(Non-)Extendability of Abel-Jacobi Maps

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 $$C^{\textrm{sm}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mtext>sm</mml:mtext> </mml:msup> </mml:math> into the Jacobian of C does not extend to any larger (separated, geometrically reduced) curve containing $$C^{\textrm{sm}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mtext>sm</mml:mtext> </mml:msup> </mml:math> 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 $$\textbf{P}^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> .