▁x y ▁= ▁zw ▁in ▁aff ine ▁ 4 - space ▁over ▁a ▁field . ▁Then ▁the ▁plane ▁D ▁in ▁X ▁defined ▁by ▁x ▁= ▁z ▁= ▁ 0 ▁cannot ▁be ▁defined ▁in ▁X ▁by ▁one ▁equation ▁near ▁the ▁origin , ▁even ▁as ▁a ▁set . ▁It ▁follows ▁that ▁D ▁is ▁not ▁Q - Cart ier ▁on ▁X ; ▁that ▁is , ▁no ▁positive ▁multiple ▁of ▁D ▁is ▁Cart ier . ▁In ▁fact , ▁the ▁divis or ▁class ▁group ▁Cl ( X ) ▁is ▁is omorphic ▁to ▁the ▁integers ▁Z , ▁generated ▁by ▁the ▁class ▁of ▁D . ▁ ▁The ▁canonical ▁divis or ▁ ▁Let ▁X ▁be ▁a ▁normal ▁variety ▁over ▁a ▁perfect ▁field . ▁The ▁smooth ▁loc us ▁U ▁of ▁X ▁is ▁an ▁open ▁subset ▁whose ▁complement ▁has ▁cod im ension ▁at ▁least ▁ 2 . ▁Let ▁j : ▁U ▁→ ▁X ▁be ▁the ▁inclusion ▁map , ▁then ▁the ▁restriction ▁hom omorphism : ▁ ▁is ▁an ▁isomorphism , ▁since ▁X ▁− ▁U ▁has ▁cod im ension ▁at ▁least ▁ 2 ▁in ▁X . ▁For ▁example , ▁one ▁can ▁use ▁this ▁isomorphism ▁to ▁define ▁the ▁canonical ▁divis or ▁K X ▁of ▁X : ▁it ▁is ▁the ▁We il ▁divis or ▁( up ▁to ▁linear ▁equivalence ) ▁corresponding ▁to ▁the ▁line ▁bundle ▁of ▁differential ▁forms ▁of ▁top ▁degree ▁on ▁U . ▁Equ ival ently , ▁the ▁she af ▁ ▁on ▁X ▁is ▁the ▁direct ▁image ▁she af ▁ ▁where ▁n ▁is ▁the ▁dimension ▁of ▁X . ▁ ▁Example : ▁Let ▁X ▁= ▁P n