Extending the double ramification cycle by resolving the Abel-Jacobi map
arXiv:1707.02261 · doi:10.1017/S1474748019000252
Abstract
Over the moduli space of smooth curves, the double ramification cycle can be defined by pulling back the unit section of the universal jacobian along the Abel-Jacobi map. This breaks down over the boundary since the Abel-Jacobi map fails to extend. We construct a `universal' resolution of the Abel-Jacobi map, and thereby extend the double ramification cycle to the whole of the moduli of stable curves. In the non-twisted case, we show that our extension coincides with the cycle constructed by Li, Graber, Vakil via a virtual fundamental class on a space of rubber maps.
35 pages, 1 figure. v2:Exposition heavily revised (and hopefully improved). Main results unchanged. There was a gap in the proof of lemma 3.14; it is replaced by the construction in lemma 6.1. v3: Extended to include the `k-twisted' case, where one allows powers of the relative dualising sheaf. v4. Final version, to appear in J. Inst. Math. Jussieu. Comments still very welcome