In my talk I will discuss the problem of irreducibility of families of curves of given
degree and genus on toric surfaces. Such families, called Severi varieties, have been
intensively studied due to a variety of applications of their geometry to the study of
moduli spaces of curves, and to various enumerative problems. After reviewing briefly
known irreducibility results, I'll describe examples of toric surfaces admitting
reducible Severi varieties, and introduce certain topological and tropical invariants
that allow one to distinguish between different irreducible components.
The talk is based on a joint work with Lionel Lang.
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