# Line in projective space is an example of a curve of genus $0$?

Let $L$ be a line in the projective space $\mathbb{P}^n$ over a field $k$. Is a line $L$ an example of a curve of genus $0$ in $\mathbb{P}^n$. I was wondering if I could verify this with someone, because I just started learning this topic and wasn't too sure.

I would also appreciate if someone could point me to a reference where I can find how to compute genus of curves. Thank you very much!

Yes, it is.

You can choose homogeneous coordinates $(X_0: \dots : X_n)$ such that $L$ is given by the equations $X_i =0$, $i\geq 2$. This gives a isomorphism to $\mathbb{P}^1$, hence they have the same genus.

Take a look at the first chapter of Hartshorne's book, Algebraic Geonetry.

If $n =2$, there is a formula for the genus of a smooth degree d curve in the plane: https://en.wikipedia.org/wiki/Genus%E2%80%93degree_formula
A "geometric" proof (without using adjunction formula) of this genus-degree formula, using Riemann-Hurwitz, is given in Andreas Gathman's notes on algebraic geometry. However, I couldn't find it in my PDF - at any rate the idea as I remember it is to choose a sufficiently generic point in the complement of the curve, so that the projection from that point onto $\mathbb{P}^1$ has very simple ramification, and then use Riemann-Hurwitz. (The details of this are not clear to me.)