Geometry of Genus One Fine Compactified Universal Jacobians



Pagani, Nicola and Tommasi, Orsola
(2023) Geometry of Genus One Fine Compactified Universal Jacobians. International Mathematics Research Notices, 2023 (10). pp. 8495-8543.

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Abstract

<jats:title>Abstract</jats:title> <jats:p>We introduce a general abstract notion of fine compactified Jacobian for nodal curves of arbitrary genus. We focus on genus $1$ and prove combinatorial classification results for fine compactified Jacobians in the case of a single nodal curve and in the case of the universal family $\overline {{\mathcal {C}}}_{1,n} / \overline {{\mathcal {M}}}_{1,n}$ over the moduli space of stable pointed curves. We show that if the fine compactified Jacobian of a nodal curve of genus $1$ can be extended to a smoothing of the curve, then it can be described as the moduli space of stable sheaves with respect to some polarisation. In the universal case we construct new examples of genus $1$ fine compactified universal Jacobians. Then we give a formula for the Hodge and Betti numbers of each genus $1$ fine compactified universal Jacobian $\overline {{\mathcal {J}}}^{d}_{1,n}$ and prove that their even cohomology is algebraic.</jats:p>

Item Type: Article
Divisions: Faculty of Science and Engineering > School of Physical Sciences
Depositing User: Symplectic Admin
Date Deposited: 09 Sep 2021 13:09
Last Modified: 19 May 2023 10:09
DOI: 10.1093/imrn/rnac094
Open Access URL: https://academic.oup.com/imrn/advance-article/doi/...
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3136201