In this paper, we use the perspective of linear series, and in particular results following from the degeneration tools of limit linear series, to give a number of new results on the existence and non-existence of tamely branched covers of the projective line in positive characteristic. Our results are both in terms of ramification indices and the sharper invariant of monodromy cycles, and the first class of results are obtained by intrinsically algebraic and positive-characteristic arguments.
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