= Picard scheme of a curve
{c}
{title2=$\operatorname{Pic}^d(C)$}
For a smooth projective <algebraic curve> of genus $g$, this smooth proper $g$-dimensional <scheme> represents degree-$d$ <line bundles> in families. Charts come from nonspecial degree-$g$ bundles: their first <sheaf cohomology> vanishes, so the <Riemann-Roch theorem> gives exactly one independent <global section>, whose zero <Cartier divisor> varies algebraically. Twisting by fixed <line bundles> gives charts covering every degree-$d$ class. For degree zero the <tensor product> gives the <Jacobian variety>; other components are its torsors.
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