[Paper Review] Logarithmic double ramification cycles

David Holmes, Samouil Molcho+3 more|arXiv (Cornell University)|Jul 14, 2022

Advanced Differential Equations and Dynamical Systems5 citations

TL;DR

This paper establishes an explicit formula for the logarithmic double ramification cycle in the logarithmic Chow ring of the moduli space of stable curves, extending Pixton's formula to the logarithmic setting using piecewise polynomials and stability conditions. The key contribution is a wall-crossing formula for log DR cycles via universal Jacobian constructions and log intersection theory.

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.

Motivation & Objective

• To extend Pixton's tautological formula for the classical double ramification cycle to the logarithmic setting.
• To resolve indeterminacies in the Abel-Jacobi map via log blow-ups and define a canonical lift to the logarithmic Chow ring.
• To establish a universal formula for the logarithmic DR cycle using stability conditions and piecewise polynomial structures.
• To provide computational tools for higher and logarithmic DR cycles, including a Sage implementation.
• To explore relations in the logarithmic tautological ring through GL₂(ℤ)-invariance and wall-crossing phenomena.

Proposed method

• Lift the classical double ramification cycle to the logarithmic Chow ring using log modifications and stacky fans.
• Apply the criterion of Holmes-Schwarz to universal line bundles over the moduli space of stable curves.
• Construct the log DR cycle via piecewise polynomial functions associated to stability conditions.
• Use subdivision of tropical curves and tropical Abel-Jacobi theory to algebraize the log cycle.
• Express the formula in terms of graph sums over stable graphs, with contributions defined by constant terms of polynomials.
• Implement the formula computationally in Sage for explicit calculations of log DR cycles.

Experimental results

Research questions

• RQ1 How can the classical double ramification cycle be canonically lifted to the logarithmic Chow ring?
• RQ2 What is the structure of the logarithmic double ramification cycle in terms of piecewise polynomials and stability conditions?
• RQ3 How does the log DR cycle behave under wall-crossing as the stability condition varies?
• RQ4 What are the tautological relations in the logarithmic tautological ring arising from GL₂(ℤ)-invariance?
• RQ5 Can the log DR cycle be computed explicitly for higher genus and higher-order cycles?

Key findings

• The paper provides a complete explicit formula for the logarithmic double ramification cycle in the logarithmic Chow ring, generalizing Pixton’s formula.
• The formula is expressed as a sum over stable graphs, with each contribution given by the constant term of a piecewise polynomial associated to the graph and the vector A.
• The construction depends on a stability condition, and the formula admits a wall-crossing study as the condition varies.
• The authors compute explicit examples of logarithmic and higher double ramification cycles, including in genus 1.
• A Sage implementation is provided for computing log DR cycles, enabling practical computation of higher-order and logarithmic cases.
• The GL₂(ℤ)-invariance of the double-double ramification cycle yields a new construction of tautological relations in the logarithmic tautological ring.