A pseudo-effective divisor is a real Cartier divisor whose numerical class belongs to the closure of the cone generated by effective divisor classes on a projective algebraic variety. Here two Cartier divisors are numerically equivalent when they have the same intersection numbers with every integral algebraic curve. Effective divisors are finite nonnegative linear combinations of prime Weil divisors (Kollár and Mori 1998).
The zero divisor and every effective Cartier divisor are pseudo-effective. Pseudo-effectivity is weaker than nefness. For example, an
exceptional algebraic curve of self-intersection
on a smooth projective surface is effective but is not
a nef divisor. The minimal
model program distinguishes the pseudo-effective case from the Mori
fiber space case.
