On the rigidity of moduli of weighted pointed stable curves

Abstract

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,n and Mg,n respectively, obtained by assigning rational weights A=(a1,...,an), 0<ai≤1 to the markings; they are defined over Z, and therefore over any field. We study the first order infinitesimal deformations of M‾g,A[n] and M‾g,A[n]. In particular, we show that M‾0,A[n] is rigid over any field, if g≥1 then M‾g,A[n] is rigid over any field of characteristic zero, and if g+n>4 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.