Question

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 ?

Answer 1

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.

Answer 2

In characteristic 0 this seems to be open, but conjecturally true: a reference is Conjecture I.2.1 in this paper by Harbourne. As mentioned in that paper, it is definitely false in positive characteristic, as one might expect.

Update: Surprisingly, this long-standing conjecture has recently been disproved! Theorem A in this paper (by many authors) says the following:

Theorem A: There exists a smooth projective complex surface containing a sequence of negative curves whose self-intersections tend to $-\infty$.

The counterexamples are related to Hilbert modular surfaces.

Let me also note that in the same paper the authors prove the following complementary theorem:

Theorem B: For every integer $m>0$, there exists smooth projective complex surfaces containing infinitely many curves of self-intersection $-m$.

Update 2 (04/12): As John L. points out, now the authors have retracted the claimed Theorem A above. So the Bounded Negativity Conjecture is back on the cards.