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Every scheme here is over complex number.

Let $X \subset (\mathbb{C}^*)^n$ be a complete intersection with $X$ defined by the ideal $I \subset \mathbb{C}[x_{1}^{\pm},\dots,x_{n}^{\pm}]$ generated by $r$ equations $f_1,\dots,f_r$. Moreover, if the rank of the matrix $(\partial f_i /\partial x_j)_{ij}$ is $r$, then is $X$ a smooth variety?

This is of course true if $I$ is radical, that is, $X$ is a variety. But I don't know any information about $I$. Actually, in my problem, I just want to know whether $I$ is reduced under the condition that the generators satisfying the Jacobian matrix smooth criterion(i.e. rank$(\partial f_i /\partial x_j)_{ij}=r$).

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  • $\begingroup$ You say "of course" it is true when $I$ is radical, but how is the radical condition relevant in any justification? It is true without the radical property, due to the functorial criterion for smoothness -- e.g., see Proposition 7(c) in 2.2 of "Neron Models" -- and the fact that the functorial criterion implies the power series property for completed local rings at closed points when working with finite type schemes over an algebraically closed field. $\endgroup$
    – user29283
    Commented May 13, 2013 at 5:20
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    $\begingroup$ This is really differential geometry, not algebraic geometry (although it is true in the algebraic setting as well, when the proof is written appropriately). Your $r$ functions give a map $f:(\mathbb{C}^*)^n \to \mathbb{C}^r$. The variety $X$ is $f^{-1}(0)$, and the condition on the matrix says that $0$ is a regular value of $f$. So $X$ is a smooth manifold. Near any point $p\in X$, $X$ is the transverse intersection of the $r$ hypersurfaces $f_1=f_2=\cdots f_r = 0$, and these hypersurfaces are smooth at $p$. $\endgroup$
    – Jack Huizenga
    Commented May 13, 2013 at 6:47

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