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I appreciate the comments so far and am modifying based on something closer to the problem I'm interested in. I started out with something far too general.

This is probably easy, but I have been away from math for quite some time now so what I'm asking may be ancient history by now. Anyway, I'm wondering what conditions are needed for the following to be true.

Suppose $Y = \cap H_i$ is a sub-scheme of a normal variety $U$, let $\overline{Y}$ be the closure in some $X$, and let $D = X{\setminus} U$. Then is $\overline{Y} = \cap \overline{H_i}$ if $\overline{Y} \cap D = \overline{H_i} \cap D$? If not, can we place conditions on $D$ such as integral, normal, Cartier (or maybe all three?), and/or can we place conditions on $Y$ such as reduced or integral?

For example, I was hoping the twisted cubic $C \subset X =\mathbb{P}^3$ would yield a counter-example, but if we take $U = \mathbb{A}^3$, $D$ the line at infinity and two hypersurfaces which cut out $C$ on $\mathbb{A}$ but only set-theoretically cut out $C$ globally, we'll get a point of multiplicity $3=deg(C)$ on $C\cap D$ but a point of multiplicity 6 on on the intersection of $D$ with the two hypersurfaces (the point at infinity is a double point on the intersection of the two hypersurfaces).

The reason I am interested in this is is the following. Let $I$ be an ideal in $k[M]$, $M{\cong}{\mathbb{Z}^n},$ $w{\in}{N{=}M^{*}},$ and $I_w{=}k[M_w]{\cap}I$ where $k[M_w]$ is the subring generated by monomials with positive $w$-weight. Then $I_w$ is the ideal of the closure of $V(I)$ in the toric variety $U_w = \text{Spec}(k[M_w])$. One can check that generators of the initial ideal $\text{in}{_w}I$ (here we take the terms of minimal weight) generate the ideal of both $V(I)$ and $\overline{V(I)}\cap O$ where $O$ is the closed orbit in $U_w$.

So, if my question is true, a set $\{f_1, \ldots, f_r\} \subset I$ produce generators for $\text{in}_wI$ when closures of the associated hypersurfaces cut-out the closure of $V(I)$. This would be another example of how the tropicalization reflects the asymptotic behavior of a subvariety in the torus - to find a tropical basis for $I$, one should look for hypersurfaces which cut out $Y$ not in $T$ but in some larger toric variety.

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  • $\begingroup$ What if D is reduced but Y_1 and Y_2 have different scheme structures at D- eg Y_2 is a thickening of Y_1 along D. Depending on what you mean, that might be a counter eg. $\endgroup$
    – meh
    Commented Apr 10, 2015 at 14:23
  • $\begingroup$ @aginensky - good point. I'm thinking of something like this...If $\{ H_i \}$ is a collection of hypersurfaces in $U$ with $Y = \cap H_i$ and $\overline{H_i} \cap D = \overline{Y} \cap D$, then $\cap \overline{H_i} = \overline{Y}$. $\endgroup$
    – user71216
    Commented Apr 10, 2015 at 14:44

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Take $X = \mathbb{A}^2$ with $D = \{P\} \subset X$ being a point $P$. Let $Y_1$ and $Y_2$ be Artin subschemes of length 2 supported (set-theoretically) at $P$. Then your conditions hold, while $Y_1$ may be different from $Y_2$ (the space of such $Y$ is parameterized by the projectivization of the tangent space to $X$ at $P$).

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  • $\begingroup$ Your answer is similar to the comment made by aginesky, but as you can see in my reply, I'm really looking for something more specific. I'll reformulate the question to something less general. Thanks. $\endgroup$
    – user71216
    Commented Apr 10, 2015 at 14:47
  • $\begingroup$ An Artin subscheme of length 2 is a complete intersection of two hypersurfaces (say $x = y^2 = 0$), so I guess this still gives you a counterexample. $\endgroup$
    – Sasha
    Commented Apr 10, 2015 at 15:00
  • $\begingroup$ You're right. Did I mention I've been away from math for a few years? :) I'll try and see if I can either reformulate it or if it's just completely sunk. $\endgroup$
    – user71216
    Commented Apr 10, 2015 at 15:34
  • $\begingroup$ I thought about this some more and realized the key mistake in my original formulation was to assume $Y_1$, $Y_2$ were subschemes of $X$. Starting with something over $U$ and taking the closure is very different. Thanks for helping me clarify! $\endgroup$
    – user71216
    Commented Apr 10, 2015 at 18:19
  • $\begingroup$ I've updated to better clarify the situation I had in mind. $\endgroup$
    – user71216
    Commented Apr 10, 2015 at 18:30

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