▁( with ▁its ▁embedding ▁in ▁M X ) ▁is ▁the ▁line ▁bundle ▁associated ▁to ▁a ▁Cart ier ▁divis or . ▁More ▁precisely , ▁if ▁O ( D ) ▁is ▁invert ible , ▁then ▁there ▁exists ▁an ▁open ▁cover ▁{ U i } ▁such ▁that ▁O ( D ) ▁restrict s ▁to ▁a ▁trivial ▁bundle ▁on ▁each ▁open ▁set . ▁For ▁each ▁U i , ▁choose ▁an ▁isomorphism ▁ ▁The ▁image ▁of ▁ ▁under ▁this ▁map ▁is ▁a ▁section ▁of ▁O ( D ) ▁on ▁U i . ▁Because ▁O ( D ) ▁is ▁defined ▁to ▁be ▁a ▁sub she af ▁of ▁the ▁she af ▁of ▁rational ▁functions , ▁the ▁image ▁of ▁ 1 ▁may ▁be ▁identified ▁with ▁some ▁rational ▁function ▁fi . ▁The ▁collection ▁ ▁is ▁then ▁a ▁Cart ier ▁divis or . ▁This ▁is ▁well - defined ▁because ▁the ▁only ▁choices ▁involved ▁were ▁of ▁the ▁covering ▁and ▁of ▁the ▁isomorphism , ▁neither ▁of ▁which ▁change ▁the ▁Cart ier ▁divis or . ▁This ▁Cart ier ▁divis or ▁may ▁be ▁used ▁to ▁produce ▁a ▁she af , ▁which ▁for ▁distinction ▁we ▁will ▁not ate ▁L ( D ). ▁There ▁is ▁an ▁isomorphism ▁of ▁O ( D ) ▁with ▁L ( D ) ▁defined ▁by ▁working ▁on ▁the ▁open ▁cover ▁{ U i }. ▁The ▁key ▁fact ▁to ▁check ▁here ▁is ▁that ▁the ▁transition ▁functions ▁of ▁O ( D ) ▁and ▁L ( D ) ▁are ▁compatible , ▁and ▁this ▁amounts ▁to ▁the ▁fact ▁that ▁these ▁functions ▁all ▁have ▁the ▁form ▁▁ ▁In ▁the ▁opposite