?? so ... ?? if blow-up systematically gets along with toric structure in some nice way (still not completely convinced of it yet, but ... ?? ...) then ... ??? ought to look for (and be able to find ... ?? ...) some nice conceptual explanation of why ... ???? ... ?? perhaps at TAG theory level .... ?? .... ??well, perhaps the "free commutative monoidal action on ideal, with Z-grading ..." idea goes through in toric context, but ... ?? not clear how conceptual this is, yet ... ??? ....