div is or ▁is ▁a ▁finite ▁formal ▁linear ▁combination ▁of ▁ir reducible ▁cod im ension - 1 ▁sub vari eties ▁of ▁X ▁with ▁rational ▁coefficients . ▁( An ▁R - div is or ▁is ▁defined ▁similarly .) ▁A ▁Q - div is or ▁is ▁effective ▁if ▁the ▁coefficients ▁are ▁non negative . ▁A ▁Q - div is or ▁D ▁is ▁Q - Cart ier ▁if ▁m D ▁is ▁a ▁Cart ier ▁divis or ▁for ▁some ▁positive ▁integer ▁m . ▁If ▁X ▁is ▁smooth , ▁then ▁every ▁Q - div is or ▁is ▁Q - Cart ier . ▁ ▁If ▁▁ ▁is ▁a ▁Q - div is or , ▁then ▁its ▁round - down ▁ ▁is ▁the ▁divis or ▁ ▁where ▁ ▁is ▁the ▁greatest ▁integer ▁less ▁than ▁or ▁equal ▁to ▁a . ▁The ▁she af ▁ ▁is ▁then ▁defined ▁to ▁be ▁ ▁The ▁Gro th end ieck – L ef sch etz ▁hyper plane ▁theorem ▁The ▁Le f sch etz ▁hyper plane ▁theorem ▁implies ▁that ▁for ▁a ▁smooth ▁complex ▁project ive ▁variety ▁X ▁of ▁dimension ▁at ▁least ▁ 4 ▁and ▁a ▁smooth ▁am ple ▁divis or ▁Y ▁in ▁X , ▁the ▁restriction ▁Pic ( X ) ▁→ ▁Pic ( Y ) ▁is ▁an ▁isomorphism . ▁For ▁example , ▁if ▁Y ▁is ▁a ▁smooth ▁complete ▁intersection ▁variety ▁of ▁dimension ▁at ▁least ▁ 3 ▁in ▁complex ▁project ive ▁space , ▁then ▁the ▁Pic ard ▁group ▁of ▁Y ▁is ▁is omorphic ▁to ▁Z , ▁generated ▁by ▁the ▁restriction ▁of ▁the ▁line ▁bundle ▁O ( 1 )