to $D$ and $D'$ define the same element of $\text{Cl}(X)$. Given an effective Cartier divisor $D \subset X$ of degree $d$ the complete linear system or complete linear series $|D|$ of $D$ is the set of effective Cartier divisors $E \subset X$ which are linearly equivalent to $D$. Another way to say it is that $|D|$ is the set of closed points of the fibre of the morphism

$\gamma _ d : \underline{\mathrm{Hilb}}^ d_{X/k} \longrightarrow \underline{\mathrm{Pic}}^ d_{X/k}$

(Picard Schemes of Curves, Lemma 44.6.7) over the closed point corresponding to $\mathcal{O}_ X(D)$. This gives $|D|$ a natural scheme structure and it turns out that $|D| \cong \mathbf{P}^ m_ k$ with $m + 1 = h^0(\mathcal{O}_ X(D))$. In fact, more canonically we have

$|D| = \mathbf{P}(H^0(X, \mathcal{O}_ X(D))^\vee )$

where $(-)^\vee$ indicates $k$-linear dual and $\mathbf{P}$ is as in Constructions, Example 27.21.2. In this language a linear system or a linear series on $X$ is a closed subvariety $L \subset |D|$ which can be cut out by linear equations. If $L$ has dimension $r$, then $L = \mathbf{P}(V^\vee )$ where $V \subset H^0(X, \mathcal{O}_ X(D))$ is a linear subspace of dimension $r + 1$. Thus the classical linear series $L \subset |D|$ corresponds to the linear series $(\mathcal{O}_ X(D), V)$ as defined above.

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