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 ?



Surfaces containing curves of arbitrarily negative self-intersection
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](http://arxiv.org/abs/0907.4151%20) by Harbourne, and


[Recent developments and open problems in linear series](http://arxiv.org/abs/1101.4363) 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.


