▁be ▁the ▁degree ▁of ▁E ▁over ▁k . ▁Ext ending ▁this ▁by ▁linear ity ▁gives ▁the ▁notion ▁of ▁degree ▁for ▁a ▁divis or ▁on ▁X . ▁If ▁X ▁is ▁a ▁project ive ▁curve ▁over ▁k , ▁then ▁the ▁divis or ▁of ▁a ▁non zero ▁rational ▁function ▁f ▁on ▁X ▁has ▁degree ▁zero . ▁As ▁a ▁result , ▁for ▁a ▁project ive ▁curve ▁X , ▁the ▁degree ▁gives ▁a ▁hom omorphism ▁deg : ▁Cl ( X ) ▁→ ▁Z . ▁▁ ▁For ▁the ▁project ive ▁line ▁P 1 ▁over ▁a ▁field ▁k , ▁the ▁degree ▁gives ▁an ▁isomorphism ▁Cl ( P 1 ) ▁ <0xE2> <0x89> <0x85> ▁Z . ▁For ▁any ▁smooth ▁project ive ▁curve ▁X ▁with ▁a ▁k - r ational ▁point , ▁the ▁degree ▁hom omorphism ▁is ▁sur jective , ▁and ▁the ▁kernel ▁is ▁is omorphic ▁to ▁the ▁group ▁of ▁k - points ▁on ▁the ▁Jacob ian ▁variety ▁of ▁X , ▁which ▁is ▁an ▁ab elian ▁variety ▁of ▁dimension ▁equal ▁to ▁the ▁genus ▁of ▁X . ▁It ▁follows , ▁for ▁example , ▁that ▁the ▁divis or ▁class ▁group ▁of ▁a ▁complex ▁elli ptic ▁curve ▁is ▁an ▁un count able ▁ab elian ▁group . ▁▁ ▁General izing ▁the ▁previous ▁example : ▁for ▁any ▁smooth ▁project ive ▁variety ▁X ▁over ▁a ▁field ▁k ▁such ▁that ▁X ▁has ▁a ▁k - r ational ▁point , ▁the ▁divis or ▁class ▁group ▁Cl ( X ) ▁is ▁an ▁extension ▁of ▁a ▁finit ely ▁generated ▁ab elian ▁group , ▁the ▁N ér on – S ever i ▁group