Divisor (Algebraic Geometry)

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In geometry a divisor on an algebraic variety is a formal sum (with integer coefficients) of subvarieties.

An effective divisor is a sum with non-negative integer coefficients.

Divisors on a curve[edit]

On an algebraic curve, a divisor is a formal sum of points

${\displaystyle D=\sum _i{n}_i{P}_i}$

with degree

${\displaystyle \mathrm{deg}D=\sum _i{n}_i.}$

The support of a divisor is the set of points with non-zero coefficients in the sum.

The divisor of a function f, denoted ${\displaystyle (f)}$ or ${\displaystyle \mathrm{d}\mathrm{i}\mathrm{v}f}$, is supported on the poles and zeroes of the function, with coefficients the degree of the pole or zero, with positive sign for zeroes and negative sign for poles. The degree of the divisor of a function is zero.