On the rigidity of moduli of weighted pointed stable curves

Let M‾g,A[n]be the Hassett moduli stack of weighted stable curves, and let M‾g,A[n]be its coarse moduli space. These are compactifications of Mg,nand Mg,nrespectively, obtained by assigning rational weights A=(a1,…,an), 04 then the coarse moduli space M‾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 P4, and we prove that its family of first order infinitesimal deformations is non-singular of dimension ten, and the general deformation is smooth.