Lemma 33.40.6. Let $(A, \mathfrak m)$ be a $1$-dimensional reduced Nagata local ring. Then

$\delta \text{-invariant of }A \geq \text{number of geometric branches of }A - 1$

Proof. We may replace $A$ by the strict henselization of $A$ without changing the $\delta$-invariant (Lemma 33.39.6) and without changing the number of geometric branches of $A$ (this is immediate from the definition, see More on Algebra, Definition 15.106.6). Thus we may assume $A$ is strictly henselian and we may apply Lemma 33.40.5. $\square$

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