TITLE: Surfaces containing curves of arbitrarily negative self-intersection
QUESTION [14 upvotes]: Olivier Wittenberg and I are curious about the following :
Let $S$ be a smooth projective complex surface. Are the self-intersection numbers of integral curves on $S$ always bounded below ? Or can $S$ contain integral curves with arbitrarily high negative self-intersection ?

REPLY [17 votes]: It is a folklore conjecture that surfaces in characteristic zero has bounded negativity. For a nice account of this problem and references, see the two survey articles
Global aspects of the geometry of surfaces by Harbourne, and
Recent developments and open problems in linear series by Bauer et al.
In positive characteristic however, the situation is different and there is a nice counterexample due to Kollar (taken from the 2nd paper above):
Let $C$ be a smooth curve of genus $g\geq 2$ defined over a field
   of characteristic $p>0$ and let $X$ be the product surface $X=C\times C$.
   The graph $\Gamma_q$ of the Frobenius morphism defined by taking
   $q=p^r$--th powers is a smooth
   curve of genus $g$ and self-intersection $\Gamma_q^2=q(2-2g)$.
   With $r$ going to infinity, we obtain a sequence of smooth curves
   of fixed genus with self-intersection going to minus infinity.