A Cartier divisor on an integral scheme is specified locally by nonzero rational functions, with two such rational functions identified when their ratio is a unit. These local data record zeros and poles with integer multiplicities. On a general scheme, the local equations are invertible sections of the sheaf of total rings of fractions, also called total quotient rings.
A Cartier divisor is effective when it can locally be represented by regular functions. Equivalently, an effective Cartier divisor is a closed subscheme whose ideal sheaf is an invertible sheaf.
The exceptional divisor of an algebraic blow-up is an effective Cartier divisor.
On a projective algebraic variety, a nef divisor has nonnegative intersection with every
irreducible algebraic curve. A Cartier divisor
is big when the sections of sufficiently
large multiples define a birational transformation
onto their image. It is semiample when some positive multiple is basepoint-free,
meaning that its global sections have no common zero. It is ample when some positive
multiple defines a closed embedding into projective
space. These properties can equivalently be stated for the associated line
bundle 
(Keel 1999).
The Ambro-Kawamata effective nonvanishing conjecture concerns nef divisors whose difference from a canonical divisor is also a nef divisor and is big.
