On the bicanonical map of irregular varieties

Barja, Miguel; Lahoz, Martí; Naranjo, Juan; Pareschi, Giuseppe

doi:10.1090/S1056-3911-2011-00565-1

Authors: Miguel Angel Barja, Martí Lahoz, Juan Carlos Naranjo and Giuseppe Pareschi
Journal: J. Algebraic Geom. 21 (2012), 445-471
DOI: https://doi.org/10.1090/S1056-3911-2011-00565-1
Published electronically: July 27, 2011
MathSciNet review: 2914800
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Abstract | References | Additional Information

Abstract: From the point of view of uniform bounds for the birationality of pluricanonical maps, irregular varieties of general type and maximal Albanese dimension behave similarly to curves. In fact Chen-Hacon showed that, at least when their holomorphic Euler characteristic is positive, the tricanonical map of such varieties is always birational. In this paper we study the bicanonical map. We consider the natural subclass of varieties of maximal Albanese dimension formed by primitive varieties of Albanese general type. We prove that the only such varieties with non-birational bicanonical map are the natural higher-dimensional generalization to this context of curves of genus $2$: varieties birationally equivalent to the theta-divisor of an indecomposable principally polarized abelian variety. The proof is based on the (generalized) Fourier-Mukai transform.

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Additional Information

Miguel Angel Barja
Affiliation: Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, ETSEIB Avda. Diagonal 647, 08028 Barcelona, Spain
Email: miguel.angel.barja@upc.edu

Martí Lahoz
Affiliation: Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain
Email: marti.lahoz@ub.edu

Juan Carlos Naranjo
Affiliation: Departament d’Àlgebra i Geometria, Facultat de Matemàtiques, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spain
MR Author ID: 318430
ORCID: 0000-0003-1989-4924
Email: jcnaranjo@ub.edu

Giuseppe Pareschi
Affiliation: Dipartimento di Matematica, Università di Roma, Tor Vergata, V. le della Ricerca Scientifica, I-00133 Roma, Italy
Email: pareschi@mat.uniroma2.it

Received by editor(s): July 29, 2009
Received by editor(s) in revised form: February 12, 2010
Published electronically: July 27, 2011
Additional Notes: The first, second and third authors were partially supported by the Proyecto de Investigación MTM2006-14234. The first and second authors were also partially supported by 2005SGR-557 and the third author was partially supported by 2005SGR-787.