= Picard charts from nonspecial divisors
{c}
{title2=$H^1(C,\mathcal O(E))=0,\quad\deg E=g$}
A degree-$g$ <line bundle> with vanishing first <sheaf cohomology> has exactly one section up to scale, by <Riemann-Roch theorem>. Its zero divisor is unique and varies in families by <cohomology and base change for line bundles on a curve>. For a fixed divisor $B$ of degree $N-g$, the open subset of $C^{(g)}$ where $\mathcal O(E)$ has vanishing first cohomology gives a chart in degree $N$, by $E\mapsto\mathcal O(B+E)$. Different charts are glued by the unique residual divisor of $\mathcal O(B+E-B')$ wherever that line bundle is again nonspecial. Normalizing the universal bundles along a fixed rational point gives their canonical compatible gluing and a representing <scheme> for the <Picard functor of a curve>.
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