Exercise 111.27.8. Blowing up: Part II. Let $A = k[x, y]$ where $k$ is a field, and let $I = (x, y)$. Let $R$ be the blowup algebra for $A$ and $I$.

Show that the strict transforms of $Z_1 = V(\{ x\} )$ and $Z_2 = V(\{ y\} )$ are disjoint.

Show that the strict transforms of $Z_1 = V(\{ x\} )$ and $Z_2 = V(\{ x-y^2\} )$ are not disjoint.

Find an ideal $J \subset A$ such that $V(J) = V(I)$ and such that the strict transforms of $Z_1 = V(\{ x\} )$ and $Z_2 = V(\{ x-y^2\} )$ in the blowup along $J$ are disjoint.

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