Holmes, Molcho, Pandharipande, Pixton, Schmitt — Logarithmic Double Ramification Cycles (2024)

Let \(A=(a_1,\ldots, a_n)\) be a vector of integers which sum to \(k(2g-2+n)\). The double ramification cycle \(\mathsf{ DR}_{ g,A}\in \mathsf{ CH}\textasciicircum g(\mathcal{ M}_{ g,n} )\) on the moduli space of curves is the virtual class of an Abel-Jacobi locus of pointed curves \((C,x_1,\ldots,x_n)\) satisfying $\mathcal{ O}_C\Big(∑ _i=1\textasciicircum n a_i x_i\Big) \, ≅\, \big(ω\textasciicircum{\mathsf{ log}} _C\big)\textasciicircum k\, .$ 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 \(\mathsf{ logDR}_{ g,A}{} \in \mathsf{ logCH}\textasciicircum g(\mathcal{ M}_{ g,n} )\) 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.