Two-dimensional cycle classes on $$\overline{\mathcal {M}}_{0,n}$$

Abstract

For each $$n\ge 5$$ n ≥ 5 , we give an $$S_n$$ S n -equivariant basis for $$H_4(\overline{\mathcal {M}}_{0,n},\mathbb {Q})$$ H 4 ( M ¯ 0 , n , Q ) , as well as for $$H_{2(n-5)}(\overline{\mathcal {M}}_{0,n},\mathbb {Q})$$ H 2 ( n - 5 ) ( M ¯ 0 , n , Q ) . Such a basis exists for $$H_2(\overline{\mathcal {M}}_{0,n},\mathbb {Q})$$ H 2 ( M ¯ 0 , n , Q ) and for $$H_{2(n-4)}(\overline{\mathcal {M}}_{0,n},\mathbb {Q})$$ H 2 ( n - 4 ) ( M ¯ 0 , n , Q ) , but it is not known whether one exists for $$H_{2k}(\overline{\mathcal {M}}_{0,n},\mathbb {Q})$$ H 2 k ( M ¯ 0 , n , Q ) when $$3\le k\le n-6$$ 3 ≤ k ≤ n - 6 .