Exercise 111.61.5. Let $k$ be a field. Let $X = \mathop{\mathrm{Spec}}(k[x, y])$ be affine $2$ space. Let

Let $Y \subset X$ be the closed subscheme corresponding to $I$. Let $b : X' \to X$ be the blowing up of the ideal $(x, y)$, i.e., the blow up of affine space at the origin.

Show that the scheme theoretic inverse image $b^{-1}Y \subset X'$ is an effective Cartier divisor.

Given an example of an ideal $J \subset k[x, y]$ with $I \subset J \subset (x, y)$ such that if $Z \subset X$ is the closed subscheme corresponding to $J$, then the scheme theoretic inverse image $b^{-1}Z$ is not an effective Cartier divisor.

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