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
is a formal linear combination
of irreducible, reduced and proper curves
Numerical equivalence of 1-cycles is defined by intersections: two 1-cycles
Denote the real vector space of 1-cycles modulo numerical equivalence by
We define the cone of curves of
are irreducible, reduced, proper curves on
One useful application of the notion of the cone of curves is the Kleiman condition, which says that a (Cartier) divisor
, the closure of the cone of curves in the usual real topology.
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
The great breakthrough of the early 1980s (due to Mori and others) was to construct (at least morally) the necessary birational map from
as a sequence of steps, each of which can be thought of as contraction of a
This process encounters difficulties, however, whose resolution necessitates the introduction of the flip.
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.
be a smooth projective variety.
There are countably many rational curves
For any positive real number
, where the sum in the last term is finite.
The first assertion says that, in the closed half-space of
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.
So the cone theorem shows that the cone of curves of a Fano variety is generated by rational curves.
is defined over a field of characteristic 0, we have the following assertion, sometimes referred to as the Contraction Theorem: 3.
be an extremal face of the cone of curves on which