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Let $R = \mathbf{C}[x,y,u,v]$ be the coordinate ring of $\mathbf{C}^4$. Let $I$ be the ideal generated by $u$ and $v$, let $J$ be the ideal generated by $u$ and $y$. What are the flat deformations of the ideal $I^pJ^q$?

This is the ideal of the union of two non-transversely intersecting $2$-planes, with some nilpotent thickness on each plane. I would like to learn something about its smooth deformations as a subvariety of $\mathbf{C}^4$.

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  • $\begingroup$ One might try to embed the ideal in the simplest way in $\mathbb P^4$, compute its Hilbert polynomial, and try to understand the appropriate Hilbert scheme. $\endgroup$
    – Will Sawin
    Commented Jun 28, 2015 at 3:22
  • $\begingroup$ That ideal is, unfortunately, not Cohen-Macaulay (at least if I did my computation correctly). It has an embedded prime $\langle y,u,v\rangle$. That is unfortunate: if it were Cohen-Macaulay, then the Hilbert-Burch-(Schaps) Theorem would describe all flat deformations via variation of a matrix of polynomials (in particular, infinitesimal deformations would always be unobstructed). $\endgroup$
    – Jason Starr
    Commented Jun 28, 2015 at 18:18
  • $\begingroup$ Thanks Will and Jason. Jason, can a non-CM ideal be flatly deformed to a CM ideal? Or are you telling me that this variety has no smooth deformations at all! $\endgroup$
    – David Treumann
    Commented Jun 28, 2015 at 20:59
  • $\begingroup$ @DavidTreumann Some non-Cohen-Macaulay ideals deform to Cohen-Macaulay ideals. I am not saying anything about the ideals that can or cannot be obtained by deforming your ideal. I am simply mentioning that your ideal is not Cohen-Macaulay, which is too bad, because otherwise it would be much easier to study your problem. $\endgroup$
    – Jason Starr
    Commented Jun 28, 2015 at 22:34

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