(Non-)Extendability of Abel-Jacobi Maps

2024/04/03 by Zev Rosengarten, Rosengarten, Zev
#math.AG

paper · pdf · doi:10.48550/arxiv.2404.02766

Abstract

We investigate the "natural" locus of definition of Abel-Jacobi maps. In particular, we show that, for a proper, geometrically reduced curve C -- not necessarily smooth -- the Abel-Jacobi map from the smooth locus C^sm into the Jacobian of C does not extend to any larger (separated, geometrically reduced) curve containing C^sm except under certain particular circumstances which we describe explicitly. As a consequence, we deduce that the Abel-Jacobi map has closed image except in certain explicitly described circumstances, and that it is always a closed embedding for irreducible curves not isomorphic to P¹.

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