What is the morphism associated to a divisor?

Let $X$ be a smooth projective curve, so that we may define the group of divisors $\mathrm{Div}\ X$. For any $D\in\mathrm{Div}$, we let $L(D)$ be the vector space of functions $f\in k(X)$ such that $D+(f)$ is effective; we also let $0\in L(D)$. Next, define $l(D):=\dim_k L(D)$. We define $K$ as the canonical class of the curve $X$ (so $L(K)$ is defined up to isomorphism, and $l(K)$ is well defined).

I know that the concept of a morphism associated to a divisor exists, and I have heard about it, but I can't find it in Shafarevich, or in Gathmann, for example. As a result, what follows will be very vague.

I'm reasonably sure that there exists some rational map $\varphi_D:X\to\mathbb{P}^{l(D)-1}$, or possibly $\varphi_D:X\to\mathbb{P}^{l(K)-1}$ associated to $D$. This map has the property that if $D$ is very ample, that is to say, $l(D-p-q)=l(D)-2$ for all points $p,q\in X$, then $\varphi_D$ is an embedding (this is one application of Riemann-Roch).

What exactly is this map? How is it defined?

• See the discussion after Theorem 3.3 in Shafarevich. It is surely in Gathmann as well.
– Hoot
Aug 10, 2016 at 3:28

Given a divisor $D$, consider a basis $f_0, \dots, f_n$ of $L(D)$. Assuming the $f_i$ do not vanish at a common point $p \in X$, this gives a map $X \to \mathbb P^n$ (with $n+1=l(D)$) given by $p \mapsto [f_0(p): \dots, f_n(p)]$. Note choosing a different basis will correspond to composing the above map by an element of $PGL_n(k)$.