Logarithmic tautological rings of the moduli spaces of curves

TL;DR

The paper introduces the logarithmic Chow ring and log tautological rings for moduli spaces of stable curves, extending the classical tautological framework to the log setting via cone stacks, Artin fans, and tropical geometry. It provides a complete genus-0 computation, establishes surjectivity results and kernels in genus 1, and outlines higher-genus obstructions tied to Pixton-type relations, showing that the log tautological structure is governed by the relations in standard tautological rings. A general theory for arbitrary smooth normal crossings pairs $(X,D)$ is developed, including decorated log strata, homological piecewise polynomials, and pushforward/pullback machinery compatible with log blowups. The framework unifies log Gromov–Witten theory, tropical intersection theory, and decorated strata in a coherent algebraic setting, enabling explicit calculations (e.g., genus-0 and certain genus-1 cases) and guiding further questions about lifts of Pixton relations and higher-genus presentations. Overall, the work delivers a robust, computable, and conceptually geometric description of log tautological rings with broad applicability to logarithmic intersection theory. $\logR^ullet(X,D)$ collapses to the familiar log Chow ring in genus 0, while in higher genus it encodes refined information controlled by the standard tautological relations, offering new avenues for understanding the intrinsic log geometry of moduli spaces. $

Abstract

We define the logarithmic tautological rings of the moduli spaces of Deligne-Mumford stable curves (together with a set of additive generators lifting the decorated strata classes of the standard tautological rings). While these algebras are infinite dimensional, a connection to polyhedral combinatorics via a new theory of homological piecewise polynomials allows an effective study. A complete calculation is given in genus 0 via the algebra of piecewise polynomials on the cone stack of the associated Artin fan (lifting Keel's presentation of the Chow ring of $\overline{\mathcal{M}}_{0,n}$). Counterexamples to the simplest generalizations in genus 1 are presented. We show, however, that the structure of the log tautological rings is determined by the complete knowledge of all relations in the standard tautological rings of the moduli spaces of curves. In particular, Pixton's conjecture concerning relations in the standard tautological rings lifts to a complete conjecture for relations in the log tautological rings of the moduli spaces of curves. Several open questions are discussed. We develop the entire theory of logarithmic tautological classes in the context of arbitrary smooth normal crossings pairs $(X,D)$ with explicit formulas for intersection products. As a special case, we give an explicit set of additive generators of the full logarithmic Chow ring of $(X,D)$ in terms of Chow classes on the strata of $X$ and piecewise polynomials on the cone stack.

Figures (3)

• Figure 1: A cross section through the star of $\Gamma_0$ in the cone stack of $\overline{\mathcal{M}}\newline_{1,3}$. For better visibility, we draw the double-cover of the actual picture where the two edges with lengths $\ell_1, \ell_2$ are distinguishable. The star of $\Gamma_0$ is the quotient of the figure under reflection along the horizontal axis (the red subdivision defined by $\ell_1 = \ell_2$).
• Figure 2: The fan $\Sigma_X$ defined in Example \ref{['ex:a2z2']}
• Figure 3: A cone stack $\Sigma_X$ and its star subdivision at an object $\tau_{12}$, with new cones on the right indicated in red. We draw here a slice through the underlying $3$-dimensional picture.

Theorems & Definitions (178)

• Theorem 1
• Theorem 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Definition 7
• Proposition 8
• proof
• Theorem 9: Kee92