# Symmetric products of smooth non-proper curves over generalized Jacobians

Does anyone know a written reference for the following fact?

For large n, $\operatorname{Sym}^n X \to \operatorname{Jac}^nX$ is a vector bundle, where $X$ is a smooth, non-proper curve, and $\operatorname{Jac}X$ is its generalized Jacobian, so $\operatorname{Jac}^nX = \operatorname{Pic}^n X^+$ where $X^+$ is the one-point compactification given by the quotient of the smooth compactification $Xc$ by $Xc -X$.

(I know how to prove it-- I would like to be able to cite a reference for it.)

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## 1 Answer

I think I should have said "affine bundle" instead of "vector bundle." I still haven't found a reference, but I wrote a proof in the appendix of:

http://www.math.harvard.edu/~kwickelg/papers/delta2real.pdf

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