Hi. I have some questions concerning tropical geometry:

1) If I'm correct, there is a notion of 'tropical morphism' between tropical manifolds. What about a notion of 'rational map' between such varieties? In fact, I wonder if some authors have already introduced/considered tropical notions analogous to the standard notions of classical birational geometry (birational map, blow up, etc.). Of course, I'm aware that 'tropical modification' that is 1-codimensional blow-ups do not have to be considered as birational transforms but as isomorphisms.

2) Is there a theory of linear systems on tropical surfaces?

3) What about the notion of "canonical class" of a smooth tropical surface?

Thanks for the answers.