Irreducible Curve (Variety)?

Question

I want to construct an irreducible variety of a plane curve. I would like it to be of the curve $f(x,y)=xy^s$, where s is an arbitrary integer. Since the ideal of this would be generated by one element, and that element a monomial, am I guaranteed that this is irreducible?

Answer 1

It is not clear what you mean by "the curve" $f(x,y)=xy^s$. Do you mean the zero set of the of $f(x,y)$ in $\mathbb A ^2$? In that case, the ideal of the curve would be $I=\langle x \rangle \cap \langle y^s \rangle$, which is clearly not irreducible.

However, could it be that what you really mean is the image of $\mathbb A^1 \ni x \mapsto (x,x^s) \in \mathbb A^2$? In that case, the ideal is given by $I= (x^s-y)$, and this is certainly irreducible.

Answer 2

You'd need the polynomial $f(x,y) = xy^s$ to be an irreducible polynomial which isn't true ($x, y$ both divide $xy^s$).