Morphism of varieties defined by the greatest common divisor I'm reading the paper On the Chow Bunches for Different Projective Embeddings of a Complete Variety, Hoyt (1966). In this paper the author wants prove that the Chow monoind $\mathcal{C}_p(X)=\bigsqcup_{d\geq 0}\mathcal{C}_{p,d}(X)$ given by the disjoint union of Chow varieties $\mathcal{C}_{p,d}(X)$ does not depend from the choice of the embedding of the projective variety $X\subseteq\mathbb{P}^n_\mathbb{C}$. In order to do this he needs to prove the following lemma.


I don't understand how the condition (ii) defines a closed set and, similarly, the condition (i) defines an open set. Moreover  the author says that $h$ is a morphism, but it's unclear to me its "proof" of this fact.
Any help to understand these arguments is very well appreciated.
 A: First, note that the text and the question at the time of writing disagree. I'll be explaining the text and ignoring the perceived error in the question.
We make one obvious reduction: we may ignore all indices $i$ such that $A_i$ are zero, and assume that $A_i\neq 0$ for all $i$.
Define $Y_{d}$ to be the projectivization of the space of polynomials in indeterminants $\mathfrak{X}$ of degree less than $d$, and define $Z_{d}$ to be the projectivization of the space of homogeneous polynomials in indeterminants $\mathfrak{X}$ of degree exactly $d$.
Consider the two maps from $L\times \prod_{i\in I} Y_{\deg A_i - d}\to \prod_{i,j\in I, i<j} Y_{\deg A_i+\deg A_j}$ given by $((A_1,\cdots,A_t),C_1,\cdots,C_t)\mapsto A_iC_j$ in the $(i,j)$ factor and $((A_1,\cdots,A_t),C_1,\cdots,C_t)\mapsto A_jC_i$ in the $(i,j)$ factor. The locus of agreement of the two morphisms is closed, as the target is separated. Thus the preimage of the complement of this locus of agreement under either morphism is open in the LHS. The image of this set in $L$ under the projection is again open. Thus "GCD of degree less than $d$" defines an open condition on $L$, and "GCD of degree exactly $d$" defines a locally closed condition on $L$, as it's the intersection of "GCD degree $<d+1$" and the complement of "GCD degree $<d$".
Now, we consider iii). $A_i=aB_iG$ is equivalent to $(A_1,\cdots,A_t)$ being in the image of $\Phi:Z_d \times \prod_{i\in I} Z_{\deg B_i} \to L$ defined by $(G,z_1,\cdots,z_t)\mapsto (Gz_1,\cdots,Gz_t)$, and since the source is irreducible, its image must be as well. Anything in the image of this map must have GCD of degree at least $d$.
Since $U$ is the intersection of $\operatorname{im} \Phi$, which is irreducible, and "GCD less than $d+1$", which is open, it is again irreducible. By the previous discussion, it is also locally closed. We've shown what we needed to show about $U$.
The construction that gives $h$ is morally the same as how one constructs a blowup, up to perhaps some cosmetic changes. The point is that once you have something satisfying an equation like $A_iB_j=A_jB_i$, there should be good tools to recover what $B_i,B_j$ should look like if you know $A_i,A_j$. Another proof that $g$ is a morphism may be given by looking at the projection to $Z_d$ of the preimage of $(A_1,\cdots,A_t)$ under $\Phi$ and verifying it consists of one point.
