In classical algebraic geometry the adjunction formula relates the canonical class a divisor to the canonical class of the original variety. For smooth curves in surfaces this formula can be written using the genus of the curve, and in the case of singular curves it defines the arithmetic genus. Using Mikhalkin's definition of the canonical class of an abstract tropical surface together with tropical intersection theory we may consider a tropical version of this formula and give sufficient conditions for it to hold. However, examples show these conditions are not necessary.

It is equally interesting to consider situations where the tropical adjunction formula fails. In joint work with E. Brugallé, we use the adjunction formula and a correspondence between complex and tropical intersections of curves to obtain local obstructions to the approximation of tropical curves in surfaces. Applying these results to the pathological lines in smooth tropical surfaces discovered by Vigeland, we discovered a peculiar phenomenon in tropical geometry; smoothness as it is defined in the tropical world is not an intrinsic property of varieties.