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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Stacks of trigonal curves


Authors: Michele Bolognesi and Angelo Vistoli
Journal: Trans. Amer. Math. Soc. 364 (2012), 3365-3393
MSC (2010): Primary 14H10; Secondary 14A20, 14C22
Published electronically: February 17, 2012
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Abstract: In this paper we study the stack $ \mathcal {T}_g$ of smooth triple covers of a conic; when $ g \geq 5$ this stack is embedded $ \mathcal {M}_{g}$ as the locus of trigonal curves. We show that $ \mathcal {T}$ is a quotient $ [U_{g}/\Gamma _{g}]$, where $ \Gamma _g$ is a certain algebraic group and $ U_g$ is an open subscheme of a $ \Gamma _g$-equivariant vector bundle over an open subscheme of a representation of $ \Gamma _g$. Using this, we compute the integral Picard group of $ \mathcal {T}_g$ when $ g > 1$. The main tools are a result of Miranda that describes a flat finite triple cover of a scheme $ S$ as given by a locally free sheaf $ E$ of rank two on $ S$, with a section of $ \mathrm {Sym}^{3}E\otimes \mathrm {det}\,E^\vee $, and a new description of the stack of globally generated locally free sheaves of fixed rank and degree on a projective line as a quotient stack.


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Additional Information

Michele Bolognesi
Affiliation: Dipartimento di Matematica, Università di Roma Tre, Largo San Leonardo Murialdo, 1 00146 Roma, Italy
Email: bolognesi.michele@gmail.com

Angelo Vistoli
Affiliation: Scuola Normale Superiore, Università degli Studi di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy
Email: angelo.vistoli@sns.it

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05370-0
PII: S 0002-9947(2012)05370-0
Received by editor(s): February 19, 2010
Received by editor(s) in revised form: April 16, 2010
Published electronically: February 17, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.