Powers of the theta divisor and relations in the tautological ring

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.

v2: agrees with published version, 27 pages

References in corpus (6)

• Double ramification cycles on the moduli spaces of curves
• Relations on ${\overline{M}}_{g,n}$ via equivariant Gromov-Witten theory of ${\mathbb{P}}^1$
• Pixton's double ramification cycle relations
• Extensions of the universal theta divisor
• Compactifications of the universal Jacobian over curves with marked points
• Compactified universal jacobian and the double ramification cycle

Cited by in corpus (5)

• A new cohomology class on the moduli space of curves
• Double ramification cycles with target varieties
• Pixton's formula and Abel-Jacobi theory on the Picard stack
• Topological recursion relations from Pixton's formula
• The tautological ring of ${\mathcal{M}}_{g,n}$ via Pandharipande-Pixton-Zvonkine $r$-spin relations