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On the automorphisms of Hassett's moduli spaces

Authors: Alex Massarenti and Massimiliano Mella
Journal: Trans. Amer. Math. Soc. 369 (2017), 8879-8902
MSC (2010): Primary 14H10, 14J50; Secondary 14D22, 14D23, 14D06
Published electronically: May 30, 2017
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Abstract: Let $ \overline {\mathcal {M}}_{g,A[n]}$ be the moduli stack parametrizing weighted stable curves, and let $ \overline {M}_{g,A[n]}$ be its coarse moduli space. These spaces have been introduced by B. Hassett, as compactifications of $ \mathcal {M}_{g,n}$ and $ M_{g,n}$, respectively, by assigning rational weights $ A = (a_{1},\dots ,a_{n})$, $ 0< a_{i} \leq 1$ to the markings. In particular, the classical Deligne-Mumford compactification arises for $ a_1 = \dots = a_n = 1$. In genus zero some of these spaces appear as intermediate steps of the blow-up construction of $ \overline {M}_{0,n}$ developed by M. Kapranov, while in higher genus they may be related to the LMMP on $ \overline {M}_{g,n}$. We compute the automorphism groups of most of the Hassett spaces appearing in Kapranov's blow-up construction. Furthermore, if $ g\geq 1$ we compute the automorphism groups of all Hassett spaces. In particular, we prove that if $ g\geq 1$ and $ 2g-2+n\geq 3$, then the automorphism groups of both $ \overline {\mathcal {M}}_{g,A[n]}$ and $ \overline {M}_{g,A[n]}$ are isomorphic to a subgroup of $ S_{n}$ whose elements are permutations preserving the weight data in a suitable sense.

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Additional Information

Alex Massarenti
Affiliation: IMPA, Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, Brazil
Address at time of publication: Universidade Federal Fluminense - UFF, Rua Mario Santos Braga, 24020-140, Niteroi, Rio de Janeiro, Brazil

Massimiliano Mella
Affiliation: Dipartimento di Matematica e Informatica, Università di Ferrara, Via Machiavelli 35, 44100 Ferrara, Italy

Keywords: Moduli space of curves, Hassett's moduli spaces, fiber type morphism, automorphisms
Received by editor(s): October 13, 2015
Received by editor(s) in revised form: April 20, 2016, and April 29, 2016
Published electronically: May 30, 2017
Additional Notes: This work was partially supported by Progetto PRIN 2010 “Geometria sulle varietà algebriche” MIUR and GRIFGA. This work was done while the first author was a Post-Doctorate at IMPA, funded by CAPES-Brazil.
Article copyright: © Copyright 2017 American Mathematical Society

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