On the automorphisms of Hassett’s moduli spaces
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- by Alex Massarenti and Massimiliano Mella PDF
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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 $A = (a_{1},\dots ,a_{n})$, $0< a_{i} \leqslant 1$ 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 $g\geqslant 1$ we compute the automorphism groups of all Hassett spaces. In particular, we prove that if $g\geqslant 1$ and $2g-2+n\geqslant 3$, 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.References
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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
- MR Author ID: 961373
- Email: alexmassarenti@id.uff.br
- Massimiliano Mella
- Affiliation: Dipartimento di Matematica e Informatica, Università di Ferrara, Via Machiavelli 35, 44100 Ferrara, Italy
- Email: mll@unife.it
- 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.
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 8879-8902
- MSC (2010): Primary 14H10, 14J50; Secondary 14D22, 14D23, 14D06
- DOI: https://doi.org/10.1090/tran/6966
- MathSciNet review: 3710647