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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Twists of Mukai bundles and the geometry of the level $ 3$ modular variety over $ \overline{\mathcal{M}}_{8}$


Author: Gregor Bruns
Journal: Trans. Amer. Math. Soc. 370 (2018), 8359-8376
MSC (2010): Primary 14H10, 14H45; Secondary 14E08, 14H40, 14K10
DOI: https://doi.org/10.1090/tran/7239
Published electronically: June 7, 2018
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Abstract: For a curve $ C$ of genus $ 6$ or $ 8$ and a torsion bundle $ \eta $ of order $ \ell $ we study the vanishing of the space of global sections of the twist $ E_C \otimes \eta $ of the rank $ 2$ Mukai bundle $ E_C$ of $ C$. The bundle $ E_C$ was used in a well-known construction of Mukai which exhibits general canonical curves of low genus as sections of Grassmannians in the Plücker embedding.

Globalizing the vanishing condition, we obtain divisors on the moduli spaces $ \overline {\mathcal {R}}_{6,\ell }$ and $ \overline {\mathcal {R}}_{8,\ell }$ of pairs $ [C, \eta ]$. First we characterize these divisors by different conditions on linear series on the level curves, afterwards we calculate the divisor classes. As an application, we are able to prove that $ \overline {\mathcal {R}}_{8,3}$ is of general type.


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Additional Information

Gregor Bruns
Affiliation: Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
Email: math@gregorbruns.eu

DOI: https://doi.org/10.1090/tran/7239
Received by editor(s): November 1, 2016
Received by editor(s) in revised form: March 14, 2017, and March 17, 2017
Published electronically: June 7, 2018
Article copyright: © Copyright 2018 American Mathematical Society

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