▁defined ▁to ▁be ▁the ▁quot ient ▁of ▁ ▁by ▁the ▁action ▁of ▁the ▁multip lic ative ▁group ▁k × . ▁Each ▁point ▁of ▁P ( V ) ▁therefore ▁corresponds ▁to ▁a ▁copy ▁of ▁k × , ▁and ▁these ▁copies ▁of ▁k × ▁can ▁be ▁assemble d ▁into ▁a ▁k × - bundle ▁over ▁P ( V ). ▁k × ▁diff ers ▁from ▁k ▁only ▁by ▁a ▁single ▁point , ▁and ▁by ▁ad jo ining ▁that ▁point ▁to ▁each ▁fi ber , ▁we ▁get ▁a ▁line ▁bundle ▁on ▁P ( V ). ▁This ▁line ▁bundle ▁is ▁called ▁the ▁t aut ological ▁line ▁bundle . ▁This ▁line ▁bundle ▁is ▁sometimes ▁denoted ▁ ▁since ▁it ▁corresponds ▁to ▁the ▁dual ▁of ▁the ▁Ser re ▁tw isting ▁she af ▁. ▁ ▁Maps ▁to ▁project ive ▁space ▁Suppose ▁that ▁X ▁is ▁a ▁space ▁and ▁that ▁L ▁is ▁a ▁line ▁bundle ▁on ▁X . ▁A ▁global ▁section ▁of ▁L ▁is ▁a ▁function ▁ ▁such ▁that ▁if ▁ ▁is ▁the ▁natural ▁projection , ▁then ▁= ▁id X . ▁ ▁In ▁a ▁small ▁neighborhood ▁U ▁in ▁X ▁in ▁which ▁L ▁is ▁trivial , ▁the ▁total ▁space ▁of ▁the ▁line ▁bundle ▁is ▁the ▁product ▁of ▁U ▁and ▁the ▁underlying ▁field ▁k , ▁and ▁the ▁section ▁s ▁restrict s ▁to ▁a ▁function ▁. ▁However , ▁the ▁values ▁of ▁s ▁depend ▁on ▁the ▁choice ▁of ▁trivial ization , ▁and ▁so ▁they ▁are ▁determined ▁only ▁up ▁to ▁multiplication ▁by ▁a ▁now here - van ishing ▁function . ▁ ▁Global ▁sections ▁determine ▁maps ▁to ▁project ive ▁spaces