This question comes from Huybrechts' lecture notes on K3 surfaces, more specifically, chapter 2.

Let $ X $ be a K3 surface (over an algebraically closed field $ k $) and $ L $ a line bundle on $ X $. The base locus of the linear system $ |L| $ is defined as a closed subscheme of $ X $ by $$ \text{Bs} (L) := \cap_{s \in H^0(X,L)} Z(s) $$ where $ Z(s) $ is the zero locus of the section $ s $.

On a surface $ X $, the base locus may have components of dimension zero and one. Let $ F $ be the one-dimensional part, called the fixed part of $ L $.

(1) Why is $ F $ a divisor on $ X $?

I fail to see why $ F $ should necessarily be a divisor, the one dimensional component may have something bad like embedded points. But provided that it is one,

(2) Why is $ h^0(X,F) = 1 $?

Huybrechts' doesn't really give any explanation and uses this 'fact' later to prove that $ F $ is a sum of smooth rational curves. I would really appreciate any help.