10 MASAYOSHI NAGAT A Another importan t subvariet y i n W corresponding t o a given subvariet y i n U is the on e called th e prope r transform , whic h is defined a s follows i n the birationa l case . I f U and W are birational, the n th e se t F o f point s o n U at whic h th e rationa l ma p U — W is not bireg - ular form s a proper close d subse t o f U. If a subvariety C of U is not containe d i n F, the n C - ( C n F) correspond s biregularl y t o a subset C 1 of W and th e closur e C * o f C f i s a sub- variety o f W . C * i s called th e proper transform of C on W. Thi s notion ma y b e generalize d to th e cas e where W —• U is not birational , bu t w e omit th e case . For detail s o f th e topic s in thi s section, see Zariski [24] .

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