3
$\begingroup$

Let X be a reduced, reducible curve over $\mathbb{C}$ with locally planar singularities, and let $\widetilde{X}$ be its normalization. I am interested in the Jacobian varieties $\mathrm{Jac}(X)$ and $\mathrm{Jac}(\widetilde{X})$, along with the compactification $\mathrm{Jac}(X)$ by rank 1 torsion-free sheaves. One of the questions I have is whether we can still write $\mathrm{Jac}(X)$ as an extension of $\mathrm{Jac}(\widetilde{X})$ by a product of $\mathbb{G}_a$'s and $\mathbb{G}_m$'s. Literature surrounding this setup seems sparse, and what I've come across tends to focus solely on the case where $X$ is nodal. Any references/results would be appreciated!

Improve this question
$\endgroup$
1
  • 3
    $\begingroup$ Yes. The kernel of the map from $\mathrm{Jac}(X)$ to $\mathrm{Jac}(\widetilde{X})$ is a connected commutative affine algebraic group and any such group over $\mathbb{C}$ is a product of additive and multiplicative groups. $\endgroup$
    – naf
    Commented Dec 5 at 4:11

0

Reset to default

You must log in to answer this question.