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Fine compactified Jacobians of reduced curves


Authors: Margarida Melo, Antonio Rapagnetta and Filippo Viviani
Journal: Trans. Amer. Math. Soc. 369 (2017), 5341-5402
MSC (2010): Primary 14D20, 14H40, 14H60; Secondary 14H20, 14F05, 14K30, 14B07
DOI: https://doi.org/10.1090/tran/6823
Published electronically: March 6, 2017
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Abstract: To every singular reduced projective curve $ X$ one can associate many fine compactified Jacobians, depending on the choice of a polarization on $ X$, each of which yields a modular compactification of a disjoint union of a finite number of copies of the generalized Jacobian of $ X$. We investigate the geometric properties of fine compactified Jacobians focusing on curves having locally planar singularities. We give examples of nodal curves admitting nonisomorphic (and even nonhomeomorphic over the field of complex numbers) fine compactified Jacobians. We study universal fine compactified Jacobians, which are relative fine compactified Jacobians over the semiuniversal deformation space of the curve $ X$. Finally, we investigate the existence of twisted Abel maps with values in suitable fine compactified Jacobians.


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Additional Information

Margarida Melo
Affiliation: Departamento de Matemática, Universidade de Coimbra, Largo D. Dinis, Apartado 3008, 3001 Coimbra, Portugal – and – Dipartimento di Matematica, Università Roma Tre, Largo S. Leonardo Murialdo 1, 00146 Roma, Italy
Email: mmelo@mat.uc.pt

Antonio Rapagnetta
Affiliation: Dipartimento di Matematica, Università di Roma II - Tor Vergata, 00133 Roma, Italy
Email: rapagnet@mat.uniroma2.it

Filippo Viviani
Affiliation: Dipartimento di Matematica, Università Roma Tre, Largo S. Leonardo Murialdo 1, 00146 Roma, Italy
Email: viviani@mat.uniroma3.it

DOI: https://doi.org/10.1090/tran/6823
Keywords: Compactified Jacobians, locally planar singularities, Abel map.
Received by editor(s): June 15, 2014
Received by editor(s) in revised form: May 8, 2015, and August 24, 2015
Published electronically: March 6, 2017
Article copyright: © Copyright 2017 American Mathematical Society