▁of ▁a ▁differenti able ▁manifold , ▁where ▁the ▁corresponding ▁line ▁bundle ▁is ▁the ▁determin ant ▁bundle ▁of ▁the ▁tang ent ▁bundle ▁( see ▁below ). ▁ ▁The ▁M ö b ius ▁strip ▁corresponds ▁to ▁a ▁double ▁cover ▁of ▁the ▁circle ▁( the ▁ θ ▁→ ▁ 2 θ ▁mapping ) ▁and ▁by ▁changing ▁the ▁fi ber , ▁can ▁also ▁be ▁viewed ▁as ▁having ▁a ▁two - point ▁fi ber , ▁the ▁unit ▁interval ▁as ▁a ▁fi ber , ▁or ▁the ▁real ▁line . ▁ ▁Complex ▁line ▁bund les ▁are ▁closely ▁related ▁to ▁circle ▁bund les . ▁ ▁There ▁are ▁some ▁celebrated ▁ones , ▁for ▁example ▁the ▁H opf ▁f ibr ations ▁of ▁spher es ▁to ▁spher es . ▁ ▁In ▁algebraic ▁geometry , ▁an ▁invert ible ▁she af ▁( i . e ., ▁locally ▁free ▁she af ▁of ▁rank ▁one ) ▁is ▁often ▁called ▁a ▁line ▁bundle . ▁ ▁Every ▁line ▁bundle ▁ar ises ▁from ▁a ▁divis or ▁with ▁the ▁following ▁conditions ▁ ▁( I ) ▁If ▁X ▁is ▁reduced ▁and ▁ir reducible ▁scheme , ▁then ▁every ▁line ▁bundle ▁comes ▁from ▁a ▁divis or . ▁ ▁( II ) ▁If ▁X ▁is ▁project ive ▁scheme ▁then ▁the ▁same ▁statement ▁holds . ▁ ▁The ▁t aut ological ▁bundle ▁on ▁project ive ▁space ▁ ▁One ▁of ▁the ▁most ▁important ▁line ▁bund les ▁in ▁algebraic ▁geometry ▁is ▁the ▁t aut ological ▁line ▁bundle ▁on ▁project ive ▁space . ▁The ▁project iv ization ▁P ( V ) ▁of ▁a ▁vector ▁space ▁V ▁over ▁a ▁field ▁k ▁is