I've been reading about parametrization of algebraic curves recently and the idea of the "genus of a curve" appears quite often (my impression is that a curve is parametrizable exactly when it has genus 0), but I can't seem to find a definition for it, much less an intuitive idea of what this means. I'd appreciate if anyone could explain what the genus of an algebraic curve is.
More specifically, papers often say something like this (where $\mathcal{C}$ is our curve):
$\mathcal{C}$ has singularities at $P_1=(1:0:0),P_2=(0:1:0),P_3=(0:0:1),P_4=(1:1:1)$, where $P_1,P_2,P_3$ are 5-fold points and $P_4$ is a 4-fold point. So the genus of $\mathcal{C}$ is 0. [For reference, $\mathcal{C}$ is a curve of degree 10 in $\mathbb{P}^2(\mathbb{C})$.]
$\mathcal{C}$ has a triple point $P_1$ at the origin $(0,0)$, and double points $P_2=(0,1), P_3=(1,1)$, $P_4=(1,0)$. So the genus of $\mathcal{C}$ is 0. [$\mathcal{C}$ is a curve of degree 5 in $\mathbb{A}^2(\mathbb{C})$.]
I don't understand how to make the leap from singular points to genus in these examples. Can someone explain?
Best Answer
Hmm. I was hoping someone who actually knows algebraic geometry would write an answer to this question.
First, some intuition. As it turns out, complex projective non-singular algebraic curves are the same thing as (connected) compact Riemann surfaces, which topologically are compact oriented surfaces. By the classification of compact surfaces, such surfaces are uniquely classified by a single number, their genus $g$, which counts how many holes there are in the surface. More precisely, for every $g$ there is an oriented surface $S_g$ which is the connected sum of $g$ tori (that is, it's a "doughnut with $g$ holes"), and every (connected) compact orientable surface is homeomorphic to $S_g$ for a unique $g$.
The genus $g$ has several equivalent definitions, and some of these generalize to algebraic geometry where we do not have direct access to topological information. Unfortunately, none of them are particularly easy to describe. An excellent introduction to this subject can be found in Fulton's Algebraic Curves.
So the idea is clear for non-singular curves. However, the genus turns out to be a birational invariant of curves (in particular, invariant under deletion of finitely many points), so it is possible to extend the definition of the genus to singular curves by declaring the genus of a singular curve to be the genus of a non-singular curve birational to it.
Example. Consider the singular curve $y^2 = x^3$ of degree $3$ in $\mathbb{P}^2(\mathbb{C})$ (equivalently $\mathbb{A}^2(\mathbb{C})$). It has a singular point at the origin of order $2$. Now, a non-singular curve of degree $3$ in $\mathbb{P}^2(\mathbb{C})$ has genus $1$ (see elliptic curve), but this curve doesn't: in fact, using the birational map $t \mapsto (t^2, t^3)$ we see that this curve is birational to $\mathbb{P}^1(\mathbb{C})$, hence has genus $0$.
So we see from the above that, roughly speaking, singularities decrease the "expected" genus of a curve (where "expected" means the number $\frac{(d-1)(d-2)}{2}$ that one gets from the genus-degree formula). Exactly how much singularities decrease the expected genus appears to me to be a somewhat complicated question and I am not the one to discuss it in detail. However, for "ordinary" singular points (I am not sure exactly what this means) of order $r$ it seems that the genus gets decreased by $\frac{r(r-1)}{2}$. So the genus in your first example is $$\frac{9 \cdot 8}{2} - 3 \frac{5 \cdot 4}{2} - \frac{4 \cdot 3}{2} = 36 - 30 - 6 = 0$$
and the genus in your second example is $$\frac{4 \cdot 3}{2} - \frac{3 \cdot 2}{2} - 3 \frac{2 \cdot 1}{2} = 6 - 3 - 3 = 0.$$