On a non-singular projective variety, the complete linear system $|D|$ of a divisor $D$ is the set of all effective divisors linearly equivalent to $D$.
Often we speak of the dimension $\mathrm{dim}|D|$, which is the number of parameters.
Consider in particular a complex projective variety, with an effective divisor $D$. Can there be disconnected components of $|D|$? For example, can it be that $\mathrm{dim}|D|=0$ while there are multiple elements in $|D|$?
Personally I would be particularly interested in examples of divisors $D$ on a complex surface where the self-intersection is negative, $D^2 < 0$: here $\mathrm{dim}|D|=0$ but I wonder if there may be several elements in $|D|$.
Best Answer
In fact, $|D|$ is a projective space — namely, $\Bbb P(H^0(D)))$. The linear system is coming from "continuous" variation of the divisor, as it consists of divisors that are linearly equivalent to the given divisor. Linear equivalence, in particular, is a specific sort of homotopy, and the divisors can be "connected" in a continuous way.