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The double ramification cycle and the theta divisor


Authors: Samuel Grushevsky and Dmitry Zakharov
Journal: Proc. Amer. Math. Soc. 142 (2014), 4053-4064
MSC (2010): Primary 14H10; Secondary 14H51
Published electronically: August 14, 2014
MathSciNet review: 3266977
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Abstract: We compute the classes of universal theta divisors of degrees zero and $ g-1$ over the Deligne-Mumford compactification $ {\overline {\mathcal {M}}_{g,n}}$ of the moduli space of curves, with various integer weights on the points, in particular reproving a recent result of Müller.

We also obtain a formula for the class in $ CH^{g}({\mathcal {M}_{g,n}^{ct}})$ (moduli of stable curves of compact type) of the double ramification cycle, given by the condition that a fixed linear combination of the marked points is a principal divisor, reproving a recent result of Hain.

Our approach for computing the theta divisor is more direct, via test curves and the geometry of the theta divisor, and works easily over all of $ \overline {\mathcal {M}}_{g,n}$. We used our extended result in another paper to study the partial compactification of the double ramification cycle.


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Additional Information

Samuel Grushevsky
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651
Email: sam@math.sunysb.edu

Dmitry Zakharov
Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651
Email: dvzakharov@gmail.com

DOI: https://doi.org/10.1090/S0002-9939-2014-12153-8
Received by editor(s): July 2, 2012
Received by editor(s) in revised form: January 25, 2013
Published electronically: August 14, 2014
Additional Notes: The research of the first author was supported in part by the National Science Foundation under grant DMS-10-53313.
Communicated by: Lev Borisov
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.