Logarithmic intersections of double ramification cycles

Abstract

Let A = (a 1 , . . . , a n ) be a vector of integers which sum to k(2g − 2 + n). The double ramification cycle DR g,A ∈ CH g (M g,n ) on the moduli space of curves is the virtual class of an Abel-Jacobi locus of pointed curves (C, x 1 , . . . , x n ) satisfyingThe Abel-Jacobi construction requires log blow-ups of M g,n to resolve the indeterminacies of the Abel-Jacobi map. Holmes [29] has shown that DR g,A admits a canonical lift logDR g,A ∈ logCH g (M g,n ) to the logarithmic Chow ring, which is the limit of the intersection theories of all such blow-ups.The main result of the paper is an explicit formula for logDR g,A which lifts Pixton's formula for DR g,A . The central idea is to study the universal Jacobian over the moduli space of curves (following Caporaso [12], Kass-Pagani [37], and Abreu-Pacini [3]) for certain stability conditions. Using the criterion of Holmes-Schwarz [34], the universal double ramification theory of Bae-Holmes-Pandharipande-Schmitt-Schwarz [5] applied to the universal line bundle determines the logarithmic double ramification cycle. The resulting formula, written in the language of piecewise polynomials, depends upon the stability condition (and admits a wall-crossing study). Several examples are computed.

Rubber tori in the boundary of expanded stable maps

“…We now use the approach developed in Section 2 to define spaces of expanded maps to higher rank rubber targets. The virtual fundamental classes of these spaces will produce toric contact cycles [18,19,29,33]. Consider a normal crossings pair (𝑋|𝐷) together with a torus action 𝑇 ↷ (𝑋|𝐷), that is, an action 𝑇 ↷ 𝑋 that sends 𝐷 to itself.…”

“…We now use the approach developed in Section 2 to define spaces of expanded maps to higher rank rubber targets. The virtual fundamental classes of these spaces will produce toric contact cycles [18, 19, 29, 33].…”

The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves

“…There are now three proofs of the other half of the conjecture via three different approaches. The first two are by Abel–Jacobi theory in [HS22] and by controlling the difference between and in an appropriate blow-up of in [MR21]. The third, presented in [HMPPS22], proves the conjecture directly by giving a formula for (a representative of) in terms of -classes and piecewise polynomials.…”

Smooth Compactifications of the Abel-Jacobi Section

“…For these problems, the methods of [JPPZ17] have not been successfully adapted. To approach them, it has been understood ([HPS19], [Ran19], [HS22], [MR21], [Her19]) that one should study these problems in the context of logarithmic geometry. In this context, it is more natural to study instead the logarithmic double ramification cycle This is a certain refinement of , but it does not live on – or, better, it does not lie in , but rather in the logarithmic Chow ring ([Bar18], [MPS21]).…”

“…The idea here is that, since no best possible choice for exists, one might as well choose one that is fine enough that avoids as many pathologies as possible: choose an that is smooth, whose strata do not self-intersect, and so on. These approaches sufficed to prove soft properties of the , which depend on the form of the class; it was proven, for instance, that it is tautological [MR21, HS22]. But choosing least pathological models relies on abstract use of resolution of singularities, which makes the problem of finding an explicit formula essentially impossible.…”