On an integral scheme, a Cartier divisor is specified locally by nonzero rational functions, with two local 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.
