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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2$\begingroup$ Any (nonconstant) elliptic surface with infinite Mordell-Weil group should do. $\endgroup$– Noam D. ElkiesSep 4, 2011 at 4:14
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2$\begingroup$ Related question: mathoverflow.net/questions/2179/… $\endgroup$– Jorge Vitório PereiraSep 4, 2011 at 5:11
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$\begingroup$ So I asked a redundant question. $\endgroup$– anonymousSep 4, 2011 at 17:49
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3$\begingroup$ 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/… $\endgroup$– J.C. OttemSep 4, 2011 at 18:01
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1 Answer
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Blow up $\mathbb P^2$ at 9 points. See e.g. Hartshorne exercise 5.4.15e) and the reference there.
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2$\begingroup$ 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. $\endgroup$ Sep 4, 2011 at 10:31