coordinate ring of curve

Question

Assume one has a curve $X$ in $\mathbb{P}^2(k)$ defined by a polynomial in $k[x_0,x_1,x_2]$. By intersecting it with the hyperplanes defined by $x_0 \neq 0$ and $x_2 \neq 0$ one obtains an affine open cover of it which consists of two affine curves, in this case the zero sets $Z(y^n-x^{n-1}+1)$ and $Z(v^n-u+u^n)$ respectively. I think these have coordinate rings given by $\mathbb{C}[x,y]/(y^n-x^{n-1}+1)$ when $x_0 \neq 0$ and $\mathbb{C}[u,v]/(v^n-u+u^n)$ when $x_2 \neq 0$ respectively. How can I find the coordinate ring of the intersection of these affine curves, i.e. when I impose both $x_0$ and $x_2$ to be non zero?

Answer 1

In the affine plane $x_0\ne 0$, $\Bbb{C}[x_1,x_2]$ the curve is given by $x_1^n-x_2^{n-1}+1$, to impose $x_2\ne 0$, we just localize at $x_2$ to get that the intersection has coordinate ring $\Bbb{C}[x_1,x_2,x_2^{-1}]/(x_1^n-x_2^{n-1}+1)$. Alternatively in the other open affine $x_2\ne 0$, it has coordinate ring $\Bbb{C}[x_0,x_1,x_0^{-1}]/(x_1^n-x_0+x_0^n)$.