▁in ▁the ▁following ▁way : ▁Cho osing ▁ ▁not ▁all ▁zero ▁points ▁in ▁a ▁fi ber ▁of ▁L ▁cho oses ▁a ▁fi ber ▁of ▁the ▁t aut ological ▁line ▁bundle ▁on ▁Pr , ▁so ▁choosing ▁ ▁non - sim ult ane ously ▁van ishing ▁global ▁sections ▁of ▁L ▁determ ines ▁a ▁map ▁from ▁X ▁into ▁project ive ▁space ▁Pr . ▁This ▁map ▁sends ▁the ▁fib ers ▁of ▁L ▁to ▁the ▁fib ers ▁of ▁the ▁dual ▁of ▁the ▁t aut ological ▁bundle . ▁More ▁specifically , ▁suppose ▁that ▁ ▁are ▁global ▁sections ▁of ▁L . ▁In ▁a ▁small ▁neighborhood ▁U ▁in ▁X , ▁these ▁sections ▁determine ▁k - valu ed ▁functions ▁on ▁U ▁whose ▁values ▁depend ▁on ▁the ▁choice ▁of ▁trivial ization . ▁However , ▁they ▁are ▁determined ▁up ▁to ▁simultane ous ▁multiplication ▁by ▁a ▁non - zero ▁function , ▁so ▁their ▁r ati os ▁are ▁well - defined . ▁That ▁is , ▁over ▁a ▁point ▁x , ▁the ▁values ▁ ▁are ▁not ▁well - defined ▁because ▁a ▁change ▁in ▁trivial ization ▁will ▁multiply ▁them ▁each ▁by ▁a ▁non - zero ▁constant ▁ λ . ▁But ▁it ▁will ▁multiply ▁them ▁by ▁the ▁same ▁constant ▁ λ , ▁so ▁the ▁hom ogeneous ▁coordinates ▁[ s 0 ( x ) ▁: ▁... ▁: ▁sr ( x )] ▁are ▁well - defined ▁as ▁long ▁as ▁the ▁sections ▁ ▁do ▁not ▁simultaneously ▁van ish ▁at ▁x . ▁ ▁Therefore , ▁if ▁the ▁sections ▁never ▁simultaneously ▁van ish , ▁they ▁determine ▁a ▁form ▁[ s 0 ▁: ▁... ▁: