TY - JOUR
T1 - Powers of the theta divisor and relations in the tautological ring
AU - Clader, Emily
AU - Grushevsky, Samuel
AU - Janda, Felix
AU - Zakharov, Dmitry
N1 - Funding Information:
E.C. acknowledges the generous support of Dr Max Rössler, the Walter Haefner Foundation, and the ETH Foundation. The authors would like to thank Aaron Pixton and Ravi Vakil for useful discussions, and Dimitri Zvonkine for pointing out a gap in an earlier version of the argument deducing the vanishing result for the tautological ring of Mg,n from the theta relations. Our collaboration started during a visit by the first and third authors to Columbia University, where the second author was visiting on sabbatical. The authors would like to thank Columbia University for the hospitality.
Funding Information:
This study was partially supported by the National Science Foundation under the grants [DMS-12-01369 and DMS-15-01265 to S.G.], and by a Simons Fellowship in Mathematics (Simons Foundation grant #341858 to S.G.).
Publisher Copyright:
© The Author(s) 2017. Published by Oxford University Press. All rights reserved.
PY - 2018/12/12
Y1 - 2018/12/12
N2 - We show that the vanishing of the (g + 1)-st power of the theta divisor in the cohomology and Chow rings of the universal abelian variety implies, by pulling back along a collection of Abel–Jacobi maps, the vanishing results in the tautological ring of Mg,n of Looijenga, Ionel, Graber–Vakil, and Faber–Pandharipande. We also show that Pixton’s double ramification cycle relations, which generalize the theta vanishing relations and were recently proved by the first and third authors, imply Theorem of Graber and Vakil. Moreover, our proof provides an algorithm for expressing any tautological class on Mg,n of sufficiently high codimension as a tautological class supported on the boundary.
AB - We show that the vanishing of the (g + 1)-st power of the theta divisor in the cohomology and Chow rings of the universal abelian variety implies, by pulling back along a collection of Abel–Jacobi maps, the vanishing results in the tautological ring of Mg,n of Looijenga, Ionel, Graber–Vakil, and Faber–Pandharipande. We also show that Pixton’s double ramification cycle relations, which generalize the theta vanishing relations and were recently proved by the first and third authors, imply Theorem of Graber and Vakil. Moreover, our proof provides an algorithm for expressing any tautological class on Mg,n of sufficiently high codimension as a tautological class supported on the boundary.
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U2 - 10.1093/imrn/rnx115
DO - 10.1093/imrn/rnx115
M3 - Article
AN - SCOPUS:85062025366
SN - 1073-7928
VL - 2018
SP - 7725
EP - 7754
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 24
ER -