If two varieties are isomorphic so is their blow-up. I'm following Gathmann notes on Algebraic Geometry. In Problem 10.17, we are to prove the affine curves $X_k = V(x_2^2 - x_1^{2k+1})\subset \mathbb{A}^2_{\mathbb{C}}$ are not isomorphic for different $k$.
The last step in the proof I'm struggling with is showing that if $X_k$ is isomorphic to $X_l$, then $\widetilde{X_k}$ is isomorphic to $\widetilde{X_l}$. These are their strict transforms in the blow-up of $\mathbb{A}^2$ at the origin (this part is necessary to find a contradiction, after blowing these up $l$ times).
I use the affine Jacobi criterion to find that the only singular point in both is the origin. This shows a supposed isomorphism $f: X_k\to X_l$ maps origin to origin. This already shows that there's a natural mapping $\widetilde{X_k}\setminus \pi^{-1}(0)\to \widetilde{X_l}\setminus \pi^{-1}(0)$ where $\pi:\widetilde{\mathbb{A}^2}\to \mathbb{A}^2$ is the blow-up, since $\widetilde{X}\setminus\pi^{-1}(0)\cong X\setminus \{0\}$. I'm just not sure how we can finish this isomorphism on the exceptional set part.
 A: Lemma 9.15 of Gathmann shows that the blow-up of $X$ at some functions $f_1, ..., f_r\in A(X)$ depends only on the ideal $(f_1, ..., f_r)\unlhd A(X)$.
This shows that if we have an isomorphism $f: X\to Y$, and if the ideal $J\unlhd A(Y)$ is sent to $I\unlhd A(X)$ under the induced isomorphism $f^*: A(Y)\to A(X)$, then the blow-up of $X$ at $I$ will be isomorphic to the blow-up of $Y$ at $J$.
We can show with the Affine Jacobi Criterion (Prop. 10.11) that the only singular points are $0$ in both $X_k$ and $X_l$. In our case this is equivalent to the tangent space having dimension at least 2 (since the local dimension is always $1$ for these irreducible curves). Since the dimension of the tangent space depends solely on the stalk $\mathscr{O}_{X,a}\cong \mathscr{O}_{Y,f(a)}$, we know it must be preserved under isomorphisms. We conclude if $f:X_k\to X_l$ is an isomorphism then the origin goes to the origin.
This means $I = I(\{0\})\unlhd A(X_l)$ is sent to $f^*I = I(\{f(0)\})\unlhd A(X_k)$.
Therefore, the blow-up $\tilde{X_k}$ of $X_k$ at the origin, i.e., at $I(\{0\})$, is isomorphic to the blow-up $\tilde{X_l}$ of $X_l$ at the origin, i.e., at $I(\{f(0)\})$.
