On the unirationality of moduli spaces of pointed curves

We show that Mg,n, the moduli space of smooth curves of genus g together with n marked points, is unirational for g= 12 and 2 ≤ n≤ 4 and for g= 13 and 1 ≤ n≤ 3 , by constructing suitable dominant families of projective curves in P1× P2 and P3 respectively. We also exhibit several new unirationality results for moduli spaces of smooth curves of genus g together with n unordered points, establishing their unirationality for g= 11 , n= 7 and g= 12 , n= 5 , 6.