d=ax, e=ay, f=az, g=by, h=bz .... dg=ee, dh=ef, eh=fg ... ?? specializing to d=e=g=0 .... ????as giving a projective line, which is supposed to be the blown-up point .... specializing (?? ...) to d invertible .... ????? ??? both a and x being invertible .... ???? ...... ???well, maybe a isn't too important here ... ??? .....
?? well, when d is invertible, then g and h can be eliminated, co-eliminating the first two relations, and making the third one tautological .... ???? ..... ??? so that seems to help make it plausible that this really is the projective plane with a point blown-up .... ???? ......

?? so.... is this toric ?? ..... ?? .... ?? coming from fan ???? .... ??? ...

?? first try getting torus from it ... ??? .... ?? so, tame stable 2-group given by 3d lattice mapping to 1d .... ????...... ??? then we're supposed to take the kernel or something??? .....

?? just as a wild guess, maybe this will be something like ... ???? 2d lattice divided into 3 cones, one of them checkerboard ... ????? ....... ??hmmm, but ... ??? non-singular toric variety can't be made out of singular pieces, can it ??? ....

?? well, so for a perhaps somewhat different wild guess, maybe there's more cones in the fan here .... say maybe 5 ... ???...... ?? hmm, what dimension (?? ...) cones am i talking about here ??? ...

?? well, so what are the "localizations" (?? ...) here .... ??? also, what are "the localizations of a fan" ??? ..... ?? ....

?? stuff you can invert .... d,f,g,h,df,dg,fh,gh,dfgh ... ???? hmmm, sounds sort of like face-lattice of square ... ????? dfhg ... ???? ..... ???so are we getting some sort of square-ish fan here ??? ...

?? so maybe what's happening, roughly, is that we start with the fan for the projective plane ... and one of the three sectors corresponds to the point that's being blown up .... and that sector is getting "bisected", which somehow corresponds to the point getting blown-up, aka "(functorially ... ??? ...) replaced by a projective line" .... ???????? ......

??idea of resolving singularities by "triangulation" of cones in fan ???? ..... ???? .... .... ?? but what about functoriality of blow-up ... ?? ... ?? ....

?? cone / cocone confusion in trying to understand blow-up / "bisection" ... functoriality thereof ... ????? .....

???hmmm..... ??? so consider projective plane with all 3 apartment points blown up .... ??? is this ess just the a2 kaleidoscope toric variety ??? ..... ??? if so, then what's going on here ???? ......

?? hmmm ... ?? maybe more than one way to interpret kaleidoscope as fan ... ???? idea that for certain (maybe hopefully somewhat obvious ?? ...) way, resulting toric variety is _non_-singular ??? .... ???? weyl chamber as ... "orthant-ish" ... ???? .... ??? weight lattice vs root lattice ??? ..... ????? ....