?? to what extent did we ever work out and / or legitimately reject idea of "filter" approach to fan ??? ...

subdivision comparison morphism seems to actually exist ... from subdivided to original ...

?? -> ...

x |-> a

y |-> ab

?? ....

?? "given x,y, what are the solutions for a,b to x=a,y=ab?" ... ???? .....

?? aka "solutions for b to y=xb" .... ???? ....

?? "subring tend to correspond to quotient space" .... ??? .... ?? except that the exceptional case tends to become the usual case .... ???? .....

?? b = y/x .... ???? one solution, except none when x = 0 but y =/= 0 ... (?? when working over a field ??? ... ??? ... ?? ...) ...?? hmmm, but also _lots_ of solutions when both x = 0 and y = 0 .... ... ... ??? .... ??? so not "dense open subspace" ..... ??? ...... ???? has "blow-up" aspect to it ???? ..... ???? ..... hmmmm ....... ???? ......

?? image neither open nor closed ??? ....

?? ingredient (?? ...) in blow-up but not complete blow-up by itself .... ?? .... ?? sort of like "transplanting skin from elsewhere on your body ..." .... ???? ...... "skin graft" ... ??? "auto-skin-graft" ?? ....

?? so should be interesting trying to see actual blow-up as glued together from these things ..... ???? ...... ?? "two ribbons, each with its own third dimension, giving four dimensions total, glued together to form blow-up" .... ??? ....

??? was going to mention before contrast between "voodoo limit" and "blow-up limit" ..... ???? .....

?? so birational isomorphism as somewhat more general than dense zariski open inclusion .... well, hmmm, we really should have / secretly (?? from ourself ??) did know that, because blow-up comparisons qualify too ... but now we're seeing these blow-up / inclusion hybrids (?? simply composites, in some case ... ??? .... ?? hmmm, maybe yes ... thinking in "fan subdivision" picture ... ?? ... usual danger of over-generalizing too much from 2d pictorial intuition .... ??? ..... ?? inclusion of toric open into blow-up, followed by blow-down .... ??? ....) .... ??? ......

?? mental image of "ribbon twist" here .... ????? ..... ?? ribbon very wide though ??? ..... line-of-sight singularity .... ???? ....

?? "infinite subdivision" ... ??? ... ??? "cauchy completion of filter poset" ??? ... ???? ... ????? .... ??? ...

??? "taking affine wedge and not subdividing it, but supplementing it with maximally subdivided wedges to fill it whole fan ...." ... ??? ...

?? confusion about non-/canonicalness of blow-up here .... ??? role of co-dimension .... ??? .... ???? ..... ?? again, to what extent did we work this out before ..... ??? .....

?? "blowing up higher codimension subvariety by blowing up codimension one subvarieties intersecting in it and then combining (?? ...) together ?? ... ??? resemblance / relationship to "removing higher co-dimension subariety by removing codimension one subvarieties intersecting in it and then combining (?? ...) together" ??? .... ???? ..... ????? .....

?? confusionabout what sort of subvariety can get blown up in "toric" case ... ?? ...

?? "toric divisor" ... ??? .... ?? higher codimension analog .... ??? .... ??? ... intersection ... ??? ....