?? so ... according to current guesses ... ??? .... subdividing a (highest-dim, say for now ....) cone does in fact always "blow-up" the same point ... namely the one corresponding to the unique coset of the linear span of that cone ... but the "blow-up" might not be the canonical blow-up; rather it might be the generalized blow-up corresponding to the ideal power filtration of some "non-radical" (??? or whatever the sensible terminology is here ... ?? ....) ideal whose radical (??? ....) is the maximal ideal corresponding to the special point ... ???? ..... ?? but then there should be just _one_ choice of subdivision (??? ....) which _does_ give the canonical blow-up .... ??? but seems likely that this choice is "lattice-dependent" ??? .... ???? ....

?? "choice of subdivision" vs "choice of subdividing ray" ... ????? ..... ????? .... possibility of filtration more general than ideal power filtration here ???? ..... ???? ......