The complete linear system of a Cartier divisor 
on an integral projective
variety 
over an algebraically closed field
is the projective space of its nonzero global
sections modulo nonzero scalar,
with each section identified with its effective
Cartier divisor linearly equivalent to 
. It is written 
when nonzero sections
exist. A Cartier divisor 
is rationally effective when a positive integer
multiple 
is linearly equivalent to an effective Cartier
divisor, equivalently when 
for some 
.
An anticanonical linear system is 
. An algebraic
torus action on 
that lifts to the line bundle 
makes its space of global sections
a direct sum of group character eigenspaces. A
nonzero invariant section lies in the trivial group
character eigenspace.
