I am working on a computation of a blowup and got stuck at a point, I hope there is somebody to help me.
Consider $V=V(y^2-x^3-x^2) \subseteq \mathbb{A}^2_k$ for some algebraic closed field $k$. I want to blow up $V$ in the origin, the corresponding ideal is $I=(x,y)$. Let $R=k[V]=k[x,y]/(y^2-x^3-x^2)$ and $X=\mathrm{Spec}R$. I got that far to say, that the blowup $\tilde{X}$ of $X$ along $I$ is given by $$\tilde{X}=\mathrm{Proj\ } k[x,y,X,Y]/(yX-xY,y^2-x^3-x^2, yY-x^2X-xX, Y^2-xX^2-X^2),$$
but now I want to show that $$\mathrm{Proj\ }k[x,y,X,Y]/(yX-xY,y^2-x^3-x^2, yY-x^2X-xX, Y^2-xX^2-X^2), \cong \mathrm{Spec} k[X]$$ (which is well-known result). But I was not able to do this. Does anybody have a hint or solution for me?
Best regards and a merry christmas to you.
by now I tried to solve it on my own, I don't know whether it's right, so I would be glad if somebody could briefly check it:
I consider the affine chart $D_+(X)$. So with notation $S_({X})$ for the degree-zero-elements of the localization of $S$ by the ideal generated by $X$ I get
\begin{alignat*}{5} D_+(X)\ &=&&\ \left( k[x,y,X,Y] / (y^2-x^3-x^2, yX-xY, yY-x^2X-xX, Y^2-xX^2-X^2) \right)_{(X)}\\ &=&& \ \mathrm{spec} \left( k[x,y,Y]/(y^2-x^3-x^2, y-xY, yY-x^2-x, Y^2-x-1) \right)\\ &=&& \ \mathrm{spec} \left(k[x,Y]/((xY)^2-x^3-x^2, (xY)Y-x^2-x, Y^2-x-1) \right)\\ &=&& \ \mathrm{spec} \left(k[x,Y]/(x^2(Y^2-x-1), x(Y^2-x-1), Y^2-x-1) \right)\\ &=&& \ \mathrm{spec} \left(k[x,Y]/(Y^2-x-1) \right)\\ &=&& \ \mathrm{spec} \ k[Y] \end{alignat*}
and analogously for the other chart. Thanks for helping me. Arthur