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Suppose we have generic surface in a Toric variety (say an elliptically fibered 3-fold), it can be branched over a divisor too, and suppose we actually know the complete set of divisors over this surface (where the number them is the same as the Picard number), and their intersection form. So can anyone help me to see how can I find a new set of basis such that every effective divisor can be expended in terms of them with positive integer coefficients (i.e. to find it's Mori Cone, if I'm right).

Thanks a lot,

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In the generality you have written the question, I do not think there is a good answer. There is no algorithm, that can take the Néron--Severi lattice of a surface and output a set of generators of the Mori cone. (I am aware that you have some extra geometric conditions, but I don't think they will help much.) Instead, let me try to explain some general ideas that help to compute the Mori cone of a surface. Maybe they will be useful to you, or to other readers.

If you know some extra information about your surface, then things become more tractable. For example, if you know it is an abelian surface, then the Mori cone is just one "half" of the positive cone of the intersection form. (Of course, a surface in a smooth toric 3-fold can't be abelian, so this doesn't help for your special case). If you know it is a K3 surface, then its Mori cone is spanned by the positive cone plus the classes of all $(-2)$-curves, and these may be possible to find explictly. There are similar descriptions for Enriques surfaces.

Apart from these special cases, the main tool available is the Cone Theorem. That tells you that the extremal rays of the cone in the $K$-negative part of $NS(X)$ are spanned by $(-1)$-curves. If your surface is e.g. rational, and not too complicated, that might be enough to find the whole cone. For example, this idea lets you determine the Mori cones of all del Pezzo surfaces, plus (with a few extra arguments) the Mori cone of the blow-up of $\mathbf P^2$ in 9 very general points. (Note that in the latter case, the cone has infinitely many extremal rays.)

Beyond this, the only technique I know is to guess and check. This means the following. Suppose you have some collection $\{[C_i]\}_{i \in I}$ of effective curve classes. Then their span $\Gamma = \bigoplus_{i \in I} \mathbf R_+ [C_i]$ is a subcone of the Mori cone. That means that the dual cone $\Gamma^\vee$ contains the nef cone $Nef(X)$. If, by fair means or foul, you can prove that every class in $\Gamma^\vee$ is actually nef, then you can conclude that $\Gamma^\vee=Nef(X)$ and hence that $\Gamma$ is the Mori cone. How could you prove that? For example by finding the extremal rays of $\Gamma^\vee$, and showing that each one is spanned by a basepoint-free divisor. This requires understanding the geometry of your surface.

Section 1.5 of the book Positivity in Algebraic Geometry by Lazarsfeld is a good reference for these ideas.

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  • $\begingroup$ This is a great answer. (+1) $\endgroup$ – Simone Weil Sep 9 '16 at 16:37
  • $\begingroup$ @Harambe: thanks, glad you like it! $\endgroup$ – Nefertiti Sep 12 '16 at 14:58

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