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This question is sort of an extension to this previous question of mine,

Hyperellipticity (or not!) of a Riemann surface and the singularities of the curve

  • If one knows the multiplicity of a certain point on the algebraic curve then what does it tell about the Riemann surface normalizing it? (..beyond just saying that the genus of that is now strictly lower than the upper bound of $\frac{(d-1)(d-2)}{2}$ where $d$ is the degree of the curve..)

  • If a point on the algebraic curve has a non-trivial multiplicity then what can be said about its pre-image in the Riemann surface normalizing it?

  • To may be help sharpen the above question - if $p$ is one of the points on the Riemann surface which is mapped to the point with some given multiplicity on the algebraic curve then can something be said about $i(p)$ or $l(p)$ (..the two ways in which points can enter the Riemann-Roch theorem..)

Or if $\{p_i\}$ are the set of points on the Riemann surface which go down to this point with multiplicity on the curve then I wonder if I should be looking at $i(\sum p_i)$ and $l(\sum p_i)$ instead of the above..

  • If say on the algebraic curve $f=0$, $q$ is one such point with non-trivial multiplicity (..say $n$...) then what is the intersection number of $f$ and $\frac{\partial f}{\partial y}$ at $q$ - i.e what is $(f.\frac{\partial f}{\partial y})_q$? (..I would think that this quantity is interesting since that would I guess is the amount by which each such mupltiplicity point reduced the genus up there..)

And lastly if anything can be said about hyperellipticity of the Riemann surface by looking at the curve's equation and multiplicities. This is most confusing to me since a priori it seemed that the notions of hyperellipticity and their classification was based directly on the Riemann surface and not in terms of the algebraic curve which it normalizes! I want to know what I am missing.

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