▁domain . ▁ ▁An ▁algebraic ▁cycle ▁is ▁a ▁higher - cod im ension ▁general ization ▁of ▁a ▁divis or ; ▁by ▁definition , ▁a ▁We il ▁divis or ▁is ▁a ▁cycle ▁of ▁cod im ension ▁ 1 . ▁ ▁Div is ors ▁on ▁a ▁Riemann ▁surface ▁A ▁Riemann ▁surface ▁is ▁a ▁ 1 - dimensional ▁complex ▁manifold , ▁and ▁so ▁its ▁cod im ension - 1 ▁sub man if olds ▁have ▁dimension ▁ 0 . ▁The ▁group ▁of ▁divis ors ▁on ▁a ▁compact ▁Riemann ▁surface ▁X ▁is ▁the ▁free ▁ab elian ▁group ▁on ▁the ▁points ▁of ▁X . ▁ ▁Equ ival ently , ▁a ▁divis or ▁on ▁a ▁compact ▁Riemann ▁surface ▁X ▁is ▁a ▁finite ▁linear ▁combination ▁of ▁points ▁of ▁X ▁with ▁integer ▁coefficients . ▁The ▁degree ▁of ▁a ▁divis or ▁on ▁X ▁is ▁the ▁sum ▁of ▁its ▁coefficients . ▁ ▁For ▁any ▁non zero ▁mer omorphic ▁function ▁f ▁on ▁X , ▁one ▁can ▁define ▁the ▁order ▁of ▁van ishing ▁of ▁f ▁at ▁a ▁point ▁p ▁in ▁X , ▁ord p ( f ). ▁It ▁is ▁an ▁integer , ▁negative ▁if ▁f ▁has ▁a ▁pole ▁at ▁p . ▁The ▁divis or ▁of ▁a ▁non zero ▁mer omorphic ▁function ▁f ▁on ▁the ▁compact ▁Riemann ▁surface ▁X ▁is ▁defined ▁as ▁ ▁which ▁is ▁a ▁finite ▁sum . ▁Div is ors ▁of ▁the ▁form ▁( f ) ▁are ▁also ▁called ▁principal ▁divis ors . ▁Since ▁( fg ) ▁= ▁( f ) ▁+ ▁( g ), ▁the ▁set ▁of ▁principal ▁divis ors ▁is ▁a ▁subgroup ▁of