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We construct a well-behaved compactification of finite covers of a stacky curve using admissible cover degenerations. Using our construction, we compactify the space of tetragonal curves on Hirzebruch surfaces. As an application, we explicitly describe the boundary divisors of the closure in $\overline{M}_6$ of the locus of smooth plane quintic curves.

Covers of stacky curves and limits of plane quintics