?? comparison morphism of blow-up ... from blown-up to original .... toric case ... fan picture .... ??? "subdivision" .... comparison map from subdivided to original ...... ????? ......

?? problematicness (?? ...) of " ...... ?????? .....


?? hmmm, confusion here between .... "subdividing" and "adding on extra chambers" .... ??? ... ???? relationship(s ?? ...) ...

??? "dense zariski open inclusion" =?= "birational isomorphism" ???? .....

(thinking about "birational geometry as guide to glueing" here ... ??? ...)

??? "universal completion of toric variety" ... ??? ...(also non-toric case ...) ... ??? as maybe problematic in way relating to problematicness of "universal blow-up" ???? ..... ???? .....

?? still confused about which of these things really are problematic .... ??? .... ?? as long as there's a genuine comparison morphism, then doesn't that let you do stuff here .... ???? ....

?? "toric algebraic completion" .... "forgetting lattice" ??? .... ????.... ?? q vs r ??? ... ??? .... ?? confusion about whether there's galois symmetry here .... ???? ....... ???? .... ?? ... "cover" .... ??? ... vs "canonicalness / uniqueness of roots" .... ????? ..... ???? roots of unity .... ???? .... fan picture vs .... ??? ...... ????? .....

?? hmmm, maybe a lot of the confusion that i seem to be experiencing here (?? this post in general ?? ....) has to do with ... ??? having at least temporarily totally forgotten how the "poincare duality" aspect of fans works .... ?? open / closed reversal .... ???? ..... ???? ....

?? actually not sure we did completely forget that ... because whole (?? ...) point of question as ... ?? involving _other_ geometric picture, where open / closed is not reversed .... ???? ....

?? anyway, lots of stuff to be straightened out here ...