A simple branched covering is a branched covering with branching points of degree at most 2, in some context, it is also required to have at most one branching point in each fiber.
My question is one of the exercises from Donaldson's Riemann Surface: every Riemann surface can be realized as a simple branched covering over sphere.
Donaldson mentioned this result once again in chapter 14 when he talked about Hurwitz moves. And Donaldson said it was by Riemann-Roch theorem.
There is one proof in section 8 of http://www.jstor.org/stable/pdfplus/1970748.pdf?acceptTC=true&jpdConfirm=true, but it used more than Riemann-Roch.