On the automorphisms of Hassett's moduli spaces

2013/07/24 by Alex Massarenti, Massimiliano Mella, Massarenti, Alex +1 · 2 citations
Mathematics · #14D06 #14D23 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Primary 14H10 #Secondary 14D22

paper · pdf · doi:10.48550/arxiv.1307.6828

openalex publication_date 2013/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M_g,A[n] be the moduli stack parametrizing weighted stable curves, and let M_g,A[n] be its coarse moduli space. These spaces have been introduced by B. Hassett, as compactifications of M_g,n and M_g,n respectively, by assigning rational weights A = (a₁,...,a_n), 0< a_i ≤ 1 to the markings. In particular, the classical Deligne-Mumford compactification arises for a₁ = ... = a_n = 1. In genus zero some of these spaces appear as intermediate steps of the blow-up construction of M_0,n developed by M. Kapranov, while in higher genus they may be related to the LMMP on M_g,n. We compute the automorphism groups of most of the Hassett's spaces appearing in the Kapranov's blow-up construction. Furthermore, if g≥ 1 we compute the automorphism groups of all Hassett's spaces. In particular, we prove that if g≥ 1 and 2g-2+n≥ 3 then the automorphism groups of both M_g,A[n] and M_g,A[n] are isomorphic to a subgroup of S_n whose elements are permutations preserving the weight data in a suitable sense.

Cited by

• On the rigidity of moduli of curves in arbitrary characteristic
• Remarks on two problems by Hassett

Related

• On the automorphisms of moduli spaces of curves
• On the rigidity of moduli of weighted pointed stable curves
• Existence of good moduli spaces for A_k-stable curves
• Stable maps to quotient stacks with a properly stable point
• The automorphisms group of M_0,n