Describing the compactified Jacobian of a nodal curve

Question

$\DeclareMathOperator{\Pic}{Pic}$Let $C$ be an integral projective curve over $\mathbb C$, which is smooth except for a single node $x\in C$. Let $M$ be the moduli space of stable torsion-free sheaves on $C$ of rank 1 and degree $d$, so $M$ naturally contains $\Pic^d C$.

Question. How to describe the structure of $M$ in detail? Is there a good reference? I'm not asking for a proof of existence, which is handled in [1], but rather a good description of $M$'s structure.

Let $$\nu: \tilde C \to C$$ be the normalization map. Here are my thoughts up to now:

1. Every pure sheaf $F$ of rank $1$ is stable, since it does not admit any proper quotients $F \to F'' \to 0$, which are also pure. This means there are no strictly semistable sheaves, which implies that $M$ is projective. So $M$ is a compactification of $\Pic^d C$.

2. Additionally to line bundles $L \in \Pic^d(C)$, there is also the ideal sheaf of the node, $I_x \subset \mathcal O_C$, which is pure of rank $1$, but is not a line bundle. One my also consider it's dual $I_x^\vee$, and also tensor any line bundle $L$ with $I_x$ and $I_x^\vee$. If I understand correctly, $I_x$ has degree $-2$, because $\nu^*I_x = \mathcal O(-p-q)$, where $p,q \in \tilde C$ sit above $x$. Is that correct? Are there any other pure sheaves of rank $1$?

3. The pull-back morphism $\nu^*: \Pic^d C \to \Pic^d \tilde C$ is a $\mathbb C^*$-bundle, because descending a line bundle $\tilde L \in \Pic\tilde C$ to $C$ means identifying $L_p$ with $L_q$, so there is a $\mathbb C^*$-choice of descend data.

4. Is it true that $\nu^*$ extends to a map $M \to \Pic^d \tilde C$? If I'm not mistaken, pulling-back a pure sheaf of rank $1$ should give a line bundle on $\tilde C$, so this is promising. On the other hand, [2, part 2 Good singular fibres] claims that only the normalisation $\tilde M$ maps to $\Pic^d \tilde C$, and that the map $\tilde M \to M$ identifies points of different fibers, so there is no change to get a map $M \to \Pic^d \tilde C$. I tried to take a look at [3], which is given in [2] as a reference, but I could not get into that 90 page paper.

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[1] Huybrechts, Lehn; The Geometry of Moduli Spaces of Sheaves.

[2] Sawon; On the Discriminant Locus of a Lagrangian fibration

[3] Oda, Seshadri; Compactifications of the Generalized Jacobian Variety

Answer 1

$\DeclareMathOperator{\Tors}{Tors}\DeclareMathOperator{\Pic}{Pic}$Here is what I got from discussing with my advisor.

Basic construction Let $L$ be a line bundle on $\tilde C$, and let $\pi: (\nu_* L)\otimes \mathcal O_x \to \mathcal O_x$ be a quotient, where $\mathcal O_x$ is the structure sheaf of the point $x \in C$. This datum defines a stable sheaf of rank $1$ by considering the short exact sequence $$0 \to E \to \nu_* L \xrightarrow{\pi} \mathcal O_x \to 0.$$

Note that $\nu_* L \otimes \mathcal O_x = \nu_* \mathcal O_p \oplus \nu_* \mathcal O_q$, so there is a $\mathbb P^1$ of choices of $\pi$. Now one verifies the following properties:

1. $E$ is a line bundle if and only if $\pi$ is non-zero on both summand $\nu_* \mathcal O_p$ and $\nu_* \mathcal O_q$.

   Clearly $E$ is a line bundle on $C \setminus \{x\}$. At $x$, one may complete the local ring $\mathcal O_{C,p}$ to check if $E$ is locally free. That means considering the rings $$A = \mathbb C[[x,y]]/(xy) \hookrightarrow \mathbb C[[x]] \times \mathbb C[[y]] = \tilde A.$$ Wlog, $L$ is of the form $\mathcal O_{\tilde C}(ap+bq)$, and so corresponds to $$\{\, (f,g) \in \mathbb C((x)) \times \mathbb C((y)) : x^{-a}f, y^{-b}g \text{ have no poles}\,\}.$$ If $\pi$ is zero on one factor, say on $\nu_* \mathcal O_p$, then $E$ corresponds to $\mathcal O_{\tilde C}((a+1)p + bq)$, hence $E$ is not locally free over $A$, and actually $E \cong \nu_* \nu^* E$. If $\pi = (\lambda, \mu)$ for $\lambda, \mu \neq 0$, then $E$ is generated (even over $A$) by $(\mu x^a, -\lambda y^{b})$.

2. Every stable sheaf of rank $1$ is of that type.

   If $E$ is stable of rank $1$, then $\nu^*E / \Tors(\nu^*E)$ is locally free, so consider the natural map $\varphi: E \to \nu_*(\nu^* E / \Tors(\nu^*E))$. If $\varphi$ has cokernel $\mathcal O_x$, we are done. Otherwise, $\varphi$ is an isomorphism. Set $L = \nu^*E / \Tors(\nu^*E) \otimes \mathcal O(p)$, and consider the sequence $$0 \to \nu^*E / \Tors(\nu^* E) \to L \to \mathcal O_p \to 0.$$ Pushing down gives the desired sequence.

The two parts also show: If $E$ is a line bundle, $L$ and $\pi$ are uniquely determined. If $E$ is not a line bundle, there are two bundles $L$ and $L \otimes \mathcal O(q-p)$ which both can be used to define $E$.

To turn this construction into a description of $M_C(1, \chi)$, consider the Poincaré bundle $\mathcal P$ on $\Pic^{\chi+1}(\tilde C) \times \tilde C$. Here $\Pic^{\chi+1}$ is the Picard variety of line bundles of Euler characteristic $\chi$ (which is a bit more convenient than to convert degrees into Euler characteristics all the time). On $\Pic^{\chi+1}(\tilde C)$ consider the $\mathbb P^1$-bundle $$X = \mathbb P(\mathcal P|_{\Pic \times \{p\}} \oplus \mathcal P|_{\Pic \times \{q\}}) \to \Pic^{\chi+1}(\tilde C).$$ Now a point on $X$ consists exactly of the datum $(L, \pi)$ used to define a stable sheaf $E$ of Euler characteristic $\chi$. This provides the morphism $$f: X \to M_C(1,\chi).$$ Formally, one would push-down the quotient of $\mathcal P$ which exists on on $X \times \tilde C$ to $X \times C$. The kernel should then provide a family of stable sheaves.