= Picard functor of a curve
{c}
{title2=$\operatorname{Pic}^d_{C/k}$}
For a smooth projective geometrically connected <algebraic curve> $C/k$, this <functor> assigns to a <scheme> $T$ the <line bundles> on $C\times T$ having degree $d$ on each geometric fibre, modulo bundles pulled back from $T$. One generally takes the associated sheaf for faithfully flat covers. A rational point of $C$ permits normalization of every <line bundle> along that point; normalized bundles have no automorphisms preserving their normalization and satisfy effective descent. The resulting <functor> is represented by the degree-$d$ component of the <Picard scheme of a curve>.
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