Logarithmic double ramification cycles by D. Holmes & S. Molcho & R. Pandharipande & A. Pixton & J. Schmitt ISBN 101007/S0022202501318Z instant download

Logarithmic double ramification cycles by D. Holmes & S. Molcho & R. Pandharipande & A. Pixton & J. Schmitt ISBN 101007/S0022202501318Z instant download

AbstractLet A = (a1,...,an) be a vector of integers which sum to k(2g − 2 + n). The doubleramification cycle DRg,A ∈ CHg(Mg,n) on the moduli space of curves is the virtualclass of an Abel-Jacobi locus of pointed curves (C,x1,...,xn) satisfyingOC(︂naixi)︂≃ (︁ωlogC)︁k .∑︂i=1The Abel-Jacobi construction requires log blow-ups of Mg,n to resolve the indeterminacies of the Abel-Jacobi map. Holmes (J. Inst. Math. Jussieu 2019) has shownthat DRg,A admits a canonical lift logDRg,A ∈ logCHg(Mg,n) to the logarithmicChow ring, which is the limit of the intersection theories of all such blow-ups. Themain result of the paper is an explicit formula for logDRg,A which lifts Pixton’s formula for DRg,A. The central idea is to study the universal Jacobian over the modulispace of curves (following Caporaso (Am. Math. Soc. 7(3):589–660 1994), KassPagani (Trans. Am. Math. Soc. 372:4851–4887 2019), and Abreu-Pacini (Adv. Math.378:107520 2021)) for certain stability conditions. Using the criterion of HolmesSchwarz (Algebr. Geom. 9(5):574–605 2022), the universal double ramification theory of Bae-Holmes-Pandharipande-Schmitt-Schwarz (Acta Math. 230(2):205–3192023) 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). Severalexamples of logarithmic and higher double ramification cycles are computed.