The irreducibility of Hurwitz spaces and Severi varieties on toric surfaces

In 1969, Fulton introduced classical Hurwitz spaces parametrizing simple d-sheeted coverings of the projective line in the algebro-geometric setting. He established the irreducibility of these spaces under the assumption that the characteristic of the ground field is greater than d, but the irreducibility problem in smaller characteristics remained open. We resolve this problem in the current paper and prove that the classical Hurwitz spaces are irreducible over any algebraically closed field. On the way, we establish the irreducibility of Severi varieties in arbitrary characteristic for a rich class of toric surfaces, including all classical toric surfaces. Our approach to the irreducibility problems comes from tropical geometry, and the paper contains two more results of independent interest - a lifting result for parametrized tropical curves and a strong connectedness property of the moduli spaces of parametrized tropical curves.