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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.

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Yes, such things do exist. For a general introduction see:

http://homepages.warwick.ac.uk/staff/D.Maclagan/papers/TropicalBook.pdf (Maclagan and Sturmfels) (see page 9, and on)

A nice paper which discusses some of the questions you are asking is:

http://www.mathematik.uni-kl.de/~boehm/tropical%20riemann%20roch%20and%20tropical%20abel%20jacobi.pdf (tropical linear systems and tropical jacobian, Janko Bohm).

The Maclagan-Sturmfels book talks about blow-ups some.

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