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I've been doing some reading on deformation theory and one way it is used is to study singularities on varieties while perturbing the varieties. I would actually like to use deformation theory to construct a flat family linking the following two subschemes of $\mathbb{A}_{\mathbb{C}}^8$.

I would like to thank David Speyer who helped me to formulate this question from my other post.

Does there exist a closed subset of $\mathbb{A}_{\mathbb{C}}^8\times \mathbb{A}_{\mathbb{C}}^1$, which is flat over $\mathbb{A}_{\mathbb{C}}^1$, whose fiber over $0$ is the set $ Z(x_1 y_1 + z_1 y_2, x_2 y_2 +z_3^2, x_3 y_3-z_3 y_2, x_1-x_2+x_3) $ and whose fiber over $1$ is the set $ Z(x_1 y_1 + z_1 y_2,x_2 y_2, x_3 y_3-z_3 y_2, x_1-x_2+x_3)? $

Note that the only difference between the two sets is that the second polynomial is a monomial.

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    Of course, and I will! =) Thank you for your time.2012-07-05

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