?? so given dimensional theory, consider .... ???? ..... commutative monoid obtained as follows :

?? take ab gp of iso classes of dimensions ...

?? take subset of "occupied" dimensions ...

?? take submonoid generated by it ...

??? .....

??? conceptual / geometric interpretation supposed to be ... ?? affine toric variety (?? ...) given by .... ???? ..... ????? .....

??? well, this started out as just a straightforward attempt to extract an affine toric variety in the form of a commutative monoid from a [torus acting on an affine variety, equipped with a point of the variety ...] in the form of a [dimensional theory equipped with a model ... ??? ...] ... ?? but then the model started seeming a bit superfluous to me ... though lack of it may in fact be screwing up / complicating conceptual /geometric interpretation ... ??? ....

??? "picard stack" and toric variety .... ????? ......

??? "jacobian ..." .... ?? ... "albanese ..." .... ???? ....

??? elliptic curve .... ???? .....

?? underlying real variety of complex variety .... ????? .....


?? kaleidoscope case?? .... ??? hmmm, as what more or less prompted the idea in the first place ??? .... ?? and maybe it's in fact tying in in a nice way here .... ???? .... ???? ......

?? issue of how monoid algebras can sometimes be twisted .... ???? .....