▁each ▁point . ▁Again , ▁the ▁analog ous ▁statement ▁fails ▁for ▁higher - cod im ension ▁sub vari eties . ▁As ▁a ▁result ▁of ▁this ▁good ▁property , ▁much ▁of ▁algebraic ▁geometry ▁studies ▁an ▁arbitrary ▁variety ▁by ▁analyz ing ▁its ▁cod im ension - 1 ▁sub vari eties ▁and ▁the ▁corresponding ▁line ▁bund les . ▁ ▁On ▁singular ▁vari eties , ▁this ▁good ▁property ▁can ▁fail , ▁and ▁so ▁one ▁has ▁to ▁distinguish ▁between ▁cod im ension - 1 ▁sub vari eties ▁and ▁vari eties ▁which ▁can ▁locally ▁be ▁defined ▁by ▁one ▁equation . ▁ ▁The ▁former ▁are ▁We il ▁divis ors ▁while ▁the ▁latter ▁are ▁Cart ier ▁divis ors . ▁ ▁Top olog ically , ▁We il ▁divis ors ▁play ▁the ▁role ▁of ▁hom ology ▁classes , ▁while ▁Cart ier ▁divis ors ▁represent ▁coh om ology ▁classes . ▁ ▁On ▁a ▁smooth ▁variety ▁( or ▁more ▁generally ▁a ▁regular ▁scheme ), ▁a ▁result ▁analog ous ▁to ▁Po inc ar é ▁du ality ▁says ▁that ▁We il ▁and ▁Cart ier ▁divis ors ▁are ▁the ▁same . ▁ ▁The ▁name ▁" div is or " ▁goes ▁back ▁to ▁the ▁work ▁of ▁D ed ek ind ▁and ▁We ber , ▁who ▁showed ▁the ▁relev ance ▁of ▁D ed ek ind ▁domains ▁to ▁the ▁study ▁of ▁algebraic ▁curves . ▁The ▁group ▁of ▁divis ors ▁on ▁a ▁curve ▁( the ▁free ▁ab elian ▁group ▁generated ▁by ▁all ▁divis ors ) ▁is ▁closely ▁related ▁to ▁the ▁group ▁of ▁fraction al ▁ide als ▁for ▁a ▁D ed ek ind