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Suppose $C$ is a (singular) rational curve whose normalization $p: \mathbb P^1 \to C$ is a set-theoretic bijection.

Can one understand how the compactified Jacobian of $C$ looks like?

For example, the answer to this question reads "The compactified Jacobian is a homeomorphic to a product of the Jacobian of the normalization times some local factors from the singularities." How do the local factors coming from the singularities look like?

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    $\begingroup$ The compactified Jacobian is isomorphic to $C$. This is quite classical by now, you can look for instance at Kass' paper "Singular curves and their compactified Jacobians" in the Harris volume (Clay math. Proc., vol. 18), or better, do the exercise yourself. $\endgroup$
    – abx
    Commented Sep 25, 2022 at 13:30
  • $\begingroup$ @abx Sorry, I meant to ask about the compactified Jacobian not just for the rational curve with one cusp (a planar cuspidal cubic curve), but about any curve $C$ whose normalization is ${\mathbb P}^1$ and whose normalization ${\mathbb P}^1 \to C$ is a set-theoretic bijection. Do you think I should reformulate my question to make this clear? $\endgroup$
    – IntegrableSystemsEnthusiast
    Commented Sep 25, 2022 at 14:49
  • $\begingroup$ @abx Sorry, just to clarify, does your comment answer the question only for a planar rational cubic curve or for any rational curve $C$ whose normalization is a set-theoretic bijection? $\endgroup$
    – IntegrableSystemsEnthusiast
    Commented Sep 25, 2022 at 15:00

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Section 3 of the following paper of Beauville should answer your question: https://arxiv.org/abs/alg-geom/9701019

In particular, it is shown there that, up to replacing the compactified Jacobian by a homeomorphic variety, it embeds in a Grassmannian.

For example, the completion of the curve $y^3 =x^4$ is worked out in the introduction of this paper: https://www.degruyter.com/document/doi/10.1515/crelle-2012-0093/html?lang=en (Note that you really want the published version, since the arxiv version doesn't contain this example.)

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  • $\begingroup$ Thank you very much! Do you know if it is also possible to understand the structure of the compactified Jacobian as an algebraic variety, not just up to homeomorphism? $\endgroup$
    – IntegrableSystemsEnthusiast
    Commented Sep 26, 2022 at 9:49
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    $\begingroup$ I'm not sure what you're looking for. The above for example shows that if $C$ has ADE singularities then the compactified Jacobian is stratified into affine spaces (namely the orbits of the open Jacobian on the compactified one). The paper arxiv.org/abs/1406.2299 should contain some more useful info if you're willing to assume that singularities are planar; otherwise it doesn't behave so well. (Can be reducible for example) $\endgroup$
    – Jef
    Commented Sep 26, 2022 at 18:44

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