On the unirationality of moduli spaces of pointed curves

Abstract

AbstractWe show that $$\mathcal {M}_{g,n}$$ M g , n , the moduli space of smooth curves of genus g together with n marked points, is unirational for $$g=12$$ g = 12 and $$2 \le n\le 4$$ 2 ≤ n ≤ 4 and for $$g=13$$ g = 13 and $$1 \le n \le 3$$ 1 ≤ n ≤ 3 , by constructing suitable dominant families of projective curves in $$\mathbb {P}^1 \times \mathbb {P}^2$$ P 1 × P 2 and $$\mathbb {P}^3$$ P 3 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 $$g=11, n=7$$ g = 11 , n = 7 and $$g=12, n =5,6$$ g = 12 , n = 5 , 6 .