  Starting from certain rational varieties blown-up from (P^1)^N, we construct
a tropical, i.e., subtraction-free birational, representation of Weyl groups as
a group of pseudo isomorphisms of the varieties. Furthermore, we develop an
algebro-geometric framework of tau-functions as defining functions of
exceptional divisors on the varieties. In the case where the corresponding root
system is of affine type, our construction yields a class of (higher order)
q-difference Painleve equations and its algebraic degree grows quadratically.
