, ▁by ▁the ▁group ▁of ▁k - points ▁of ▁a ▁connected ▁group ▁scheme ▁ ▁For ▁k ▁of ▁characteristic ▁zero , ▁ ▁is ▁an ▁ab elian ▁variety , ▁the ▁Pic ard ▁variety ▁of ▁X . ▁ ▁For ▁R ▁the ▁ring ▁of ▁integers ▁of ▁a ▁number ▁field , ▁the ▁divis or ▁class ▁group ▁Cl ( R ) ▁:= ▁Cl ( Spec ▁R ) ▁is ▁also ▁called ▁the ▁ideal ▁class ▁group ▁of ▁R . ▁It ▁is ▁a ▁finite ▁ab elian ▁group . ▁Under standing ▁ideal ▁class ▁groups ▁is ▁a ▁central ▁goal ▁of ▁algebraic ▁number ▁theory . ▁▁ ▁Let ▁X ▁be ▁the ▁quad ric ▁cone ▁of ▁dimension ▁ 2 , ▁defined ▁by ▁the ▁equation ▁x y ▁= ▁z 2 ▁in ▁aff ine ▁ 3 - space ▁over ▁a ▁field . ▁Then ▁the ▁line ▁D ▁in ▁X ▁defined ▁by ▁x ▁= ▁z ▁= ▁ 0 ▁is ▁not ▁principal ▁on ▁X ▁near ▁the ▁origin . ▁Note ▁that ▁D ▁can ▁be ▁defined ▁as ▁a ▁set ▁by ▁one ▁equation ▁on ▁X , ▁namely ▁x ▁= ▁ 0 ; ▁but ▁the ▁function ▁x ▁on ▁X ▁van ishes ▁to ▁order ▁ 2 ▁along ▁D , ▁and ▁so ▁we ▁only ▁find ▁that ▁ 2 D ▁is ▁Cart ier ▁( as ▁defined ▁below ) ▁on ▁X . ▁In ▁fact , ▁the ▁divis or ▁class ▁group ▁Cl ( X ) ▁is ▁is omorphic ▁to ▁the ▁cy clic ▁group ▁Z / 2 , ▁generated ▁by ▁the ▁class ▁of ▁D . ▁▁ ▁Let ▁X ▁be ▁the ▁quad ric ▁cone ▁of ▁dimension ▁ 3 , ▁defined ▁by ▁the ▁equation