In
mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
is a
combinatorial invariant of importance to the
birational geometry
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational ...
of
.
Definition
Let
be a
proper
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map for ...
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
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
on
. Denote the
real vector space
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
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