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Fix an algebraically closed field of characteristic $0$ and let $X$ be the variety corresponding to the ideal $I_X$ of $k[x_1,\ldots, x_n]$ generated by $$A_X=\{f_i(x_1,\ldots,x_n)\vert i=1,\ldots,k\}.$$ $k[x_1,\ldots,x_n]\leq k[x_1,\ldots, x_n,y_1,\ldots,y_m]$, $A_X$ generates an ideal $I_{X'}$ in $k[x_1,\ldots,x_n,y_1,\ldots,y_n]$ and denote the corresponding variety by $X'$. Now define the variety $Y$ corresponding to the ideal $I_Y$ generated by $$A_Y=A_X\cup\{g_j(x_1,\ldots,x_n,y_1,\ldots,y_m)\vert j=1,\ldots,l \text{ and } g_j\not\in k[x_1,\ldots,x_n]\}.$$ $I_{X'}\leq I_Y$ and $Y\subset X'$.

Question: How should I refer to the relationship between $Y$ and $X$?

In some sense, it seems reasonable to feel that I have constructed $Y$ as an extension of $X$. If I am not mistaken there should exist an infinite family of subvarieties $X^*\subset X'$ such that $X \cong X^*$ and that if $Y$ is non-empty there exists a subfamily $X^{**}$ such that $X^{**}\cap Y$ is non-empty and $(\bigcup X^{**})\cap Y=Y$. Additionally, the coordinate ring $A(X)$ should be a subring of $A(Y)$ and so $A(Y)$ can be viewed as an extension of $A(X)$.

I would imagine that someone has studied this sort of thing before, but in my ignorance I do not know where to look or what to search for. I have been trying to consult Hartshorne to get a handle on this, and I very much like it, but the book seems rather mum on the subject.

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  • $\begingroup$ As long as we're being careful: I would not say that $I_X$ is an ideal in $k[x,y]$ $\endgroup$
    – Hoot
    May 29, 2015 at 22:27
  • $\begingroup$ To take a step back: geometrically, what are you trying to accomplish with this construction? $\endgroup$
    – Hoot
    May 29, 2015 at 22:30
  • $\begingroup$ What I'm interested in studying are fusion categories and structures (braided, pivotal) which may be added to them. Given some input data I can obtain a fusion category by solving a system of polynomial equations known as pentagon equations. Given a fusion category and solution to the pentagons, pivotal/braided structures correspond to solutions of additional polynomial equations involving variables from the pentagons and new ones. Details can be found in 1305.2229 but the construction is generalized by the procedure described in the question. $\endgroup$ May 30, 2015 at 6:37
  • $\begingroup$ Oh, gosh, I was asking on a simpler level -- just in terms of classical algebraic geometry. Oddly enough I know Orit... $\endgroup$
    – Hoot
    May 30, 2015 at 6:47

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