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Largely I want to know as to how does one say anything about the hyperellipticity or the genus of the Riemann surface by looking at the algebraic curve and its singularities.


  • To give a specific example, what is the meaning of the statement that, "a curve of genus 2 can be expressed as a fourth degree plane curve possessing one double point" ?

Does this mean that any Riemann surface of genus 2 is a normalization of a fourth degree algebraic curve in $\mathbb{P}^2$ with one double point?

In general the proof says that any compact hyperelliptic Riemann surface of genus $g$ is a normalization of a an algebraic curve of degree $2g+2$ of the form $y^2 = \prod _{i = 1}^{2g+2} (x-a_i)$

So I would have naively thought that a genus $2$ Riemann surface (which is always hyperelliptic) will need a $2\times 2 +2 = 6$ degree algebraic curve. Hence I am not clear as to what to read of the quoted statement. Is something very special happening for genus $2$? Is the general theorem not a sharp statement?

  • The general statement seems to tell me that the $a_i$ being distinct guarantees the smoothness of the algebraic curve except may be at the points at infinity. Now if there is a lower degree curve that can equally well represent the genus $2$ surface then is that necessarily going to be a curve with singularities?

  • If the general statement is not a sharp statement and one can in cases do with lower degree curves than $2g+2$ then how does one derive the genus of the Riemann surface by looking at the algebraic curve and may be its singularities. Is there a "generalized" genus formula that works always?

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    @GottfriedLeibniz I currently have access to only the book on Riemann surfaces by Griffiths. Thats the book that I am reading. It would be great if you can sketch the argument or give enough hints on these matters on which I can build on.2011-11-25

1 Answers 1

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A smooth degree $4$ plane curve has genus $3$. If you now degenerate this curve so that it obtains one ordinary double point, this corresponds to pinching off one loop on the genus $3$ Riemann surface. If you visualize this, you will see that it now looks like a genus $2$ Riemann surface with two points identified. Desingularizing this, you obtain a genus $2$ curve.

Now hyperelliptic curves are also traditionally represented as $y^2 = f(x)$, where $f(x)$ has degree $2g+1$ or $2g+2$. When the degree is $> 3$, these are singular equations, but the singularity is not an ordinary double point. So this is simply a different way of representing a hyperelliptic curve (which also puts the hyperelliptic involution in evidence: it is the map $(x,y) \mapsto (x,-y)$). Note that in the model of the first paragraph, the hyperelliptic involution is not so evident.

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    Thanks for your reply. I am finding it to be very non-intuitive that a change of coordinates can change the degree of the curve! I don't know how to go about looking for the hyperelliptic involution of degree$4$curve when I don't even know that curve. All I have is that degree$6$curve. May be you can also see this other question of mine where I have gone into some more details of the associated things that I am stuck with - http://math.stackexchange.com/questions/85834/relation-between-n-tuple-points-on-an-algebaric-curve-and-its-pre-image-in-the-n2011-11-26