Let $C$ be any nonsingular curve in $A^3_{\mathbb C}$. Can a point be an irreducible component of $C$? I am not able to find an example of such $C$.
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What is your definition of curve? I would say that a subvariety that has a point as an irreducible component is not a curve, simply because a point is 0-dimensional. – Zhen Lin May 26 '13 at 07:38
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A curve is an affine scheme of finite type over a field $k$ of dimension one. Dimension of a scheme is the maximum of the dimensions of its finitely many irreducible components. – A.G May 26 '13 at 07:55
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First, note that a point is an irreducible component of a variety if and only if its complement is closed, i.e. if and only if the point is isolated. So any example would be a disjoint union of another non-singular curve and a point.
The following equations cut out the disjoint union of a line and a point in the affine plane: \begin{align} x y & = 0 \\ x^2 - x & = 0 \end{align} Thus it is seen that the coordinate ring of this affine variety is isomorphic to $\mathbb{C}[y] \times \mathbb{C}$, which is certainly of dimension 1. It is a regular variety in the sense that every local ring is regular, but perhaps you would not count it as non-singular, because the tangent space at the isolated point is 0-dimensional instead of 1-dimensional.
Zhen Lin
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