▁is ▁intersection ▁with ▁multip licity ▁two ) ▁is ▁one ▁quadratic ▁condition , ▁so ▁determined ▁a ▁quad ric ▁in ▁P 5 . ▁However ▁the ▁linear ▁system ▁of ▁divis ors ▁consisting ▁of ▁all ▁such ▁quadr ics ▁is ▁not ▁without ▁a ▁base ▁loc us . ▁In ▁fact ▁each ▁such ▁quad ric ▁contains ▁the ▁Ver on ese ▁surface , ▁which ▁paramet riz es ▁the ▁con ics ▁ ▁( a X ▁+ ▁b Y ▁+ ▁c Z ) 2 ▁= ▁ 0 ▁ ▁called ▁' double ▁lines '. ▁This ▁is ▁because ▁a ▁double ▁line ▁intersect s ▁every ▁line ▁in ▁the ▁plane , ▁since ▁lines ▁in ▁the ▁project ive ▁plane ▁intersect , ▁with ▁multip licity ▁two ▁because ▁it ▁is ▁double d , ▁and ▁thus ▁satisfies ▁the ▁same ▁intersection ▁condition ▁( inter section ▁of ▁multip licity ▁two ) ▁as ▁a ▁non de generate ▁con ic ▁that ▁is ▁tang ent ▁to ▁the ▁line . ▁ ▁The ▁general ▁Bé z out ▁theorem ▁says ▁ 5 ▁general ▁quadr ics ▁in ▁ 5 - space ▁will ▁intersect ▁in ▁ 3 2 ▁= ▁ 2 5 ▁points . ▁But ▁the ▁relevant ▁quadr ics ▁here ▁are ▁not ▁in ▁general ▁position . ▁From ▁ 3 2 , ▁ 3 1 ▁must ▁be ▁subtract ed ▁and ▁attributed ▁to ▁the ▁Ver on ese , ▁to ▁leave ▁the ▁correct ▁answer ▁( from ▁the ▁point ▁of ▁view ▁of ▁geometry ), ▁namely ▁ 1 . ▁This ▁process ▁of ▁att rib uting ▁inter sections ▁to ▁' de generate ' ▁cases ▁is ▁a ▁typical ▁geometric ▁introduction ▁of ▁a ▁' f ud ge ▁factor '.