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Let's suppose we have a non-hyperelliptic Riemann surface $X$ of genus $3$ (without lost of generality, embedded in $\mathbb{P}^2$ as a smooth plane curve) and we consider its Theta Divisor. Is well known that the Gauss Map of the Theta Divisor can be identified with the map which associates to the divisor $P+Q$ on $X$ the line in $\mathbb{P}^2$ spanned by those two points, and to the divisor $2P$ the tangent line to $X$ at the point $P$.

What about the differential of this map? What are the points of the Theta divisor for which the differential is not injective? Do they have a geometrical description?

Thanks in advice

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  • $\begingroup$ Today, a colleague anonymously downvotes even well formulated questions like this one. I don't understand! $\endgroup$ – Jean Marie Oct 25 '16 at 14:43
  • $\begingroup$ What you don't understand? $\endgroup$ – Lucke Oct 25 '16 at 15:09
  • $\begingroup$ And sorry JeanMarie: where is the question you mean? There are a lot of thinks that I don't understand... $\endgroup$ – Lucke Oct 25 '16 at 15:10
  • $\begingroup$ What I don't understand ? I don't understand the downvote that maybe you have not noticed because I have upvoted just after. Besides, it's difficult for me to understand your question because it's not my field of competence. I only see that there is a connection with groups on elliptic curves. $\endgroup$ – Jean Marie Oct 25 '16 at 15:21
  • $\begingroup$ Sorry, now I understand what you mean. You say that is not your field, but seriously you make me really curious: in another context I've seen a connection with "groups on elliptic curves" that you meen. Only I don't see this connection here. Could you say me something more, or just where I can find more information on this topic? I would be really greatful. $\endgroup$ – Lucke Oct 25 '16 at 18:27

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