On the rigidity of moduli of weighted pointed stable curves

Let (M) over bar (g,A[n]) be the Hassett moduli stack of weighted stable curves, and let (M) over barg(,)A[n] be its coarse moduli space. These are compactifications of M-g,(n) and M-g,M-n respectively, obtained by assigning rational weights A = (a(1), ... ,a(n)), 0 < a(i) <= 1 to the markings, they are defined over Z, and therefore over any field. We study the first order infinitesimal deformations of (M) over bar (g,A[n]) and (M) over bar (g,A[n]). In particular, we show that (M) over bar (o,A[n]) is rigid over any field, if g >= 1 then (M) over bar (g,A[n]) is rigid over any field of characteristic zero, and if g + n > 4 then the coarse moduli space (M) over bar (g,A[n]) is rigid over an algebraically closed field of characteristic zero. Finally, we take into account a degeneration of Hassett spaces parametrizing rational curves obtained by allowing the weights to have sum equal to two. In particular, we consider such a Hassett 3-fold which is isomorphic to the Segre cubic hypersurface in P-4, and we prove that its family of first order infinitesimal deformations is non-singular of dimension ten, and the general deformation is smooth. (C) 2017 Elsevier B.V All rights reserved.