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3 added 167 characters in body

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.

2 update with new reference

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 $-m$.

1

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.