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I am trying to find a (smooth) compact complex surface $X$ so that the set of irreducible curves $C$ on $X$ for which $C.C<0$ is infinite. Do any of you know of an example. Thanks.

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Any (nonconstant) elliptic surface with infinite Mordell-Weil group should do. –  Noam D. Elkies Sep 4 '11 at 4:14
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Related question: mathoverflow.net/questions/2179/… –  jvp Sep 4 '11 at 5:11
    
So I asked a redundant question. –  anonymous Sep 4 '11 at 17:49
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No not redundant, just quite well-known. Look at references for when the cone of curves is not finitely generated. For example, Kovacs' paper 'The cone of curves on a K3 surface' is a nice starting point. In particular, you'll see K3 surfaces with infinitely many $(-2)$-curves. The Fermat quartic $x^4+y^4+z^4+w^4=0$ is one example. Also, if a surface has a large automorphism group and contains one negative curve it typically contains infinitely many - this happens for example for Enriques surfaces, see eg mathoverflow.net/questions/52397/… –  J.C. Ottem Sep 4 '11 at 18:01

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up vote 4 down vote accepted

Blow up $\mathbb P^2$ at 9 points. See e.g. Hartshorne exercise 5.4.15e) and the reference there.

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Some more details: if you blow up the plane at 9 in general position, then acting by Cremona transformations gives a sequence of rational curves of arbitrarily high degree through the 9 points that pull back to divisors of self intersection -1. See Artie Prendergast's note iag.uni-hannover.de/~prendergast/Papers/coneofcurves.pdf for more details. –  J.C. Ottem Sep 4 '11 at 10:31

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