The stack \bar{\cal{M}}_{g,n} parametrizing Deligne-Mumford stable curves and its coarse moduli space \bar{M}_{g,n} are defined over Z, and therefore over any field. We investigate the rigidity of \bar{\cal{M}}_{g,n}, both in the sense of the absence of first order infinitesimal deformations and of automorphisms not coming form the permutations of the marked points. In particular, we prove that over any perfect field, \bar{M}_{0,n} does not have non-trivial first order infinitesimal deformations and we apply this result to show that, over any field, Aut(\bar{M}_{0,n})≅Sn for n≥5. Furthermore, we extend some of these results to the stacks \bar{\cal{M}}_{g,A[n]} parametrizing weighted stable curves and to their coarse moduli spaces \bar{M}_{g,A[n]}. These spaces have been introduced by Hassett as compactifications of g,n and Mg,n respectively, by assigning rational weights A=(a1,...,an), 0<ai≤1 to the markings. In particular we prove that M⎯⎯⎯0,A[n] is rigid over any perfect field and that ⎯⎯⎯⎯⎯g,A[n] for g≥1 is rigid over any field of characteristic zero. Finally, we study the infinitesimal deformations of the coarse moduli space M⎯⎯⎯g,A[n]. We prove that over any field of characteristic zero M⎯⎯⎯g,A[n] does not have locally trivial first order infinitesimal deformations if g+n≥4. Furthermore we show that M⎯⎯⎯1,2 does not admit locally trivial deformations and that it is not rigid.
