Powers of the Theta Divisor and Relations in the Tautological Ring. (15th June 2017) - Clader, Emily; Grushevsky, Samuel; Janda, Felix; Zakharov, Dmitry
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.