In section III.3.4 of Eisenbud & Harris's "The Geometry of Schemes," we/they construct an infinite family of double structures on $\mathbb{P}^1 \subset \mathbb{P}^3$ that are distinguished from each other by their genus. Here is the construction (let's just worry about $\mathbb{C}$ for now):


Let $d$ be a non-negative integer, and let $S = \mathbb{C}[u,v,x,y]/(x^2, xy, y^2, u^dx - v^dy)$. 


Then $X\_d = Proj$ $S$ is a double line with arithmetic genus $-d$.


Eisenbud and Harris then go on to say that "every projective double line of genus $-d$, with $d \geq 0$, is isomorphic to $X\_d$."


My question is: where can I find a proof of this statement?


More generally: if you fix a curve $C$ of genus $g$, how can I describe the moduli of genus $d$ double curves lying over top of $C$?



Classification of fat projective lines?
(Note: I don't assume $C$ is a curve, or even that it is smooth.)


Suppose $C'$ is a 2-fold thickening of $C$. Then $C'$ has the same underlying topological space as $C$. On that space, we have a short exact sequence of $\newcommand{\O}{\mathcal O}\O\_{C'}$-modules
$$\newcommand{\L}{\mathcal L}\tag{$\dagger$}
0\to \L\to \O\_{C'}\to \O\_C\to 0
$$
with $\O\_{C'}\to \O\_C$ a ring homomorphism.


Since $C'$ is a square-zero thickening, the sheaf of ideals $\L$ squares to zero, so it is actually an $\O\_C$-module. Since $C'$ is a 2-fold thickening, $\L$ is actually a line bundle on $C$ (thus the suggestive notation). In fact, a 2-fold thickening of $C$ is no more than an extension of sheaves of rings $\O\_{C'}\to \O\_C$ whose kernel is a line bundle. An isomorphism of thickenings will respect the map to $\O\_C$, but need not induce the identity map on $\L$, so we care about parameterizing such extensions of algebras "up to scalar" in some sense.


Given a line bundle $\L$, the set of extensions of algebras of the form $(\dagger)$ is parameterized by $Ext^1\_{\O\_C}(L\_C,\L)$, where $L\_C$ is the cotangent complex of $C$. This is a special case of Theorem 1.2.3 in Chapter III of Illusie's [Complexe Cotangent et Deformations I](http://rads.stackoverflow.com/amzn/click/3540056866). The "up to scalar" in the previous paragraph should correspond to looking for elements of this $Ext^1$ up to scalar. If $C$ is smooth, then $L\_C\cong \Omega^1\_C$ is locally free, so $Ext^1(L\_C,\L)\cong H^1(C,\mathcal{T}\otimes \L)$, where $\mathcal T = (\Omega^1\_C)^\vee$ is the tangent bundle of $C$.


**Upshot:** A 2-fold thickening is given by choosing some $\L\in Pic(C)$ and some element (up to scalar) of $Ext^1(L\_C,\L)$. Note that in such a thickening $C'$, $\L$ is the conormal bundle of $C$ in $C'$.


In the case $\newcommand{\P}{\mathbb P}C=\P^1$ and $\L=\O(d)$ ($d\ge 0$), we have that $\mathcal T=\O(2)$, and $H^1(\P^1,\O(2+d))=0$, so there is a unique extension for a given $d$. If I haven't made a mistake in this answer, there should be two non-isomorphic thickenings of $\P^1$ with conormal bundle $\L=\O(-4)$, and an infinite number of non-isomorphic thickenings with conormal bundle $\L=\O(d)$ if $d<-4$.


**Special case:** The element $0\in Ext^1(L\_C,\L)$ corresponds to the first infintessimal neighborhood of $C$ in the total space $\newcommand{\V}{\mathbb V}\V(\L^\vee)$. Here, $C$ is thought of as the zero section of $\V(\L^\vee)$.


