Lemma 54.9.5 (0B63)—The Stacks project

Table of contents
Part 3: Topics in Scheme Theory
Chapter 54: Resolution of Surfaces
Section 54.9: Rational singularities
Lemma 54.9.5 (cite)

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 $n \geq 0$.

Proof. Since $\mathcal{O}_ X(1)$ is very ample by construction, we see that its restriction to the special fibre $E$ is very ample as well. By Lemma 54.9.4 the scheme $X$ is normal. Then $E$ is Cohen-Macaulay by Divisors, Lemma 31.15.6. Lemma 54.9.3 applies and we obtain (4) and (5) from the exact sequences

\[ 0 \to \mathcal{I}^{n + 1} \to \mathcal{I}^ n \to i_*\mathcal{O}_ E(n) \to 0 \]

and the long exact cohomology sequence. In particular, we see that

\[ \deg (\mathcal{O}_ E(1)) = \chi (E, \mathcal{O}_ E(1)) - \chi (E, \mathcal{O}_ E) = \dim (\mathfrak m/\mathfrak m^2) - 1 \]

by Varieties, Definition 33.44.1. Thus (3) follows as well. $\square$

The Stacks project

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