# Intersection of projective varieties

Is there any way to study varieties of the form below

$$X=V\left(\{ f_i,g_j|i\in I,j \in J \right\}) \subset \mathbb{P}^{m+n+1},$$

where $$f_i \in k\left[ x_0, \cdots , x_m \right],\ g_j\in k\left[ y_0, \cdots , y_n \right].$$

I want to know the relationship between $$X$$ and $$X_1= V\left(\{ f_i|i\in I \}\right) \subset \mathbb{P}^{m}$$ , $$X_2=V\left(\{ g_j|j\in J \}\right) \subset \mathbb{P}^{n}$$ .

For example, if $$X_1$$ and $$X_2$$ are irreducible, then so is $$X$$ or not.(eg. $$V\left( x_0x_3-x_1x_2,x_4x_7-x_5x_6 \right)$$ corresponds to $$\mathbb{P}^1 \times \mathbb{P}^1$$ )

Another example I care about is, if we know the Hilbert polynomials of $$X_1$$ and $$X_2$$, can we get the Hilbert polynomial of $$X$$

I know similar question is "product" in case of affine, but I'm confused about projective case.

I'm guessing it might be a relationship I haven't learned yet, but I haven't looked it up.I'm sorry if this question seems stupid.

Given two non-intersecting projective varieties $$X,Y\subset \mathbb{P}^d$$, their join (or more accurately, its set of closed points) is the union of all lines meeting both $$X$$ and $$Y$$. Once you have chosen the coordinates the way you did, with $$\mathbb{P}^{n+m-1}$$ having the coordinates $$(x_0:\ldots:x_n:y_0:\ldots:y_m),$$ you can check that what you describe is exactly the join of two subvarieties $$X_1, X_2$$, with $$X_1$$ embedded into $$\mathbb{P}^n\subset\mathbb{P}^{n+m-1}$$ cut out by $$y_i=0$$ and $$X_2$$ embedded into $$\mathbb{P}^m\subset\mathbb{P}^{n+m-1}$$ cut out by $$x_j=0$$.
The graded ring of the join is the tensor product of the graded rings of $$X_1,X_2$$ and the Hilbert series of the join is just the product of Hilbert series of the two varieties. But from a glance, I don't think there is a simple description of the Hilbert polynomial of the join in terms of Hilbert polynomials of $$X_1,X_2$$.