▁direction , ▁a ▁Cart ier ▁divis or ▁ ▁on ▁an ▁integral ▁No ether ian ▁scheme ▁X ▁determ ines ▁a ▁We il ▁divis or ▁on ▁X ▁in ▁a ▁natural ▁way , ▁by ▁applying ▁ ▁to ▁the ▁functions ▁fi ▁on ▁the ▁open ▁sets ▁U i . ▁ ▁If ▁X ▁is ▁normal , ▁a ▁Cart ier ▁divis or ▁is ▁determined ▁by ▁the ▁associated ▁We il ▁divis or , ▁and ▁a ▁We il ▁divis or ▁is ▁Cart ier ▁if ▁and ▁only ▁if ▁it ▁is ▁locally ▁principal . ▁ ▁A ▁No ether ian ▁scheme ▁X ▁is ▁called ▁factor ial ▁if ▁all ▁local ▁rings ▁of ▁X ▁are ▁unique ▁factor ization ▁domains . ▁( Some ▁authors ▁say ▁" loc ally ▁factor ial ". ) ▁In ▁particular , ▁every ▁regular ▁scheme ▁is ▁factor ial . ▁On ▁a ▁factor ial ▁scheme ▁X , ▁every ▁We il ▁divis or ▁D ▁is ▁locally ▁principal , ▁and ▁so ▁O ( D ) ▁is ▁always ▁a ▁line ▁bundle . ▁In ▁general , ▁however , ▁a ▁We il ▁divis or ▁on ▁a ▁normal ▁scheme ▁need ▁not ▁be ▁locally ▁principal ; ▁see ▁the ▁examples ▁of ▁quad ric ▁con es ▁above . ▁ ▁Effect ive ▁Cart ier ▁divis ors ▁Effect ive ▁Cart ier ▁divis ors ▁are ▁those ▁which ▁correspond ▁to ▁ideal ▁she aves . ▁In ▁fact , ▁the ▁theory ▁of ▁effective ▁Cart ier ▁divis ors ▁can ▁be ▁developed ▁without ▁any ▁reference ▁to ▁she aves ▁of ▁rational ▁functions ▁or ▁fraction al ▁ideal ▁she aves . ▁ ▁Let ▁X ▁be ▁a ▁scheme . ▁An ▁effective ▁Cart ier ▁divis or ▁on ▁X