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Let $X \subset \mathbb{A}_k^3:xy=xz=yz=0$ and $Y\subset \mathbb{A}_k^3:z=xy(x+y)=0$. My definition is that two algebraic sets are isomorphic if there exists mutually inverse polynomial maps between them. Are they isomorphic? My attempts to find polynomial maps was failed. And I saw in a book, they are not isomorphic because $X$ has dimension 3 tangent space, $Y$ has dimension 2 tangent space.

But if they are isomorphic, the dimension of tangent space must be preserved? And the singular the origin should go to the origin?

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up vote 1 down vote accepted

You are completely right. If they were isomorphic, tangent spaces should be preserved. So since the tangent spaces have different dimension, they're not isomorphic.

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