We show that the vanishing of the (g+1)-st power of the theta divisor on
the universal abelian variety Xgβ implies, by pulling back along a
collection of Abel--Jacobi maps, the vanishing results in the tautological ring
of Mg,nβ 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~β of Graber and
Vakil, and we provide an explicit algorithm for expressing any tautological
class on Mg,nβ of sufficiently high codimension as a
boundary class.Comment: 29 page