Extending the double ramification cycle using Jacobians

Abstract

We prove that the extension of the double ramification cycle defined by the first-named author (using modifications of the stack of stable curves) coincides with one of those defined by the last-two named authors (using an extended Brill-Noether locus on a suitable compactified universal Jacobians). In particular, in the untwisted case we deduce that both of these extensions coincide with that constructed by Li and Graber-Vakil using a virtual fundamental class on a space of rubber maps.

Extending the Double Ramification Cycle by Resolving the Abel-Jacobi Map

“…Kass and Pagani [21] have recently constructed large numbers of extensions of the DRC (for every k ∈ Z). In [18] we show that certain of their classes coincide with the one constructed in this article.…”

“…We define DR naive to be the cycle on M g,n obtained by pulling back the schematic image of σ along the unit section. Still in characteristic zero, a small argument with the projection formula (of which details can be found in [18]) shows that the resulting cycle is equal to the one constructed in this paper, and hence also to that of Li, Graber and Vakil.…”

Double ramification cycles with target varieties

“…Pixton's formula's opened new directions in the subject: new formulas for Hodge classes [35, Section 3], new relations in the tautological ring of [17], new connections to the loci of meromorphic differentials [24, Appendix], and connections to new integrable hierarchies [9]. For a sampling of the subsequent study and applications, see [6–8, 10, 12, 16, 23, 31, 32, 45, 56, 57]. We refer the reader to [35, Section 0] and [50, Section 5] for more leisurely introductions to the subject.…”

Pullbacks of universal Brill–Noether classes via Abel–Jacobi morphisms

“…Then the Poincaré dual of (4.2) is the class that  ( ) would have as its fundamental class if it were pure of the expected codimension − . The scheme (4.3) has an explicit description as a degeneracy scheme, which was already described in the proof of [9,Lem. 6] in the case = = 0.…”

“…We will start by introducing the DRC, following the perspective of [9], which is in turn based on the resolution of the indeterminacy of the Abel-Jacobi section by D. Holmes [8] (see also [15]). For more details we refer the reader to [9]. Let  0 , be the universal generalised Jacobian, or the moduli stack of multidegree zero line bundles on stable curves (equivalently, the unique semiabelian extension of the degree zero universal Jacobian over  , ).…”