?? morphism of affine toric varieties corresponding to inclusion of checkerboard lattice into chessboard lattice ... ?? ... ??? geometric interpretation .... ??? "covering" ???? .... ??? ..... ????? .....

??? relationship to kummer's chemistry analogy ?? ...????.... ??? ....

?? actually nice (... ?? ...) maps going both ways ... actually lots of maps both ways; would be nice to understand them all ... ???? .....

?? relationship between .... geometric fiber size and algebraic cokernel size .... ?? ... ??? seems like pretty reasonable guess ... ?? .... ?? kummer ... ????? .....

??? hmmm, so then what about ... "galois group of covering" here ... ?????? ...... and / or "galois correspondent" .... ????? ........ ???? abelianness here ??? ..... ???? ....... .... torus of toric variety .... stable 2-group of toric dimensional theory .... ??? .....

??? "radical extension" of toric varieties ... ??? shortage of other kinds ???? ..... ?? "snugness" and "preservation of face structure" ??? .... ?? snugness in algebraic number theory .... ??? .....

?? galois-theoretic aspect of accidental topos ... ??? ..... .... kummer ... ???? ...

??? toric ramification .... face ?? .... ?? "toric local ..." .... ??? .... "toric birational ..." (?? stable 2-group ... ??? ...) .... "toric algebraic group" .... ?? "toric langlands ..." .... ??? .....

?? "toric [closure of geometric image ...]" ??? ....

?? "sophisticated fiber" ... (?? flatness ... ??? ...) ... dimension calculation therefor, vs for "zariski tangent space at singular point" ... ?? in toric case .... ???? .... ... spectrum of cohomology ring ... ???? .... ??? ... ???? ...