# Genus of curves in projective space

• Oct 12th 2008, 05:20 PM
cduston
Genus of curves in projective space
Hello everyone,
I am working on understanding the genus of curves in projective spaces, and I have convinced myself that an elliptic curve

$\displaystyle y^2=x(x-\lambda _1)(x-\lambda _2)$

is a topologically equivalent to a torus (genus 1) by making cuts between the zeros (0, $\displaystyle \lambda _1, \lambda _2$, and $\displaystyle \infty$) and gluing up regions so that the function is single-valued. However, now I need to know about a curve C of degree d in the complex projective space $\displaystyle \mathbb{C}P$. I know the answer:

$\displaystyle g=\binom {d-1}{2}=\frac{1}{2}(d-1)(d-2).$

So it seems obvious what is happening is for d+1 zeros (degree d + infinity) there are d-1 possible connections between two adjacent points, and we want to identify them in pairs to make our manifold. But I'm not clear on how we know that this coefficient gives us the genus (ie number of holes). Could someone elaborate on that, or correct me if I am wrong about something?

Thanks very much.