# Complete linear systems of divisors

I looked up the web and did not find any good answer to this, so I thought about asking it here. Let $X$ be a smooth projective complex surface and $L$ a line bundle. The complete linear system $|L|$ associated to L is then the projectivization of the space $H^0(X,L)$ of global sections of $L$. Now if I understand correctly $|L|$ coincides with the set of effective divisors linearly equivalent to $L$. Can somebody explain me this in detail?

• This is done in Hartshorne Chap. II Prop. 7.7. – karl_christ Oct 2 '13 at 15:59

More generally, $X/k$ can be any smooth projective algebraic variety and $D$ an effective divisor. If $L\cong \mathcal O_X(D)$, as you say, we can denote by $|L|$ (or by $|D|$) the set of all effective divisors linearly equivalent to $D$. You then have the bijection $$(H^0(L)\setminus\{0\})/k^\times\longrightarrow |D|\,\,\,\,\,\,\,\,\,\,\,\,[v]\mapsto (v)+D,$$ where $(v)$ is the divisor of the section $v$. The key point is that two sections $v,v'$ share the same divisor (i.e. go to the same divisor on the right hand side) if and only if $v=cv'$ for some $c\in k^\times$ (i.e. they are equal in the left hand side). And, to be sure that this map is well-defined (i.e. you only hit effective divisors linearly equivalent to $D$), you also have to remember that two divisors are linearly equivalent if and only if they differ by the divisor of a rational function.
The left hand side is, by definition, $\mathbb P((H^0(L))^\vee)$, the projective space of lines in $H^0(L)$. Each such "line" consists of global sections which differ from one another by a nonzero constant. This gives you a structure of projective variety on $|D|=\textrm{Proj (Sym }H^0(L)^\vee)$.