Picard group of double "line" Are there any nice descriptions of the Picard group of a non-reduced double structure on $\mathbb{P}^1?$ In particular, I'm looking for a description that makes clear the difference between the Picard group of the double line $x^2=0$ in $\mathbb{P}^2$ (in my mind the Picard group should be zero-dimensional, since the curve is cut out by a quadratic equation and thus "genus zero") and the Picard group of the double structure $f(x,y,z)^2=0,$ where $f$ is a smooth conic (which should be three dimensional, since the curve is genus three).

Let $k$ denote the ground field. The paper https://www.math.columbia.edu/~bayer/papers/Ribbons_BE95.pdf of Bayer-Eisenbud studies the geometry of ribbons/double structures on $\mathbb P^1$; concretely, these are non-reduced scheme structures $C$ on $D=\mathbb P^1$ such that the ideal $\mathcal I$ of $D$ inside $C$ satisfies $\mathcal I^2 = 0$. The genus of this scheme is
$$
g = g(C) = 1 - \chi(C) = 1 - h^0(\mathcal O_C) + h^1(\mathcal O_C),
$$
and according to Prop. 4.1, $\operatorname{Pic}(C) \cong k^g \times \mathbb Z$.
