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We show that the complex hyperbolic metrics defined by Deligne-Mostow andThurston on ${\mathcal{M}}_{0,n}$ are singular K\"ahler-Einstein metrics when${\mathcal{M}}_{0,n}$ is embedded in the Deligne-Mumford-Knudsencompactification $\overline{\mathcal{M}}_{0,n}$. As a consequence, we obtain aformula computing the volumes of ${\mathcal{M}}_{0,n}$ with respect to thesemetrics using intersection of boundary divisors of$\overline{\mathcal{M}}_{0,n}$. In the case of rational weights, following anidea of Y. Kawamata, we show that these metrics actually represent the firstChern class of some line bundles on $\overline{\mathcal{M}}_{0,n}$, from whichother formulas computing the same volumes are derived.

Complex hyperbolic volume and intersection of boundary divisors in moduli spaces of genus zero curves