Cone of curves

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In mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an algebraic variety is a combinatorial invariant of importance to the birational geometry of .

Definition

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Let   be a proper variety. By definition, a (real) 1-cycle on   is a formal linear combination   of irreducible, reduced and proper curves  , with coefficients  . Numerical equivalence of 1-cycles is defined by intersections: two 1-cycles   and   are numerically equivalent if   for every Cartier divisor   on  . Denote the real vector space of 1-cycles modulo numerical equivalence by  .

We define the cone of curves of   to be

 

where the   are irreducible, reduced, proper curves on  , and   their classes in  . It is not difficult to see that   is indeed a convex cone in the sense of convex geometry.

Applications

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One useful application of the notion of the cone of curves is the Kleiman condition, which says that a (Cartier) divisor   on a complete variety   is ample if and only if   for any nonzero element   in  , the closure of the cone of curves in the usual real topology. (In general,   need not be closed, so taking the closure here is important.)

A more involved example is the role played by the cone of curves in the theory of minimal models of algebraic varieties. Briefly, the goal of that theory is as follows: given a (mildly singular) projective variety  , find a (mildly singular) variety   which is birational to  , and whose canonical divisor   is nef. The great breakthrough of the early 1980s (due to Mori and others) was to construct (at least morally) the necessary birational map from   to   as a sequence of steps, each of which can be thought of as contraction of a  -negative extremal ray of  . This process encounters difficulties, however, whose resolution necessitates the introduction of the flip.

A structure theorem

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The above process of contractions could not proceed without the fundamental result on the structure of the cone of curves known as the Cone Theorem. The first version of this theorem, for smooth varieties, is due to Mori; it was later generalised to a larger class of varieties by Kawamata, Kollár, Reid, Shokurov, and others. Mori's version of the theorem is as follows:

Cone Theorem. Let   be a smooth projective variety. Then

1. There are countably many rational curves   on  , satisfying  , and

 

2. For any positive real number   and any ample divisor  ,

 

where the sum in the last term is finite.

The first assertion says that, in the closed half-space of   where intersection with   is nonnegative, we know nothing, but in the complementary half-space, the cone is spanned by some countable collection of curves which are quite special: they are rational, and their 'degree' is bounded very tightly by the dimension of  . The second assertion then tells us more: it says that, away from the hyperplane  , extremal rays of the cone cannot accumulate. When   is a Fano variety,   because   is ample. So the cone theorem shows that the cone of curves of a Fano variety is generated by rational curves.

If in addition the variety   is defined over a field of characteristic 0, we have the following assertion, sometimes referred to as the Contraction Theorem:

3. Let   be an extremal face of the cone of curves on which   is negative. Then there is a unique morphism   to a projective variety Z, such that   and an irreducible curve   in   is mapped to a point by   if and only if  . (See also: contraction morphism).

References

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  • Lazarsfeld, R., Positivity in Algebraic Geometry I, Springer-Verlag, 2004. ISBN 3-540-22533-1
  • Kollár, J. and Mori, S., Birational Geometry of Algebraic Varieties, Cambridge University Press, 1998. ISBN 0-521-63277-3