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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!

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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.

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The answer is yes, as Alan's answer shows.

Some comments and references:

A method that is useful for computing the genus of curves is the Riemann-Hurwitz formula, which has a very intuitive Riemann surface analogue (about lifting good cell structures via ramified coverings): https://www.google.com/search?q=the+riemann+hurwitz+formula&ie=utf-8&oe=utf-8

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.)

(Opinion: Almost any reference is preferable to Hartshorne for an introduction to algebraic geometry. Gathmann's notes, Ravi Vakil's notes, or Shafarevich's book are much gentler.)

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