On a scheme , a Cartier divisor is a global section of the sheaf , where is the multiplicative sheaf of meromorphic functions, and the multiplicative sheaf of invertible regular functions (the units of the structure sheaf).
More explicitly, a Cartier divisor is a choice of open cover of , and meromorphic functions , such that , along with two Cartier divisors being the same if the open cover of one is a refinement of the other, with the same functions attached to open sets, or if is replaced by with .
Intuitively, the only carried by Cartier divisor is where it vanishes, and the order it does there. Thus, a Cartier divisor should give us a Weil divisor, and vice versa. On “nice” (for example, nonsingular over an algebraically closed field) schemes, it does.
|Date of creation||2013-03-22 13:52:29|
|Last modified on||2013-03-22 13:52:29|
|Last modified by||mathcam (2727)|