Divisors on algebraic curves

Abstract

Let X be a smooth projective curve over k. A divisor is a formal linear combination

$$ D = \sum {npP} $$

where the sum is over all closed points of X, the coefficients are integers and are almost always zero. We can add divisors formally and obtain a group: the group of divisors Div(X). A divisor is called effective if all n_P are non-negative. The degree of a divisor is

$$ \deg (D) = \sum {np\deg (P)} $$

with deg(P) = [k_v:k] the degree of P. (Recall that k_v =k(P) is the residue field of P, see Lecture 1.) The subgroup of divisors of degree zero is denoted Div₀(X).