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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 can now be found in the proof of Corollary 4.5 in this paper, its proof is outlined in Remark 4.6.

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.

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).

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 can now be found in the proof of Corollary 4.5 in this paper, its proof is outlined in Remark 4.6.

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.

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).