?? blow-down followed by blow-up ..... ???? asf, os .... ????? .....

?? blow-up of curve in 3-space, vs "knot surgery" ... ???? .... ??? ... ??? also "cover of knot complement branched along knot" .... ???? .... ???? ....

?? "birational geometry as guide to glueing" .... ??? relationship to "divisor as holomorphic structure on standard meromorphic line bundle" .... ??? .... glueing vs dimensional doctrine (and related doctrines ...) .... ??? ...

?? (?? nice ...) structure types carried by meromorphic line bundles (= invertible vsps over field of meromorphic functions ...) ... ??? relationship to "non-toric fan" ??? .... ???? certain kind of dimensional theory over dimensional theory of meromorphic line bundles .... ???? _faithfully_ over ?? ... ??? ...

??? toric case ???? .... ??? sensible approach to "toric divisor" ?????? ... "toric holomorphic structure on standard toric meromorphic line bundle" ??? ..... ???? .....

m1 -> g1

| |
v v

m2 -> g2

m2 = another ab gp ...

g2 = trivial ....

??? simply _bi_-graded (by ab gps ...) comm monoid ??

?? faithfulness ??? .....

????? m1 -> m2 -> g1 ??? .... ?? m2 universal groupization of m1 ???? .... ???? ....

?? some confusion here ??? ... ???? kernel of groupization(m1)->g1 ??? ....

?? relationship between line bundle and toric structure ..... ????? .....

?? m2 = groupization(m1)

?? g2 = g1

.... ???? .....