Logarithmic double ramification cycles

Abstract

Let $A=(a_1,\dots ,a_n)$ 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 $(C,x_1,\dots ,x_n)$ satisfying $${\mathcal{O}}_C\left(\sum _{i=1}^n{a}_i{x}_i\right)\simeq (\ddot{I}{‰}_C^{\mathsf{log}}{)}^k.$$ 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.

97 pages, 6 figures. Substantial revision following referee reports. Comments very welcome