= Abel map of an algebraic curve
{c}
{title2=$a_d:C^{(d)}\to\operatorname{Pic}^d(C)$}
= Abelian sum map
{synonym}
The <Abel map of an algebraic curve> sends an effective <Cartier divisor> $D$ to its <line bundle> $\mathcal O_C(D)$. Its scheme fibre over a <line bundle> $L$ is the <projective space> of one-dimensional subspaces of $H^0(C,L)$. This follows in families: a relative effective <Cartier divisor> in the fixed class determines a nowhere fibre-zero section of $L$ up to a <line bundle> from the parameter space, equivalently a line subbundle of $H^0(C,L)\otimes\mathcal O_T$. Thus the scheme fibre is smooth, not merely reduced on its points. The dimensions can nevertheless jump, so the <Abel map of an algebraic curve> need not be a smooth morphism.
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