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Hartshorne proved in his thesis that if $S$ is connected, then the Hilbert scheme $\operatorname{Hilb}^p=\operatorname{Hilb}^p(\mathbb{P}^n_S/S)$ is too (where $p\in \mathbb{Q}[z]$). Can the same be ...

Let $C$ be a smooth curve of genus $g\geq 3$, embedded in its Jacobian $X=\textrm{Jac } C$ via an Abel map. Let $\textrm{Hilb}_1(X)$ be the Hilbert scheme of curves in $X$, and let $[C]\in\textrm{...