The double ramification cycle and the theta divisor
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- by Samuel Grushevsky and Dmitry Zakharov PDF
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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
- MR Author ID: 192264
- Email: sam@math.sunysb.edu
- Dmitry Zakharov
- Affiliation: Department of Mathematics, Stony Brook University, Stony Brook, New York 11794-3651
- Email: dvzakharov@gmail.com
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 4053-4064
- MSC (2010): Primary 14H10; Secondary 14H51
- DOI: https://doi.org/10.1090/S0002-9939-2014-12153-8
- MathSciNet review: 3266977