?? classification of affine toric varieties in 2 dimensions ... ??? relationship to ... ?? cokernel of 2-by-2 matrix of integers ... ???.... ??? ...... ?? certain kind of morphism of torus and/or toric variety ?? .... ??? ....

?? some sort of morphism of affine toric variety corresponding to inclusion of cones .... ???? ..... ???? ....

?? so... given endomorphism f of Z^2, consider [... ?? inverse image under inverse (presuming it exists) of g := "f tensored with R" ...]] of N^2 (as "quadrant-lattice" in source of g = target of g^[-1]) ... ????..... ?? any particular nicer way of saying this ?? ....] ??? as affine coordinat monoid of affine toric variety ... ?? ...

?? vague feeling ... stuff about invariant distributions on flag varieties and so forth .... somethng we ran into there ... ??as somehow similar/related to ... ?? how a "face" of a toric variety adheres to it .... ???? ..... ???? singularity of schubert variety .... ????? ..... ?????? ..... .... hmmmmm ..... ?????? .... ??? are we maybe finally figuring out something about singularities of affine toric varieties ?????? ...... ??????? ........

??am i claiming that cone of "quadratic" (...) projective embedding of projective line is singular ??? .... ????hmmm, maybe that would make perfect sense .... ????... .... hmmmm .... ???classic conical singularity as having "toric" structure ???? ....... ??????? ....... ??? in some sense should have already been obvious, maybe ???? .... ... ?? ... (?? ... hmmm, still some confusion between ... remembering vs forgetting grading .... ???? ...... ?????.....) .....

??? toric world as to some extent then good playground for toy examples of singularities .... ????? ...... ????? ......

?? lot of puns on cone / conic / conical here .... ????? ..... ?? mapping cone .... ???? .....

??? blowing-up of toric singularity ..... ?????? ........ ... grading ... ???? ....

??? kaleidoscope as fan here (... ???....) ?????? ......

??? kummer's chemistry analogy .... ??????? .......

?? vague memories of other weird ideas relating to ways lattice can relate to hyperplane in vector space .... ?? .... ??? 2d case ??? ..... ???? .....

??relationship between ["cohomological" aspect of affine coordinate monoid of singular affine toric variety ... ??? or is this aspect maybe somewhat special to 2d case ????? .... ??? maybe misleading to try to generalize from that case ??? .... ????? .....] and [complex of ideas relating to "deformation cohomology" and blowing up of singularities ... ??? .... and so forth ... ???] ??? ....

??? .... "singular morse theory" .... cartographic group action ... ??? ..... mapping cone ..... ????? ...... .... ???? .....