The (ir)reducibility problem for Severi varieties on toric surfaces

A classical theorem due to Joe Harris asserts that over the field of complex numbers the Severi varieties parameterizing irreducible reduced plane curves of given degree and geometric genus are irreducible. Little is known about the irreducibility problem on other surfaces. In my talk I'll consider the case of toric surfaces. I’ll discuss several cases when the Severi varieties are known to be irreducible, and will give examples of toric surfaces admitting reducible Severi varieties. Although, in these examples one can prove reducibility algebraically, the examples can be understood (and, in fact, were discovered) tropically.