Leapfrog integration: Difference between revisions

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In [[mathematics]], the '''cone of curves''' (sometimes the '''Kleiman-Mori''' cone) of an [[algebraic variety]] <math>X</math> is a [[combinatorial invariant]] of much importance to the [[birational geometry]] of <math>X</math>.


==Definition==
Let <math>X</math> be a [[Proper morphism|proper]] variety. By definition, a (real) ''1-cycle'' on <math>X</math> is a formal [[linear combination]] <math>C=\sum a_iC_i</math> of irreducible, reduced and proper curves <math>C_i</math>, with coefficients <math>a_i \in \mathbb{R}</math>. ''Numerical equivalence'' of 1-cycles is defined by intersections: two 1-cycles <math>C</math> and <math>C'</math> are numerically equivalent if <math>C \cdot D = C' \cdot D</math> for every Cartier [[divisor]] <math>D</math> on <math>X</math>. Denote the [[real vector space]] of 1-cycles modulo numerical equivalence by <math>N_1(X)</math>.


We define the ''cone of curves'' of <math>X</math> to be
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: <math>NE(X) = \left\{\sum a_i[C_i], \ 0 \leq a_i \in \mathbb{R} \right\} </math>
 
where the <math>C_i</math> are irreducible, reduced, proper curves on <math>X</math>, and <math>[C_i]</math> their classes in <math>N_1(X)</math>. It is not difficult to see that <math>NE(X)</math> is indeed a [[Cone_(linear_algebra)#Convex_cone|convex cone]] in the sense of convex geometry.
 
==Applications==
One useful application of the notion of the cone of curves is the '''[[Steven Kleiman|Kleiman]] condition''', which says that a (Cartier) divisor <math>D</math> on a complete variety <math>X</math> is [[ample line bundle|ample]] if and only if <math>D \cdot x > 0</math> for any nonzero element <math>x</math> in <math>\overline{NE(X)}</math>, the closure of the cone of curves in the usual real topology. (In general, <math>NE(X)</math> 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 model]]s of algebraic varieties. Briefly, the goal of that theory is as follows: given a (mildly singular) projective variety <math>X</math>, find a (mildly singular) variety <math>X'</math> which is [[birational]] to <math>X</math>, and whose [[canonical divisor]] <math>K_{X'}</math> is [[numerically effective|nef]]. The great breakthrough of the early 1980s (due to [[Shigefumi Mori|Mori]] and others) was to construct (at least morally) the necessary birational map from <math>X</math> to <math>X'</math> as a sequence of steps, each of which can be thought of as contraction of a <math>K_x</math>-negative extremal ray of <math>NE(X)</math>. This process encounters difficulties, however, whose resolution necessitates the introduction of the [[flip (algebraic geometry)|flip]].
 
==A structure theorem==
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 [[Shigefumi Mori|Mori]]; it was later generalised to a larger class of varieties by [[János_Kollár|Kollár]], [[Miles Reid|Reid]], [[Vyacheslav_Shokurov|Shokurov]], and others. Mori's version of the theorem is as follows:
 
'''Cone Theorem.''' Let <math>X</math> be a smooth [[projective variety]]. Then
 
1. There are [[countably many]] [[rational curve]]s <math>C_i</math> on <math>X</math>, satisfying <math>0< -K_X \cdot C_i \leq \operatorname{dim} X +1 </math>, and
 
: <math>\overline{NE(X)} = \overline{NE(X)}_{K_X\geq 0} + \sum_i \mathbf{R}_{\geq0} [C_i].</math>
 
2. For any positive real number <math>\epsilon</math> and any [[ample divisor]] <math>H</math>,
 
: <math>\overline{NE(X)} = \overline{NE(X)}_{K_X+\epsilon H\geq0} + \sum \mathbf{R}_{\geq0} [C_i],</math>
 
where the sum in the last term is finite.
 
The first assertion says that, in the [[closed half-space]] of <math>N_1(X)</math> where intersection with <math>K_X</math> 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 variety|rational]], and their 'degree' is bounded very tightly by the dimension of <math>X</math>. The second assertion then tells us more: it says that, away from the hyperplane <math>\{C : K_X \cdot C = 0\}</math>, extremal rays of the cone cannot accumulate.  
 
 
If in addition the variety <math>X</math> is defined over a field of characteristic 0, we have the following assertion, sometimes referred to as the '''Contraction Theorem''':
 
3. Let <math>F \subset \overline{NE(X)}</math> be an extremal face of the cone of curves on which <math>K_X</math> is negative. Then there is a unique [[morphism]] <math>\operatorname{cont}_F : X \rightarrow Z</math> to a projective variety ''Z'', such that <math>(\operatorname{cont}_F)_* \mathcal{O}_X = \mathcal{O}_Z</math> and an irreducible curve <math>C</math> in <math>X</math> is mapped to a point by <math>\operatorname{cont}_F</math> if and only if <math>[C] \in F</math>.
 
==References==
 
* 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
 
[[Category:Algebraic geometry]]
[[Category:Birational geometry]]

Revision as of 09:45, 27 February 2014


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