Let $A=(a_{1},\ldots , a_{n})$ be a vector of integers which sum to $k(2g-2+n)$ . The double ramification cycle $\mathsf{DR}_{g,A}\in \mathsf{CH}^{g}(\overline{\mathcal{M}}_{g,n})$ 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 \( \mathcal{O}_{C}\Big(\sum _{i=1}^{n} a_{i} x_{i}\Big) \, \simeq \, \big(\omega ^{\mathsf{log}}_{C}\big)^{k}\, . \) The Abel-Jacobi construction requires log blow-ups of $\overline{\mathcal{M}}_{g,n}$ to resolve the indeterminacies of the Abel-Jacobi map. Holmes (J. Inst. Math. Jussieu 2019) has shown that $\mathsf{DR}_{g,A}$ admits a canonical lift $\mathsf{logDR}_{g,A} \in \mathsf{logCH}^{g}(\overline{\mathcal{M}}_{g,n})$ 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 (Am. Math. Soc. 7(3):589–660 1994), Kass-Pagani (Trans. Am. Math. Soc. 372:4851–4887 2019), and Abreu-Pacini (Adv. Math. 378:107520 2021)) for certain stability conditions. Using the criterion of Holmes-Schwarz (Algebr. Geom. 9(5):574–605 2022), the universal double ramification theory of Bae-Holmes-Pandharipande-Schmitt-Schwarz (Acta Math. 230(2):205–319 2023) 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.
