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### The indeterminacy of the Abel-Jacobi maps

##### Nicola Pagani (Liverpool) Aula Beltrami - Mercoledì 25 Ottobre 2017 h.15:00

Abstract. The Abel-Jacobi morphisms are the sections of the forgetful morphism from the universal Jacobian to the corresponding moduli space of smooth pointed curves. When the source and target moduli spaces are compactified, these morphisms can be reinterpreted as rational maps, and it is natural to ask for their locus of indeterminacy. We explicitly characterize the indeterminacy locus, which depends on the chosen compactification of the universal Jacobian (for the source we fix the Deligne-Mumford compactification \bar{M}_{g,n} by means of stable curves). In particular, we deduce that for evey Abel-Jacobi map there exists a compactification such that the map extends to a well-defined morphism on \bar{M}_{g,n}. This offers an approach to define and then compute the classes of several different extensions of the "Jacobian double ramification cycles" (= the pullbacks of the zero section via the Abel-Jacobi maps). This is a joint work with Jesse Kass."

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