I have this definition:
take $ \mathcal{S} \subset|D|$ a linear system contained in the complete linear system associated to the divisor $D \in Div(S)$ where $S$ is a complex algebraic projective surface.
$\mathcal{S}$ has fixed part if there exist an effective divisor $Z$ such that
$ \forall D^{'}\in \mathcal{S}, \ D^{'}=Z+R$ with $R>0$.
So this is just the definition but i have no more properties on my notes. For example is there a relation between the fixed part and the base locus of the system $\mathcal{S}$, ($\mathcal{B}_s(\mathcal{S})$)?
If i take the map defined from my surface $S - \mathcal{B_s(S)}$ to $\mathbb{P}^N$ as
$\phi \ : S - \mathcal{B_s(S)} \rightarrow \mathbb{P}^N$
$\qquad \qquad x \mapsto(s_0(x):\ldots:s_N(x))$ where i suppose that $\mathcal{S}=\mathbb{P}(U)$ and $U \subset H^0(D)$ is a vector sub space of $H^0(D)$ generated by $(s_0,\ldots,s_N)$ what is the role of the fixed part?