Francis Brown introduced a partial compactification M-0,n(delta) of the moduli space M-0,M-n. We prove that the gravity cooperad, given by the degree-shifted cohomologies of the spaces M-0,M-n, is cofree as a nonsymmetric anticyclic cooperad; moreover, the cogenerators are given by the cohomology groups of M-0,n(delta). As part of the proof we construct an explicit diagrammatically defined basis of H.(M-0,M-n) which is compatible with cooperadic cocomposition, and such that a subset forms a basis of H.(M-0,n(delta)). We show that our results are equivalent to the claim that H-k (M-0,n(delta)) has a pure Hodge structure of weight 2k for all k, and we conclude our paper by giving an independent and completely different proof of this fact. The latter proof uses a new and explicit iterative construction of M-0,n(delta) from A(n-3) by blow-ups and removing divisors, analogous to Kapranov's and Keel's constructions of (M) over bar (0,n) from Pn-3 and (P-1)(n-3), respectively.