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 \(C^{\textrm{sm}}\) into the Jacobian of C does not extend to any larger (separated, geometrically reduced) curve containing \(C^{\textrm{sm}}\) 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 \(\textbf{P}^1\) .
