Why do we classify algebraic curves primarily by genus and not degree?

Question

I read in a text the following,

'One such classification might be according to the degree of the curve. Although this is a reasonable idea for very low degree curves, it turns out to be unsatisfactory for higher degree curves. Instead of the degree there is a more interesting quantity, namely the genus of a curve.'

What is the 'unsatisfactory' problem with classifying curves by degree when their respective degree is large?

Answer 1

As suggested by KReiser, let me develop the comment I made above. Here, "curve" means a non-singular connected projective $k$-variety of dimension $1$.

First, one should note that the degree of a projective variety $X$ is only defined for a given closed immersion $X\hookrightarrow{} \mathbf{P}^n_k$: indeed, if you take another immersion into a projective space, the degree might be different. For curves, a good example is the one of the projective line: the degree of the identity map of $\mathbf{P}^1_k$ is $1$, but you can also use the Veronese map to embed $\mathbf{P}^1_k$ into $\mathbf{P}^N_k$ for all $N$, which gives an immersion of degree $N$.

[Actually, since an immersion $X\hookrightarrow\mathbf{P}^N$ is equivalent to the data of a very ample line bundle $\mathcal{L}$ on $X$, I think that one should rather say "the degree of $\mathcal{L}$" instead of "the degree of $X$"...]

Therefore, it is more natural to look for a classification of curves relying on intrinsic characteristics of the variety: the genus is such an invariant (it does not depend on the embedding of your curve).

Using the genus actually gives a partial classification and is very useful to understand the geometry of the curve. As a summary, if $g$ denotes the genus of a geometrically connected (you may ignore this condition if you assume that $k$ is algebraically closed) curve $C$:

• if $g=0$, then it is a conic. Furhtermore, $C\simeq \mathbf{P}^1_k$ if and only if $C(k)\neq \emptyset$;
• if $g=1$ and $C(k)\neq\emptyset$, then $C$ is an elliptic curve;
• if $g\geq 2$, it is more difficult to give nice classifications: those curves are called curves of general type. For example it is known that for $g=2$ the curve $C$ is a hyperelliptic curve.

Let me add that the genus also gives some nice arithmetic informations! Gerd Falting proved in 1983/1984 the eponymous theorem which states that curves over $\mathbf{Q}$ with genus $g\geq 2$ have only finitely many rational points: in particular, Fermat curves $x^d+y^d=z^d$ have only finitely many integral solutions for $d\geq3$!

Clarification Of course, I do not intend to mean that the degree (of some embedding of $C$) may not intervene in any further classification: this would mean ignoring some geometric content of $C$ (which is that of very ample line bundles living on $C$)! I just wanted to emphasize the fact that the genus seems to give nice informations.

---

Now, if you are interested in the classification of higher dimensional varieties, you should first note that the arithmetic and geometric geni do not coincide anymore. Actually there is another invariant, called the Kodaira dimension, which seems to mimick the trichotomy we had in the case of curves (actually it gives you a $(n+2)$-tomy if you are interested in the classification of $n$-dimensional varieties).