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The Hodge-Double-Ramification conjecture and Mumford's formula on the universal Picard stack

Alessandro Chiodo, David Holmes

TL;DR

This work resolves the Hodge-DR and log-Hodge-DR conjectures by embedding the DR/LogDR problem into the universal Picard stack and using $r$-th roots, log geometry, and degeneracy-locus techniques. It develops a unified framework combining Mumford-type Chern-character formulas and Pixton-type graph expressions to compute DR/LogDR classes, both on irreducible and stable curves, and to relate them to tautological data. The paper proves the universal Hodge-DR equality via a polynomial-in-$r$ reduction and extends the LogDR theory with a robust log Chow-ring approach, including a Thom–Porteous style treatment of boundary contributions. It also generalizes prior results (e.g., Janda, Holmes, Pagani–Ricolfi–van Zelm) to a broad setting with arbitrary line bundles, roots, and stability conditions, providing computational tools for Euler characteristics and Brill–Noether-type loci in moduli problems of differentials.

Abstract

The double ramification (DR) cycle associated to a line bundle on a family of curves detects where the line bundle becomes fibrewise-trivial. The Hodge-DR Conjecture proposes a formula for powers of the first Chern class of a natural line bundle on the DR cycle, with a number of applications in the computation of Euler characteristics of strata of differentials. In this paper we prove the conjecture, as well as an analogue for the logarithmic DR cycle. The proof of the former proceeds via reduction to a localisation computation of Fan, Wu and You; the proof of the latter is based on the Thom--Porteous formula, and as a special case gives a shorter proof of a recent result of Holmes, Molcho, Pandharipade, Pixton and Schmitt. Along the way we develop an analogue of Mumford's formula for the Chern character of the universal line bundle on the universal jacobian over the moduli space of twisted curves, generalising work of Mumford, Chiodo, and Pagani--Ricolfi--van Zelm.

The Hodge-Double-Ramification conjecture and Mumford's formula on the universal Picard stack

TL;DR

This work resolves the Hodge-DR and log-Hodge-DR conjectures by embedding the DR/LogDR problem into the universal Picard stack and using -th roots, log geometry, and degeneracy-locus techniques. It develops a unified framework combining Mumford-type Chern-character formulas and Pixton-type graph expressions to compute DR/LogDR classes, both on irreducible and stable curves, and to relate them to tautological data. The paper proves the universal Hodge-DR equality via a polynomial-in- reduction and extends the LogDR theory with a robust log Chow-ring approach, including a Thom–Porteous style treatment of boundary contributions. It also generalizes prior results (e.g., Janda, Holmes, Pagani–Ricolfi–van Zelm) to a broad setting with arbitrary line bundles, roots, and stability conditions, providing computational tools for Euler characteristics and Brill–Noether-type loci in moduli problems of differentials.

Abstract

The double ramification (DR) cycle associated to a line bundle on a family of curves detects where the line bundle becomes fibrewise-trivial. The Hodge-DR Conjecture proposes a formula for powers of the first Chern class of a natural line bundle on the DR cycle, with a number of applications in the computation of Euler characteristics of strata of differentials. In this paper we prove the conjecture, as well as an analogue for the logarithmic DR cycle. The proof of the former proceeds via reduction to a localisation computation of Fan, Wu and You; the proof of the latter is based on the Thom--Porteous formula, and as a special case gives a shorter proof of a recent result of Holmes, Molcho, Pandharipade, Pixton and Schmitt. Along the way we develop an analogue of Mumford's formula for the Chern character of the universal line bundle on the universal jacobian over the moduli space of twisted curves, generalising work of Mumford, Chiodo, and Pagani--Ricolfi--van Zelm.
Paper Structure (39 sections, 23 theorems, 135 equations)

This paper contains 39 sections, 23 theorems, 135 equations.

Table of Contents

  1. Extending the DR cycle to stable curves
  2. Computing the DR and logDR cycles
  3. Extracting rth roots and setting r=0
  4. Quasi-stable twists of L
  5. The formulae of Mumford and Pixton
  6. Mumford's formula
  7. Pixton's formula
  8. Applications
  9. Proof of the Hodge-DR Conjecture
  10. Statement and proof of the log-Hodge-DR Conjecture
  11. Generalising the logDR formula
  12. Extending results of Pagani, Ricolfi and van Zelm
  13. Definitions and detailed statements of results
  14. Conventions
  15. Roots of line bundles on curves
  16. ...and 24 more sections

Key Result

Proposition 1

We restrict to the total-degree-zero part of the Picard stack. Fix a positive integer $d$. Then the constant terms of the polynomials in $r$ given by are equal in the Chow ring of $\mathfrak{Pic}^{\mathrm{tot}0}$.

Theorems & Definitions (49)

  • Proposition 1
  • Definition 1.1
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 39 more