▁she aves ▁on ▁X ▁form ▁a ▁mon oid ▁with ▁product ▁given ▁as ▁the ▁ref lex ive ▁h ull ▁of ▁a ▁tensor ▁product . ▁Then ▁ ▁defines ▁a ▁mon oid ▁isomorphism ▁from ▁the ▁We il ▁divis or ▁class ▁group ▁of ▁X ▁to ▁the ▁mon oid ▁of ▁isomorphism ▁classes ▁of ▁rank - one ▁ref lex ive ▁she aves ▁on ▁X . ▁ ▁Ex amples ▁▁ ▁Let ▁k ▁be ▁a ▁field , ▁and ▁let ▁n ▁be ▁a ▁positive ▁integer . ▁Since ▁the ▁polynomial ▁ring ▁k [ x 1 , ▁... , ▁x n ] ▁is ▁a ▁unique ▁factor ization ▁domain , ▁the ▁divis or ▁class ▁group ▁of ▁aff ine ▁space ▁An ▁over ▁k ▁is ▁equal ▁to ▁zero . ▁Since ▁project ive ▁space ▁P n ▁over ▁k ▁minus ▁a ▁hyper plane ▁H ▁is ▁is omorphic ▁to ▁An , ▁it ▁follows ▁that ▁the ▁divis or ▁class ▁group ▁of ▁P n ▁is ▁generated ▁by ▁the ▁class ▁of ▁H . ▁From ▁there , ▁it ▁is ▁straightforward ▁to ▁check ▁that ▁Cl ( P n ) ▁is ▁in ▁fact ▁is omorphic ▁to ▁the ▁integers ▁Z , ▁generated ▁by ▁H . ▁Con cret ely , ▁this ▁means ▁that ▁every ▁cod im ension - 1 ▁sub vari ety ▁of ▁P n ▁is ▁defined ▁by ▁the ▁van ishing ▁of ▁a ▁single ▁hom ogeneous ▁polynomial . ▁▁ ▁Let ▁X ▁be ▁an ▁algebraic ▁curve ▁over ▁a ▁field ▁k . ▁Every ▁closed ▁point ▁p ▁in ▁X ▁has ▁the ▁form ▁Spec ▁E ▁for ▁some ▁finite ▁extension ▁field ▁E ▁of ▁k , ▁and ▁the ▁degree ▁of ▁p ▁is ▁defined ▁to