TITLE: Hyperellipticity (or not!) of a Riemann surface and the singularities of the curve
QUESTION [5 upvotes]: 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?

REPLY [7 votes]: 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.