On the rigidity of moduli of weighted pointed stable curves

2017/01/20 by Fantechi, Barbara, Massarenti, Alex
#14D06 #14D23 #Algebraic Geometry (math.AG) #FOS: Mathematics #Primary 14H10 #Secondary 14D22

paper · doi:10.48550/arxiv.1701.05861

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 M_g,n and M_g,n respectively, obtained by assigning rational weights A = (a₁,...,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_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.

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• On the automorphisms of Hassett's moduli spaces
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