Using a result of Harer, we prove certain upper bounds for the homotopical/cohomological dimension of the moduli spaces of Riemann surfaces of compact type, of Riemann surfaces with rational tails and of Riemann surfaces with at most rational components. These bounds would follow from conjectures of Looijenga and Roth-Vakil.
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Gabriele Mondello, A remark on the homotopical dimension of some moduli spaces of stable Riemann surfaces. J. Eur. Math. Soc. 10 (2008), no. 1, pp. 231–241DOI 10.4171/JEMS/109