For every projective variety of dimension at least two that is not a rational surface, blowing up sufficiently many very general points yields a variety with no nontrivial automorphism.
On the automorphisms of moduli spaces of curves
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In the last years the biregular automorphisms of the Deligne-Mumford's and Hassett's compactifications of the moduli space of n-pointed genus g smooth curves have been extensively studied by A. Bruno and the authors. In this paper we give a survey of these recent results and extend our techniques to some moduli spaces appearing as intermediate steps of the Kapranov's and Keel's realizations of $\bar{M}_{0,n}$, and to the degenerations of Hassett's spaces obtained by allowing zero weights.
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Automorphisms of very general blow up
For every projective variety of dimension at least two that is not a rational surface, blowing up sufficiently many very general points yields a variety with no nontrivial automorphism.