?? so ... this idea that "underlying module of an ideal" more or less embodies whole "blow-up" concept still feels mostly right-track to me ... (?? though .... ?? extent to which underlying module really knows about ideal power filtration ... idealpowers vs tensor powers ... filtration .... ??? ... ??? ....) ... but there is at least one huge key aspect that i was overlooking in a couple of recent posts, evidently responsible for a certain proportion of the confusion there ...

?? ... this key aspect ... idea that "co-domension 1 subvarieties can't really be blown up" ... vs obvious fact that spectrum of symmetric algebra of invertible module has extra fiber dimension ... so it must be that it's really important to _projectivize_ to cancel out this extra dimension ... ????.....

???so symmetric algebras of different invertible modules are _not_ all the same .... and relate (of course ... ??? ...) to different projective embeddings of the "vacuous blow-up" ...

(??maybe though i should still wonder a little about what happens if you forget the grading of the symmetric algebra ... ??probably you can canonically reconstruct it though ??? ... ??? ....)

??anyway, this is probably helping in connecting recent emphasis on "underlying module of ideal as embodying blow-up" vs ... ??emphasis at other times on ... ideal power filtration and associated graded stuff and "stack" (...) interpretation ("renormalization group" ...) of all that stuff ... in understanding "blow-up" ... ??? .....

?? "inherently projective aspect of blow-up" ... ?? ... "introducing one new line bundle" ... ?? as tending to move to foreground in case where more purely blow-up aspect of blow-up is vacuous ... ???....

???some confusion here about ... ???projectivization wrt different dimensions ... ??? ..... ??? ...

???how does idea of "global sections of line bundle as forming ambient vector space of projective embedding" fit in here ??? ....

??? global sections of _new_ line bundle as ..... ????? .... ??? ....

"renormalization" .... "associated graded ..." .... "normal cone" .... ??in singular case ... ??? "conicalness of singularity" ... ????... ... ???? ..... "rees ..." .... ??? ..... deformation ...... ????? .....

??????????????? ......

???normal cone as "thing being ("scalingly" ... ???) deformed" ?? .... (or more or less equivalently but better ... ??? ... "re-scaling limit" ...) .... whole deformation thing, and "rees ..." ... ??? .....

?? "differential calculus as special case of renormalization group ...." ... ??? ... ?? "dimensional analysis" ...