▁D ▁of ▁low ▁degree ▁is ▁subt le , ▁and ▁not ▁completely ▁determined ▁by ▁the ▁degree ▁of ▁D . ▁The ▁distinct ive ▁features ▁of ▁a ▁compact ▁Riemann ▁surface ▁are ▁reflected ▁in ▁these ▁dimensions . ▁ ▁One ▁key ▁divis or ▁on ▁a ▁compact ▁Riemann ▁surface ▁is ▁the ▁canonical ▁divis or . ▁To ▁define ▁it , ▁one ▁first ▁defines ▁the ▁divis or ▁of ▁a ▁non zero ▁mer omorphic ▁ 1 - form ▁along ▁the ▁lines ▁above . ▁Since ▁the ▁space ▁of ▁mer omorphic ▁ 1 - forms ▁is ▁a ▁ 1 - dimensional ▁vector ▁space ▁over ▁the ▁field ▁of ▁mer omorphic ▁functions , ▁any ▁two ▁non zero ▁mer omorphic ▁ 1 - forms ▁yield ▁linear ly ▁equivalent ▁divis ors . ▁Any ▁divis or ▁in ▁this ▁linear ▁equivalence ▁class ▁is ▁called ▁the ▁canonical ▁divis or ▁of ▁X , ▁K X . ▁The ▁genus ▁g ▁of ▁X ▁can ▁be ▁read ▁from ▁the ▁canonical ▁divis or : ▁namely , ▁K X ▁has ▁degree ▁ 2 g ▁− ▁ 2 . ▁The ▁key ▁tr ich ot omy ▁among ▁compact ▁Riemann ▁surfaces ▁X ▁is ▁whether ▁the ▁canonical ▁divis or ▁has ▁negative ▁degree ▁( so ▁X ▁has ▁genus ▁zero ), ▁zero ▁degree ▁( gen us ▁one ), ▁or ▁positive ▁degree ▁( gen us ▁at ▁least ▁ 2 ). ▁For ▁example , ▁this ▁determ ines ▁whether ▁X ▁has ▁a ▁K äh ler ▁metric ▁with ▁positive ▁curv ature , ▁zero ▁curv ature , ▁or ▁negative ▁curv ature . ▁The ▁canonical ▁divis or ▁has ▁negative ▁degree ▁if ▁and ▁only ▁if ▁X ▁is ▁is omorphic ▁to