Lemma 54.9.5. In Situation 54.9.1. Let $X$ be the blowup of $\mathop{\mathrm{Spec}}(A)$ in $\mathfrak m$. Let $E \subset X$ be the exceptional divisor. With $\mathcal{O}_ X(1) = \mathcal{I}$ as usual and $\mathcal{O}_ E(1) = \mathcal{O}_ X(1)|_ E$ we have

$E$ is a proper Cohen-Macaulay curve over $\kappa $.

$\mathcal{O}_ E(1)$ is very ample

$\deg (\mathcal{O}_ E(1)) \geq 1$ and equality holds only if $A$ is a regular local ring,

$H^1(E, \mathcal{O}_ E(n)) = 0$ for $n \geq 0$, and

$H^0(E, \mathcal{O}_ E(n)) = \mathfrak m^ n/\mathfrak m^{n + 1}$ for $n \geq 0$.

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