▁( E uclidean ) ▁topology . ▁Like wise , ▁the ▁Pic ard ▁group ▁maps ▁to ▁integral ▁coh om ology , ▁by ▁the ▁first ▁Ch ern ▁class ▁in ▁the ▁topological ▁sense : ▁▁ ▁The ▁two ▁hom omorph isms ▁are ▁related ▁by ▁a ▁commut ative ▁diagram , ▁where ▁the ▁right ▁vertical ▁map ▁is ▁cap ▁product ▁with ▁the ▁fundamental ▁class ▁of ▁X ▁in ▁Bor el – Mo ore ▁hom ology : ▁ ▁For ▁X ▁smooth ▁over ▁C , ▁both ▁vertical ▁maps ▁are ▁is omorph isms . ▁ ▁Global ▁sections ▁of ▁line ▁bund les ▁and ▁linear ▁systems ▁A ▁Cart ier ▁divis or ▁is ▁effective ▁if ▁its ▁local ▁defining ▁functions ▁fi ▁are ▁regular ▁( not ▁just ▁rational ▁functions ). ▁In ▁that ▁case , ▁the ▁Cart ier ▁divis or ▁can ▁be ▁identified ▁with ▁a ▁closed ▁sub sch eme ▁of ▁cod im ension ▁ 1 ▁in ▁X , ▁the ▁sub sch eme ▁defined ▁locally ▁by ▁fi ▁= ▁ 0 . ▁A ▁Cart ier ▁divis or ▁D ▁is ▁linear ly ▁equivalent ▁to ▁an ▁effective ▁divis or ▁if ▁and ▁only ▁if ▁its ▁associated ▁line ▁bundle ▁O ( D ) ▁has ▁a ▁non zero ▁global ▁section ▁s ; ▁then ▁D ▁is ▁linear ly ▁equivalent ▁to ▁the ▁zero ▁loc us ▁of ▁s . ▁ ▁Let ▁X ▁be ▁a ▁project ive ▁variety ▁over ▁a ▁field ▁k . ▁Then ▁multip lying ▁a ▁global ▁section ▁of ▁O ( D ) ▁by ▁a ▁non zero ▁scalar ▁in ▁k ▁does ▁not ▁change ▁its ▁zero ▁loc us . ▁As ▁a ▁result , ▁the ▁project ive ▁space ▁of ▁lines ▁in ▁the