▁the ▁group ▁of ▁divis ors . ▁Two ▁divis ors ▁that ▁differ ▁by ▁a ▁principal ▁divis or ▁are ▁called ▁linear ly ▁equivalent . ▁ ▁On ▁a ▁compact ▁Riemann ▁surface , ▁the ▁degree ▁of ▁a ▁principal ▁divis or ▁is ▁zero ; ▁that ▁is , ▁the ▁number ▁of ▁zeros ▁of ▁a ▁mer omorphic ▁function ▁is ▁equal ▁to ▁the ▁number ▁of ▁pol es , ▁counted ▁with ▁multip licity . ▁As ▁a ▁result , ▁the ▁degree ▁is ▁well - defined ▁on ▁linear ▁equivalence ▁classes ▁of ▁divis ors . ▁ ▁Given ▁a ▁divis or ▁D ▁on ▁a ▁compact ▁Riemann ▁surface ▁X , ▁it ▁is ▁important ▁to ▁study ▁the ▁complex ▁vector ▁space ▁of ▁mer omorphic ▁functions ▁on ▁X ▁with ▁pol es ▁at ▁most ▁given ▁by ▁D , ▁called ▁H 0 ( X , ▁O ( D )) ▁or ▁the ▁space ▁of ▁sections ▁of ▁the ▁line ▁bundle ▁associated ▁to ▁D . ▁The ▁degree ▁of ▁D ▁says ▁a ▁lot ▁about ▁the ▁dimension ▁of ▁this ▁vector ▁space . ▁For ▁example , ▁if ▁D ▁has ▁negative ▁degree , ▁then ▁this ▁vector ▁space ▁is ▁zero ▁( because ▁a ▁mer omorphic ▁function ▁cannot ▁have ▁more ▁zeros ▁than ▁pol es ). ▁If ▁D ▁has ▁positive ▁degree , ▁then ▁the ▁dimension ▁of ▁H 0 ( X , ▁O ( m D )) ▁grows ▁linear ly ▁in ▁m ▁for ▁m ▁sufficiently ▁large . ▁The ▁Riemann - R och ▁theorem ▁is ▁a ▁more ▁precise ▁statement ▁along ▁these ▁lines . ▁On ▁the ▁other ▁hand , ▁the ▁precise ▁dimension ▁of ▁H 0 ( X , ▁O ( D )) ▁for ▁divis ors