- vector ▁space ▁H 0 ( X , ▁O ( D )) ▁has ▁finite ▁dimension . ▁The ▁Riemann – R och ▁theorem ▁is ▁a ▁fundamental ▁tool ▁for ▁computing ▁the ▁dimension ▁of ▁this ▁vector ▁space ▁when ▁X ▁is ▁a ▁project ive ▁curve . ▁Success ive ▁general izations , ▁the ▁Hir z ebru ch – R iemann – R och ▁theorem ▁and ▁the ▁Gro th end ieck – R iemann – R och ▁theorem , ▁give ▁some ▁information ▁about ▁the ▁dimension ▁of ▁H 0 ( X , ▁O ( D )) ▁for ▁a ▁project ive ▁variety ▁X ▁of ▁any ▁dimension ▁over ▁a ▁field . ▁ ▁Because ▁the ▁canonical ▁divis or ▁is ▁intr ins ically ▁associated ▁to ▁a ▁variety , ▁a ▁key ▁role ▁in ▁the ▁classification ▁of ▁vari eties ▁is ▁played ▁by ▁the ▁maps ▁to ▁project ive ▁space ▁given ▁by ▁K X ▁and ▁its ▁positive ▁multi ples . ▁The ▁K od ai ra ▁dimension ▁of ▁X ▁is ▁a ▁key ▁bir ational ▁invariant , ▁meas uring ▁the ▁growth ▁of ▁the ▁vector ▁spaces ▁H 0 ( X , ▁m K X ) ▁( mean ing ▁H 0 ( X , ▁O ( m K X ))) ▁as ▁m ▁increases . ▁The ▁K od ai ra ▁dimension ▁div ides ▁all ▁n - dimensional ▁vari eties ▁into ▁n + 2 ▁classes , ▁which ▁( very ▁roughly ) ▁go ▁from ▁positive ▁curv ature ▁to ▁negative ▁curv ature . ▁ ▁" Q " - div is ors ▁ ▁Let ▁X ▁be ▁a ▁normal ▁variety . ▁A ▁( We il ) ▁Q -