For a divisor $D$ on a smooth complex projective surface $X$, the *stable fixed part* is the maximal effective divisor $E$ which, for every $n \in \mathbb{N}$, is contained in every memeber of the complete linear series $|nD|$.

**Question.** *Is there a surface $X$ possessing nef and big divisors $D$ whose stable fixed part:*

(a) *has arbitrarily high degree?*

(b) *has an arbitrarily high number of components?*

Here, the surface $X$ is to be fixed, and $D$ is allowed to run through all the nef and big divisors on $X$.

ADDED following Mark's answer: What if we drop the restriction that $D$ be nef (considering all big divisors at once)? A negative answer to this would imply, in particular, the following:

**Question 2.** *Let $X$ be a surface. As $D$ runs through all big divisors on $X$, is the number of curves $C \subset X$ with $D.C < 0$ bounded?*

Note: I have changed the title to reflect this latter question, which appears to be slightly more interesting. I hope this is OK.