Logarithmic compactification of the Abel–Jacobi section

Abstract

Given a smooth curve with weighted marked points, the Abel-Jacboi map produces a line bundle on the curve. This map fails to extend to the full boundary of the moduli space of stable pointed curves. Using logarithmic and tropical geometry, we describe a modular modification of the moduli space of curves over which the Abel-Jacobi map extends. We also describe the attendant deformation theory and virtual fundamental class of this moduli space. This recovers the double ramification cycle, as well as variants associated to differentials. Contents

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

“…In general resolution of singularities is more difficult outside characteristic zero. However, in [28] (which appeared after the first version of the present article) it is shown thatM(P) is log regular (at least under mild assumptions on the base scheme), hence it is locally desingularisable. With this new reference available, the comparison results in this section are valid over Z.…”

A non-Archimedean analogue of Teichmüller space and its tropicalization

“…• to [79,81] for a non-Archimedean counting of holomorphic cylinders on Calabi-Yau surfaces, to [69,70] for a logarithmic/tropical reinterpretation of the Vakil-Zinger blow of moduli spaces of elliptic stable maps on toric varieties, and to [66] and [67] for an approach to a degeneration formula [3] and a product formula [39] in logarithmic Gromov-Witten theory; • to [58,59], to [41,42,50], as well as to [11] and [53] for an approach towards constructing a compactification of the universal Jacobian and a resolution of the universal Abel-Jacobi map; • to [14] for a construction of a compactification of a strata of abelian differentials using combinatorial data which may be translated into tropical language expanding on [57]; and • to [47] for a modular interpretation of toroidal compactifications of the moduli space A g of principally polarized complex abelian varieties.…”

Infinitesimal structure of the pluricanonical double ramification locus

“…for which it is easier to write down local charts around the general points of DRL. The precise construction of M m,1/k is given in § 2 (where we also make more concrete the relationship with the construction of Marcus and Wise [MW20]), but for us the two key properties are as follows.…”

“…The map σ does not extend naturally to M g,n , but various geometric extensions of its domain and target have been proposed that yield cycles on M g,n [HKP18, AP21, Hol21, KP19,MW20]. These constructions all produce the same cycle class on M g,n , which we denote DRC; an overview of one construction is given in § 1.2.…”