▁vers a , ▁and ▁what ▁additional ▁properties ▁ φ ▁might ▁have . ▁ ▁If ▁Z ▁is ▁a ▁prime ▁We il ▁divis or ▁on ▁X , ▁then ▁ ▁is ▁a ▁closed ▁ir reducible ▁sub sch eme ▁of ▁Y . ▁ ▁Depending ▁on ▁ φ , ▁it ▁may ▁or ▁may ▁not ▁be ▁a ▁prime ▁We il ▁divis or . ▁ ▁For ▁example , ▁if ▁ φ ▁is ▁the ▁blow ▁up ▁of ▁a ▁point ▁in ▁the ▁plane ▁and ▁Z ▁is ▁the ▁exception al ▁divis or , ▁then ▁its ▁image ▁is ▁not ▁a ▁We il ▁divis or . ▁ ▁Therefore ▁ φ * Z ▁is ▁defined ▁to ▁be ▁ ▁if ▁that ▁sub sch eme ▁is ▁a ▁prime ▁divis or ▁and ▁is ▁defined ▁to ▁be ▁the ▁zero ▁divis or ▁otherwise . ▁ ▁Ext ending ▁this ▁by ▁linear ity ▁will , ▁assuming ▁X ▁is ▁quasi - comp act , ▁define ▁a ▁hom omorphism ▁ ▁called ▁the ▁push forward . ▁ ▁( If ▁X ▁is ▁not ▁quasi - comp act , ▁then ▁the ▁push forward ▁may ▁fail ▁to ▁be ▁a ▁locally ▁finite ▁sum .) ▁ ▁This ▁is ▁a ▁special ▁case ▁of ▁the ▁push forward ▁on ▁Ch ow ▁groups . ▁ ▁If ▁Z ▁is ▁a ▁Cart ier ▁divis or , ▁then ▁under ▁m ild ▁hypoth eses ▁on ▁ φ , ▁there ▁is ▁a ▁pull back ▁ φ * Z . ▁ ▁She af - the oret ically , ▁when ▁there ▁is ▁a ▁pull back ▁map ▁, ▁then ▁this ▁pull back ▁can ▁be ▁used ▁to ▁define ▁pull back ▁of ▁Cart ier ▁divis