Tropical Surfaces

Tropical geometry has been well developed in dimension one, including the study of moduli spaces of curves, tropical versions of the Riemann-Roch theorem and of course applications to complex and real enumerative geometry. This talk will survey some tools in the study of tropical surfaces, including tropical modifications, sums, intersection theory, Chow groups and (p,q)-homology. We show that tropical versions of classical formulae for surfaces (adjunction, Noether’s formula) yield combinatorial proofs in certain cases. Using sums and (p,q)-homology, we also present some examples of non-algebraic tropical surfaces.