Intersection of projective varieties

Question

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.

Answer 1

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 join of two irreducible varieties is irreducible. This is pretty evident from the geometric picture of lines joining two irreducible varieties, and the formal argument is in an answer to 'Is the join of two irreducible varieties is irreducible?'.

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 $X_1,X_2$.