Exercise 109.57.4. Let $k$ be a field. Let $b : X \to \mathbf{A}^2_ k$ be the blow up of the affine plane in the origin. In other words, if $\mathbf{A}^2_ k = \mathop{\mathrm{Spec}}(k[x, y])$, then $X = \text{Proj}(\bigoplus _{n \geq 0} \mathfrak m^ n)$ where $\mathfrak m = (x, y) \subset k[x, y]$. Prove the following statements

the scheme theoretic fibre $E$ of $b$ over the origin is isomorphic to $\mathbf{P}^1_ k$,

$E$ is an effective Cartier divisor on $X$,

the restriction of $\mathcal{O}_ X(-E)$ to $E$ is a line bundle of degree $1$.

(Recall that $\mathcal{O}_ X(-E)$ is the ideal sheaf of $E$ in $X$.)

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