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Let $X \subset \mathbb{A}^n$ be an affine variety that is smooth in $a$ of dimension n-k and I its ideal in $K[x_1,...,x_n]$. Is it true that the localized ideal $I_{m_a} \subset \mathcal{O}_{X,a}$ is generated by k elements?

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Yes, smooth at a point implies it is a local complete intersection at that point, i.e. the defining ideal is generated by the minimum number of elements. See theorem 8.17 on page 178 of Hartshorne.

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