?? idea of fan picture as sort of "octopus's brain" .... ??? tangent space of unit point .... ???? or could think of it as translation space of torus bulk ... ?? anyway, i'm imagining this homunculus or octopunculus or whatever sitting at the nexus point ... sending out and receiving messages from the extremities ... and ... ?? each tangent vector points to (generally ...) one of the extreme points ... ???? ....

?? how this picture, if straigthened out, relates to ... ?? prescription for constructing real manifestation of toric variety from fan ??? .... ???? .... ?? hmmm, role of lattice here ???? ..... ????? ....

?? also, how it relates to ... ?? general idea of straightening out peculiar "closed / open reversal" aspect of relationship between "fan" picture and "ordinary geometric" picture of toric variety .... ??? .... ?? ambiguity in "which cone given point lies in" (??? ...) because subset whose definition involves relative complementation is ... ?? tricky .... ??? ... ??? ... ??? .... ?? ..... ??? distr lattice vs boolean alg ... ??? .... ??? ... ??? .... simplicial complex vs fan ... ??? ... ??? ... ??? .....

?? straightening out idea of "fan as filter in poset of cones" ... ?? .... ?? what's that word that i'm looking for that combinatoricists use ?? ... ?? "cut-set" ?? ... i don't think tha that's it, but something like it ... ?? .... ?? extreme limiting cases of fans, and certain such arising canonically from various things ... ?? ....

?? taking fan picture of toric variety seriously in various ways ... ?? trying to see toric line bundle "twisting" in this picture ... ??? ... ??? to what extent do total and/or fiber spaces of toric line bundle have toric structure ?? ... ??? ....