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Abstract
AbstractOver the moduli space of smooth curves, the double ramification cycle can be defined by pulling back the unit section of the universal jacobian along the Abel–Jacobi map. This breaks down over the boundary since the Abel–Jacobi map fails to extend. We construct a ‘universal’ resolution of the Abel–Jacobi map, and thereby extend the double ramification cycle to the whole of the moduli of stable curves. In the non-twisted case, we show that our extension coincides with the cycle constructed by Li, Graber, Vakil via a virtual fundamental class on a space of rubber maps.
Highlights
Fix integers g, n 0 satisfying 2g − 2 + n > 0, and integers a1 . . . an and k such that i ai = k(2g − 2)
Write J for the universal jacobian and σ for the section of J given by the line bundle ω⊗k(− i ai xi ) on Cg,n
In the case k = 0, an extension of the class to the whole of Mg,n was constructed by Li [25], [26], Graber and Vakil [12]. This class was computed in the compact-type case by Hain [15], and this was extended to tree-like curves with one loop by Grushevsky and Zakharov [13]
Summary
Marcus and Wise [28] have given another approach to resolving the Abel–Jacobi map when k = 0, rather closer in spirit to the present preprint They use logarithmic geometry to modify Mg,n, but their construction is based on stacks of stable maps rather than a universal property as in the present preprint. He did not make modifications of Mg,n, and so was not able to extend over the whole of the boundary
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