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 $[C]\in\textrm{Hilb}_1(X)$ be the corresponding point. In *Some Remarks and Examples on Continuous Systems and Moduli*, Griffiths computed the dimension of the tangent space to $\textrm{Hilb}_1(X)$ at $[C]$. The answer is

- $g$, if $C$ is non-hyperelliptic, and
- $2g-2$ if $C$ is hyperelliptic.

Say $g=3$ and $C$ is hyperelliptic. In *On the module of intermediate Jacobians*, Lieberman showed $$\textrm{Hilb}_1(X,[C])_{\textrm{red}}\cong X=\textrm{Jac } C,$$ so Griffiths' result says the Hilbert scheme component $\textrm{Hilb}_1(X,[C])\subset \textrm{Hilb}_1(X)$ is a thickening of the Jacobian (tangent spaces are $4$-dimensional everywhere).

**Question**. What is the scheme structure on $\textrm{Hilb}_1(X,[C])$, when $C$ is hyperelliptic of genus $3$? Is it known to be something like $X\times \textrm{Spec }\mathbb C[\epsilon]$, or anything more complicated?