Powers of the Theta Divisor and Relations in the Tautological Ring. (15th June 2017)

• Record Type: Journal Article
• Authors: Clader, Emily; Grushevsky, Samuel; Janda, Felix; Zakharov, Dmitry
• Is Part Of: International mathematics research notices. Volume 2018:Number 24(2018)
• Page Start: 7725
• Page End: 7754
• Publication Date: 2017-06-15
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnx115
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: 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 $\mathcal{M}_{g, n}$ 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 $\star$ of Graber and Vakil. Moreover, our proof provides an algorithm for expressing any tautological class on $\overline{\mathcal{M}}_{g, n}$ of sufficiently high codimension as a tautological class supported on the boundary.