A term for a 'product' of two varieties Let $X\subset \mathbb{P}^m$ and $Y\subset \mathbb{P}^n$ be two projective varieties defined by equations $f_1=\ldots=f_s=0$ and $g_1=\ldots=g_t=0$ respectively.
From this we can define an algebraic subset $Z$ of $\mathbb{P}^{n+m+1}$ defined by $f_1=\ldots=f_s=g_1=\ldots=g_t=0$.
It seems to me that $Z$ should be something like the projective closure of the cartesian product of the affine cones over $X$ and $Y$, but I would like to know if there is anything else one can say about it:


*

*Is there a name for this sort of operation? 

*How is $Z$ related to $X$ and $Y$?
 A: Your $Z$ is the join $Z=X\star Y$ of $X$ and $Y$.  Here are a description and some computations.  
Let $N=m+n+1$. Consider in $ \mathbb{P}^N$  with homogeneous coordinates $x_0,\ldots,x_N$ the disjoint linear subspaces $A=V(x_{m+1},\ldots,x_N)\simeq\mathbb P^m$ and $B=V(x_{0},\ldots,x_m)\simeq\mathbb P^n$.
Given $a=[a_0:\ldots:a_m:0:\dots:0]\in A$   and $b=[0:\ldots:0:b_{m+1}:\ldots:b_N]\in B$, the line $<a,b>$ joining them is given parametrically by $[sa_0:\ldots:sa_m:tb_{m+1}:\ldots:tb_N]$ where $[s:t]\in \mathbb P^1$.
The union of these lines is $\mathbb{P}^N$:    
More precisely,   given a point $c=[c_0:\ldots:c_N] \in \mathbb P^N\setminus(A\cup B)$, there is a unique line through $c$ hitting both $A$ and $B$: it is the line $<a,b>$ with $a=[c_0:\ldots:c_m:0:\ldots:0]\in A$ and $b=[0:\ldots:0:c_{m+1}:\cdots:c_N]\in B$.
(This line  $<a,b>$ is given parametrically by
$[sc_0:\ldots:sc_m:tc_{m+1}:\ldots:tc_N]$ and $c$ corresponds to $s=t=1$)  
Finally suppose you  take only  the lines $<x,y>$ with $x\in X$ and $y\in Y$:  their union is your variety $Z=X\star Y\subset \mathbb P^N$ whose equations are, just as you wanted, $$f_1(x_{0},\ldots,x_m)=\cdots=f_s(x_{0},\ldots,x_m)=g_1(x_{m+1},\ldots,x_N)=\ldots= g_t(x_{m+1},\ldots,x_N)=0$$
Edit
I forgot to mention  that $\text {dim}_: (X\star Y)=\text {dim}_: (X)+\text {dim}_: ( Y)+1$     
Second Edit
Here is the affine version of the above, inspired by a conversation with my friend André Hirschowitz.
Consider in the vector spaces $E,F$  two affine algebraic cones  $C\subset E, D\subset F$ defined respectively by the equations $f_i=0$ and $g_j=0$ obtained from two bunches of homogeneous polynomials $(f_i)$ and $(g_j)$.
These cones  have an affine join $C\star D\subset E\oplus F$ simply defined as their vector sum $C\star D=C+D=\{d+e|d\in D, e\in E\}$ .
In order to derive  equations for that join, note that our algebraic cones $D,E$ give rise to algebraic cylinders $C\oplus F, E\oplus D\subset E\oplus F$ which have the exact same equations $f_i=0$ and $g_j=0$ as $D$ and $E$.
The key observation is then that $$C\star D=C+D=(C+F)\cap(E+D)$$ so that, like all intersections, the join $C\star D$ is described by taking together the equations $f_i=0$ and $g_j=0$ of the intersectands $(C+F), (E+D)$, namely $$C\star D=V(\ldots,f_i,\ldots; \ldots,g_j,\ldots)$$.
 Needless to say the projective case I described before follows by putting $$\mathbb P^N=\mathbb P(E\oplus F), \: A=\mathbb  P(E),\:B=\mathbb  P(F),\: X=\mathbb  P(C), \:Y=\mathbb  P(D)$$ and thus $$Z=X\star Y=\mathbb P(C)\star \mathbb P(D)=\mathbb P(C\star D)$$
A: Your system of equations $f_1=\ldots=f_s=g_1=\ldots=g_t=0$ doesn't make sense, so that $Z$ is not defined,  because the $f_i$'s depend on $m+1$ variables, the $g_j$'s on $n+1$ variables and actually subvarieties of $\mathbb{P}^{n+m+1}$ should be defined by polynomials in $n+m+2$ variables! 
One can however define the product $X\times Y$ and it will be a projective variety, but that variety is embedded in a HUGE projective space: $X\times Y\subset \mathbb P^{mn+m+n}$.
 This procedure is due to Segre and you can read about it here.  
Edit
Actually the question makes excellent sense by interpreting polynomials in the $m+1$ variables   $x_0...,x_m$ as polynomials in the variables $x_0,...,x_{m+n+2}$ not depending on $x_{m+1},...,x_{m+n+2}$ !
I'm sorry for this misunderstanding and I'll answer the actual question in another post, since the new answer  is going to be a little technical.
