Logarithmic intersections of double ramification cycles

TL;DR

This paper develops a logarithmic Chow ring framework and proves that the double ramification cycle is tautological and divisorial, confirming conjectures in algebraic geometry.

Contribution

It introduces a theory of logarithmic Chow rings and proves key properties of double ramification cycles within this framework.

Findings

Double ramification cycle is in the tautological subring.

Logarithmic double ramification cycle is divisorial.

Framework generalizes piecewise-polynomial functions for algebraic stacks.

Abstract

We describe a theory of logarithmic Chow rings and tautological subrings for logarithmically smooth algebraic stacks, via a generalisation of the notion of piecewise-polynomial functions. Using this machinery we prove that the double-double ramification cycle lies in the tautological subring of the (classical) Chow ring of the moduli space of curves, and that the logarithmic double ramification cycle is divisorial (as conjectured by Molcho, Pandharipande, and Schmitt).