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Didn't get a single comment in over a day at math.SE, so maybe the question is more appropriate here.

I'm looking for a reference to a generalization of Hilbert's Nullstellensatz to the multiprojective setting. This would appear to be a rather basic fact but I'm having trouble finding a statement online or in the literature.

Consider the product $$\mathbb P=\mathbb P(\mathbb C^{n_1})\times\dots\times \mathbb P(\mathbb C^{n_k}).$$ Let $R$ be the ring of polynomials in variables $X_i^j$ with $j\in[1,k]$ and $i\in[1,n_j]$. Let $I\subset R$ be a multihomogeneous ideal, meaning that $I$ is homogeneous with respect to degree in the variables $X_1^j,\dots,X_{n_j}^j$ for every $j$. Then $I$ is seen to define a subvariety $V(I)\subset\mathbb P$ of points in which all $p\in I$ vanish. Question: where in the literature can I find a necessary and sufficient condition for $I$ to contain all polynomials vanishing on $V(I)$? (Preferably in a standard reference such as Hartshorne.)

I suspect that the condition is that $I$ is (a) multihomogeneous, (b) radical and (c) saturated with respect to the irrelevant ideals $\langle X_1^j,\dots,X_{n_j}^j\rangle$ for all $j$. I think I know how this can be proved but I'm looking for a precise reference.

Update. Here's a brief summary of what has been found so far. First of all, Balazs found this book, Theorem 2.14 there contains a condition of this sort for the case of the product $\mathbb P^1\times \mathbb P^1$. However, I'm afraid their condition is wrong, see comments. Next, I found a paper where on page 8 it says that "the assignment $V \mapsto I(V)$ is a one-to-one correspondence between..." However, I think that this claim is also wrong by the same counterexample. Finally, the latter paper cites Chapter 5 of this book which is in French but Proposition 2.17 there says that there is a correspondence between closed subschemes in $\mathbb P$ and ideals $I\subset R$ satisfying conditions (a) and (c) above. My French is nowhere near enough to tell whether a condition for subvarieties is also found in this chapter (i.e. the statement that adding radicality will ensure the subscheme being reduced).

Update 2. Shameless self-promotion time! The statement is now proved as Theorem 1.8.1 in this paper.

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    $\begingroup$ I found a reference in "Arithmetically Cohen-Macaulay Sets of Points in P^1 x P^1" by Guardo and Van Tuyl for the bigraded case, their Theorem 2.11 and thereabouts (google will show this if you search for "bigraded nullstellensatz"). But they only say "the proof is as in the graded case, so will be omitted"... $\endgroup$
    – Balazs
    Commented Aug 7, 2020 at 7:58
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    $\begingroup$ As for the case of projective space you can see $\mathbb{P}$ as the quotient of $\mathbb{A}^{n_1+\cdots+n_k}$ minus the "coordinate axes" (i.e. those subschemes obtained by replacing one $\mathbb{A}^{n_i}$ factor by 0) by an action of $\mathbb{G}_m^k$. Therefore the closed subschemes in $\mathbb{P}$ are in bijection with the closed subschemes of $\mathbb{A}^{n_1+\cdots+n_k}$ that are stable by the $\mathbb{G}_m$-action and contain the "coorodinate axes". From this the usual Nullstellensatz yields the description you're after. Probably there aren't many references because the proof is so short. $\endgroup$
    – Denis Nardin
    Commented Aug 7, 2020 at 8:06
  • $\begingroup$ @Balazs, thanks, a reputable source would be good enough for me to quote with or without a proof. But, if I'm reading this correctly, I think there is an issue here (aside from it being bigraded rather than multigraded). I'm not sure their Theorem 2.14 is correct: I don't think their notion of "projective relevance" is what is needed here. For instance, the ideal $\langle y_0(x_0-x_1),y_1(x_0-x_1)\rangle$ is projectively relevant as per Definition 2.12 but it is not the vanishing ideal of its zero set, that would be $\langle x_0-x_1\rangle$. Saturatedness is a stronger condition. $\endgroup$
    – Igor Makhlin
    Commented Aug 7, 2020 at 12:39
  • $\begingroup$ @DenisNardin, thanks, that's more or less the argument I had in mind. There is a nuance, however. You want to obtain the ideals that are the vanishing ideals of their zero sets and those are not the ideals corresponding to subvarieties in affine space containing the $k$ coordinate subspaces. Instead, you want the subvariety to contain as little from these subspaces as possible, i.e. for it to be the closure of itself minus said subspaces. This is where saturatedness comes into play. $\endgroup$
    – Igor Makhlin
    Commented Aug 7, 2020 at 13:05
  • $\begingroup$ So the argument, although not hard, is not exactly obvious. I agree, however, that it is very much the norm to sweep stuff like that under the rug and, by the looks of it, I might have to do just that. $\endgroup$
    – Igor Makhlin
    Commented Aug 7, 2020 at 13:06

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