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?
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.