On the Moduli Space of Pointed Algebraic Curves of Low Genus II ---Rationality---

We show that the moduli space M N g,1 of pointed algebraic curves of genus g with a given numerical semigroup N is an irreducible rational variety if N is generated by less than five elements for low genus (g ≤ 6) except one case.As a corollary to this result, we get a computational proof of the rationality of the moduli space M g,1 of pointed algebraic curves of genus g for 1 ≤ g ≤ 3.If g ≤ 5, we also have that M N g,1 is an irreducible rational variety for any semigroup N except two cases.It is known that such a moduli space M N g,1 is non-empty for g ≤ 7.