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The global geometry of the moduli spaces of higher spin curves and their birational classification is largely unknown for   $g \geq 2$   and   $r &gt;2$  . Using quite related geometric constructions, we almost complete the picture of the known results in genus   $g\leq 4$   showing the rationality of the moduli spaces of even and odd 4-spin curves of genus 3, of odd spin curves of genus   $4,$   and of 3-spin curves of genus 4.

On the Rationality of the Moduli of Higher Spin Curves in Low Genus