▁sr ] ▁which ▁gives ▁a ▁map ▁from ▁X ▁to ▁Pr , ▁and ▁the ▁pull back ▁of ▁the ▁dual ▁of ▁the ▁t aut ological ▁bundle ▁under ▁this ▁map ▁is ▁L . ▁In ▁this ▁way , ▁project ive ▁space ▁acqu ires ▁a ▁universal ▁property . ▁ ▁The ▁universal ▁way ▁to ▁determine ▁a ▁map ▁to ▁project ive ▁space ▁is ▁to ▁map ▁to ▁the ▁project iv ization ▁of ▁the ▁vector ▁space ▁of ▁all ▁sections ▁of ▁L . ▁In ▁the ▁topological ▁case , ▁there ▁is ▁a ▁non - van ishing ▁section ▁at ▁every ▁point ▁which ▁can ▁be ▁constructed ▁using ▁a ▁b ump ▁function ▁which ▁van ishes ▁outside ▁a ▁small ▁neighborhood ▁of ▁the ▁point . ▁Because ▁of ▁this , ▁the ▁resulting ▁map ▁is ▁defined ▁everywhere . ▁However , ▁the ▁cod om ain ▁is ▁usually ▁far , ▁far ▁too ▁big ▁to ▁be ▁useful . ▁The ▁opposite ▁is ▁true ▁in ▁the ▁algebraic ▁and ▁hol omorphic ▁settings . ▁Here ▁the ▁space ▁of ▁global ▁sections ▁is ▁often ▁finite ▁dimensional , ▁but ▁there ▁may ▁not ▁be ▁any ▁non - van ishing ▁global ▁sections ▁at ▁a ▁given ▁point . ▁( As ▁in ▁the ▁case ▁when ▁this ▁procedure ▁construct s ▁a ▁Le f sch etz ▁p enc il .) ▁In ▁fact , ▁it ▁is ▁possible ▁for ▁a ▁bundle ▁to ▁have ▁no ▁non - zero ▁global ▁sections ▁at ▁all ; ▁this ▁is ▁the ▁case ▁for ▁the ▁t aut ological ▁line ▁bundle . ▁When ▁the ▁line ▁bundle ▁is ▁sufficiently ▁am ple ▁this ▁construction ▁ver ifies ▁the ▁K od ai ra ▁embedding ▁theorem . ▁ ▁Det erm in ant