Is a torsion free sheave of rank one on a reducible curve the pushforward of a line bundle on a normalization?

Question

Let $X$ be a nodal curve, possibly reducible. Then can any torsion free sheaf of rank one on $X$ be expressed as $\pi_*(L)$, where $L$ is a line bundle on a partial normalization of $X$? This looks standard, but I just can not find a reference that precisely state this conclusion.

Answer 1

This is (exactly as stated in your question) for example in Proposition 10.1 of

Oda, Tadao; Seshadri, C. S. Compactifications of the generalized Jacobian variety. Trans. Amer. Math. Soc. 253 (1979), 1–90.

Although as the idea (by Mumford) of compactifying the Jacobian by torsion free sheaves is older, this can surely be found explicitly in older works, perhaps D'Souza.

A very much related result for families is obtained by Esteves and Pacini in

Semistable modifications of families of curves and compactified Jacobians. Ark. Mat. 54 (2016), no. 1, 55–83.

Answer 2

Short proof: A given rank $1$, torsion-free sheaf $F$ is the direct image of a sheaf $L$ on the partial normalization $Y := \operatorname{Spec} \operatorname{Hom}(F,F)$. When $X$ is nodal, $Y$ is Gorenstein, and a theorem of Vasconcelos implies that $L$ is a line bundle.

In more detail, I first claim the sheaf $\operatorname{Hom}(F,F)$ of endomorphisms is a sheaf of finite commutative $\mathcal{O}_{X}$-algebras. Since $F$ is torsion-free, $\operatorname{Hom}(F,F)$ injects into its generic fiber which is just the ring of rational functions since $F$ is generically trivial of rank $1$. We conclude that $\operatorname{Hom}(F, F)$ is commutative. The $\mathcal{O}_{X}$-algebra is also finitely generated (fix generators and track where they go) and integral (use Hamilton-Cayley to produce a polynomial that kills a given local section), so we conclude that it is finite.

Essentially by construction, $Y := \operatorname{Spec} \operatorname{Hom}(F,F)$ is a partial normalization of $X$ that carries a sheaf $L$ whose direct image is $F$. Furthermore, $L$ satisfies $\operatorname{Hom}(L,L)=\mathcal{O}_{y}$. Since $X$ was nodal, $Y$ is nodal as well, and we conclude by Theorem (3.1) of Vasconcelos's paper "Reflexive modules over Gorenstein rings".

Note: This argument applies to a curve with arbitrary singularities, although we can only conclude that $L$ is a line bundle when $Y$ is Gorenstein (otherwise we can only assert that $\operatorname{Hom}(L,L) = \mathcal{O}_{Y}$). The partial normalization $Y$ is always Gorenstein when $X$ has at worst $A_{n}$ singularities, but otherwise $X$ admits sheaves that are not direct images of line bundles. For example, if $X$ is the Zariski closure of the curve $\{x^3-y^4=0\}$, then the maximal ideal $(x,y)$ is not the direct image of a line bundle.