?? so ... given Z-filtered (nice ... ??? ...) commutative monoid x, trying to picture "rees construction" applied to it ... ??? dot (x1,j) in x X Z as answer to "is x1 born by the jth stage ... ????? ..... ??possible pos/neg confusion here ??? ... ideal power filtration .... ????? .... ?? hmm, this description makes "rees construction" sound somewhat tautological ... ??? .....

?? "convex" toric filtration ??? ..... ???? ..... ????? ..... ?? somehow stuff here reminds me of some of arnold's ideas ... ????? ..... ??? singularity ... "ray" .... "caustic" .... ??? duality ... ??? .... toric ... ???....

?? any possibility of generalizing convexness of filtration to non-toric case ??? .... ?? geometric interpretation ??? .... duality ... ??? ....

??? "toric hilbert basis theorem" .... ????? .....

??? toric case of "hilbert scheme" ..... ???? ......

?? feeling of entitlement to get nice toric variety / blow-up (... ??? ...) from poisson operad situation .... ??? so can it be pushed through somehow, or can we understand what goes wrong ??? .... ??? bit about "toric spectrum vs ordinary spectrum of toric variety" .... ???? .... ??? maybe distinction tends to vanish in "1d" case ????? ...... 1-by-1 matrix multiplication path integral as sharp (=?= toric ??? ....) .... ??????? ....... relationship to "toric" flavor of abelian class field theory ????? ..... ???? ..... transition from 1 to 2 (??...) as threshold for "non-commutativeness" and also "non-sharpness of path integral" =?= "non-toricness ??? .... ???? ..... ????? .....

(added later: ?? ...hmmm .... but there was that bit about ... ??? invertible module of comm ring r, not corresponding to any invertible action of gl(1,r) ... ????? ..... ??? ....)

??? "dynkin (vs coxeter ?? ...) kaleidoscope" (??? =?= "root system" ?? ...) as not much more than "sufficiently homogeneous (??? wrt discrete symmetry group ??? ....) toric variety" ????? ...... ????? "affine" (?? and beyond ?? ...) case ??????? ..... ???? still get perfectly good toric variety from such ????? .....

??? most obvious toric variety coming form kaleidoscope as conspicuously non-singular ??? ..... ???? vs ... ??? vague ideas about trying to relate kaleidoscopic toric variety to schubert singularities ... ???? ......

???when working on schubert singularities did we in fact venture away from the parabolic (or whatever .... ??? ....) lie subalgebra towards the rest of the weyl co-chamber ??????? ..... ??????? .......

????? so consider "N-fan" .... ??????? ...... ???? fan with extra structure ??? ..... lattice under Z, with all cones containing N+ ??? .... ?????? ...... ?? [Z = line-of-sight] perspective ...... ????? ...... wait a minute, bit of cone / cocone confusion there ... all cocones containing N+ ; all cones fitting inside a half-space .... ????? .....

?? by the way is there room for those more general "cone"s (more general than just official cones and cocones ... ??? ....) to show up here ??? ...

??? more generally, fan "respecting" another fan ??? .... respect as more general than subdivide here ??? ?? maybe still not quite general enough though ??? .... f fan .... ?? "f-fan" .... ???living on lattice mapping to lattice of f .... ????....

??? "toric etale fundamental group" ??? .....