Classify the plane projective curve $C\subset \mathbb{P}^2$ defined by $X^n+Y^n-Z^n$ with $n\in\mathbb{N}$.
I need to know how the aforementioned curve behaves when $n$ is even or odd, if it is rational or of what type it is. I know that the palne projective curve $X^2+Y^2=Z^2$ is rational, so if $n$ is even, the curve mentioned in the question will be rational or what is it? What happens when $n$ is odd? Thank you.