Logarithmic double ramification cycles

TL;DR

The paper develops a complete framework to lift the double ramification cycle to the logarithmic Chow ring, providing an explicit formula for the logarithmic DR cycle ${\mathsf{logDR}}_{g,A}$ in terms of stability data, the universal Jacobian, and piecewise polynomial functions on subdivisions of the boundary cone complex. Central to the approach is the use of small nondegenerate stability conditions to produce modular, noncanonical but computable compactifications ${\mathcal M}_{g,A}^{\theta}$ that resolve the Abel–Jacobi map; the resulting logDR class is shown to be independent of the chosen stability, and is computed via a universal DR formula on the universal Picard stack, transported through a series of almost-twistable constructions. The explicit formula employs two special piecewise polynomial constructs, $\mathfrak{P}$ and $\mathfrak{L}$, together with tautological classes $\eta$ and the map $\Phi$ that lifts piecewise polynomial data to logCH; the degree-$g$ part yields ${\mathsf{logDR}}_{g,A}$. Wall-crossing in stability conditions yields relations in $\mathsf{logR}^*(\overline{\mathcal M}_{g,n})$ and connects to Pixton-type relations via the universal DR cycle, with concrete genus-1 examples illustrating the mechanism. This work integrates log modifications, cone-stack combinatorics, tropical-ABJ theory, and the algebra of piecewise polynomials to advance logarithmic Gromov-Witten theory and its applications to toric and Hurwitz-type problems.

Abstract

Let $A=(a_1,\ldots, a_n)$ be a vector of integers which sum to $k(2g-2+n)$. The double ramification cycle $\mathsf{DR}_{g,A}\in \mathsf{CH}^g(\mathcal{M}_{g,n})$ on the moduli space of curves is the virtual class of an Abel-Jacobi locus of pointed curves $(C,x_1,\ldots,x_n)$ satisfying $$\mathcal{O}_C\Big(\sum_{i=1}^n a_i x_i\Big) \, \simeq\, \big(ω^{\mathsf{log}}_{C}\big)^k\, .$$ The Abel-Jacobi construction requires log blow-ups of $\mathcal{M}_{g,n}$ to resolve the indeterminacies of the Abel-Jacobi map. Holmes has shown that $\mathsf{DR}_{g,A}$ admits a canonical lift $\mathsf{logDR}_{g,A} \in \mathsf{logCH}^g(\mathcal{M}_{g,n})$ 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 $\mathsf{logDR}_{g,A}$ which lifts Pixton's formula for $\mathsf{DR}_{g,A}$. The central idea is to study the universal Jacobian over the moduli space of curves (following Caporaso, Kass-Pagani, and Abreu-Pacini) for certain stability conditions. Using the criterion of Holmes-Schwarz, the universal double ramification theory of Bae-Holmes-Pandharipande-Schmitt-Schwarz 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 of logarithmic and higher double ramification cycles are computed.

Figures (6)

• Figure 1: A curve $(C,x_1, x_2) \in \overline{\mathcal{M}}_{5,2}$ and the associated stable graph $\Gamma$
• Figure 2: The cone stack $\Sigma_{\overline{\mathcal{M}}_{1,2}}$ with a subdivision (in red) and the corresponding log modification $\widehat{{\mathcal{M}}}$ of $\overline{\mathcal{M}}_{1,2}$, which replaces the self-intersection of the boundary divisor of irreducible curves with a chain of rational curves of length $2$. For the cone stack (on the left), we draw the double cover of the upper (stacky) cone, and correspondingly, the local pictures around the self-intersection of $\delta_\text{irr}$ on the right represent an étale double cover of the neighbourhood of this point. If $X$ and $Y$ are the étale local coordinates around the node corresponding to $x$ and $y$, then the three affine patches of the subdivision are given by ${\mathbb{C}}[X,Y,t]/(X - tY^2)$, ${\mathbb{C}}[X,Y,u,v]/(Y^2-uX, X^2-vY, XY-uv)$, and ${\mathbb{C}}[X,Y,s]/(Y - sX^2)$ respectively.
• Figure 3: A quasi-stable curve $C'$ with stabilization $C$, and an admissible line bundle $\mathcal{L}$ on it.
• Figure 4: Formulas for the cycles $\mathsf{DR}_{1,(3,-3)} = \frac{9}{2}(\psi_1+\psi_2)-\frac{1}{12}\delta_0$ and $\mathsf{logDR}_{1,(3,-3)}$, each consisting of a linear combination of classes $\psi_1, \psi_2$ from $\overline{\mathcal{M}}_{1,2}$ and a contribution from a piecewise linear function on a subdivision $\widetilde{\Sigma}$ of $\Sigma_{\overline{\mathcal{M}}_{1,2}}$ (the trivial subdivision in the case of the regular DR cycle). See Figure \ref{['fig:M12logblow-up']} for an explanation of the coordinates $x,y$ on the cone stack, where we now only draw the coarse space of the stacky cone.
• Figure 5: A list of stable graphs $\Gamma$ for $\overline{{\mathcal{M}}}_{1,2}$ and the corresponding strict piecewise polynomials $f_\Gamma$ such that $\Phi(f_\Gamma)$ is the fundamental class of the associated stratum closure in $\overline{{\mathcal{M}}}_{1,2}$

Theorems & Definitions (99)

• Theorem A
• Definition 1
• Theorem B
• Theorem C
• Definition 2
• Definition 3: Cavalieri2020-Conestack
• Definition 4
• Remark 5
• Remark 6
• Definition 7