Example of cone of numerically effective curves which is not polyhedral - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T14:40:36Z http://mathoverflow.net/feeds/question/86124 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86124/example-of-cone-of-numerically-effective-curves-which-is-not-polyhedral Example of cone of numerically effective curves which is not polyhedral Jesus Martinez Garcia 2012-01-19T18:17:14Z 2012-04-02T08:58:47Z <p>I think I have seen more than one reference in which the cone of numerically effective curves can be 'not polyhedral', i.e. with an infinite number of extremal rays</p> <p>I cannot remember where I read that the usual example is the blow-up of the plane in the $9$ points of intersection of $2$ general cubics. This should give an infinite number of $-1$-curves, but I don't manage to see why it should be infinite! As far as I see the strict transform $C$ of any curve in the pencil would give me $C^2=\pi*(C)+ \sum E_i^2=9-9=0$ where $E_i$ are the exceptional divisors.</p> <p>What am I missing?</p> http://mathoverflow.net/questions/86124/example-of-cone-of-numerically-effective-curves-which-is-not-polyhedral/86125#86125 Answer by Artie Prendergast-Smith for Example of cone of numerically effective curves which is not polyhedral Artie Prendergast-Smith 2012-01-19T18:35:54Z 2012-01-20T21:41:54Z <p>Yes, this is the standard example of a variety whose cone of curves has infinitely many extremal rays. (A reference is Koll\'ar--Mori, p.22).</p> <p>To see why there are infinitely many (-1)-curves: each of the 9 points you blow up gives a (-1)-curve E_i, as you know. But the E_i are also sections of the elliptic fibration determined by the pencil. So one can use the group structure of the generic fibre (which is an elliptic curve over the function field k(P^1)) to translate any of the E_i to any other section of the fibration; since the action is by automorphisms of the surface, the other sections must be (-1)-curves also. As long as there are infinitely many sections (which is guaranteed by the assumption that the cubics are general) one gets infinitely many (-1)-curves this way.</p> <p><strong>Edit:</strong> It's worth mentioning that there are other examples of surfaces with non-finitely generated cone of curves, which differ from the example above in an interesting way. For example, suppose X is a K3 surface which has Picard rank at least 3 and no (-2) curves. Then the (closed) cone of curves of X consists of all classes in N^1(X) satisfying x^2 &ge; 0 and x.H &ge; 0 for any fixed ample class H. This is the standard "round cone" in N^1, with <em>uncountably</em> many extremal rays. (A similar example is obtained by taking X to be an abelian surface with Picard rank at least 3.) </p> <p>As mentioned in the comments, Volume 1 of Lazarsfeld's great book <em>Positivity in Algebraic Geometry</em> is the best reference for these kinds of questions. </p>