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Let M_{7,n} be the (coarse) moduli space of smooth curves of genus 7 with nmarked points defined over the complex field. We denote by M^1_{7,n;4} thelocus of points inside M_{7,n} representing curves carrying a g^1_4. It isclassically known that M^1_{7,n;4} is irreducible of dimension 17+n. We provein this paper that M^1_{7,n;4} is rational for 0<= n <= 11.

Birational properties of some moduli spaces related to tetragonal curves of genus 7