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Let $C$ be a smooth projective curve of genus $g \geq 2$ over an algebraically closed field and $J_C$ its Jacobian variety, of dimension $g$. The Abel-Jacobi map given by $P \mapsto [P] - [O]$ defines an embedding $C \rightarrow J_C$. How do we show that this map is a closed immersion?

Now if we were to remove a finite number of points from $C$ to obtain an affine curve $X$ (of genus $g$), we have an open immersion $X \rightarrow C$. Would the composition $X \rightarrow J_C$ fit the definition of an immersion? Are there things to check?

oleout
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  • Often, "embedding" is a synonym for "immersion". Since it appears it isn't for you, could you explain a little bit more about how you're using that term? – KReiser Nov 11 '22 at 05:01
  • @KReiser For the first question, do you mean $C \rightarrow J_C$ is a closed immersion simply because we are embedding into $J_C$ a closed subscheme $C$? For the second question, I remember seeing definition of an immersion as an open immersion followed by a closed immersion (or the other way around), and I don't think $X \rightarrow J_C$ is a closed or open immersion, so I'm wondering in this case could it then be the immersion according to the definition I just gave? – oleout Nov 11 '22 at 05:09
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    You say "The Abel-Jacobi map ... defines an embedding $C\to J_C$. How do we show that this map is a closed immersion?" I'm asking you to define your term "embedding" - if it's a synonym for immersion, then since $C$ is proper you know any map out of it is closed and so one may appeal to the fact that an immersion with closed image is a closed immersion. If it's not, perhaps there is more work to do, and since you're confused here I want to try and pin down where that confusion is coming from. – KReiser Nov 11 '22 at 05:15
  • @KReiser You are right, I'm interchanging it with immersion. I just wasn't sure about the closed part, but your link helped and now it's clear. Thank you! About the map $X \rightarrow J_C$, am I right to say that it is just an immersion then? Neither open nor closed. – oleout Nov 11 '22 at 05:20
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    Yes, that's correct, $X\to J_C$ would be a (locally closed) immersion. For a reduced scheme or a quasi-compact morphism (the latter of which applies if the source is noetherian, like you have here), you can interchange the order of open/closed immersions without worry (ref). – KReiser Nov 11 '22 at 05:28
  • @KReiser Thank you for your time again! – oleout Nov 11 '22 at 05:49
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    For the first question, see Prop. 2.3, p. 172 in Milne, "Jacobian varieties", in the volume Arithmetic Geometry ed. by Cornell and Silverman. – Damian Rössler Nov 11 '22 at 15:14

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