?? "toric divisor" as "way of shifting the walls" ??? ....

?? "hydraulic press" and weylotope degeneration and kaleidoscope fan ..... ???? .... ?? toric and tropical .... ????? .....

?? principal toric divisor of dot in algebraic lattice as ... ??? moving the walls so that they all go through that dot ??? .... ?????.....

?? toric divisors (and/or equivalence of them ... ???) as somewhat screwed up in singular case ... ????...

??? equivalence relations on divisors that people talk about ... ?? why ??? ... ?? try to understand in toric context ???




filtration (??? ...) on toric divisor class group associated with .... ??? progression toric birational, toric hartog .... ???? ..... ??? any relationship to "hodge filtration" ??? .... ???? .... ??? but .... ?? "singularity" .... ????? ....???? ...... ????? "stratification" ????? ..... ?????? ......


?? explaining fan ... ?? start with nice commutative monoid, giving affine toric variety .... draw it's (P,+)-spectrum .... so, cone dual to cocone ... then explain that glueing together affine toric varieties corresponds to glueing together these cones ... into "fan" ....

?? lots of details to soft-pedal here ??? .... ?? why single ambient lattice .... ?? "localization" ... ??? ..... ??"poincare duality" aspect .... (not _too_ soft, i guess ..... glueing along open intersections vs along closed intersections ...) ????? ..... ?? indicate that not claiming things have to be this way, in greatest generality .... more like, this is one particularly nice way they can be ... one nice way to do/allow glueing ... ???? ....??? ....