For a striking example of a classical result in algebraic geometry given a tropical proof, see A tropical proof of the Brill-Noether Theorem by Cools, Draisma, Payne, and Robeva. The original proof of this theorem (by Griffiths and Harris) involves subtle transversality arguments, which they are able to circumvent in this "combinatorial" proof. The new proof is also valid in all characteristics.
Matthew Kahle
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