Harris' AG ex 2.24: projective variety under regular map. I am trying to show that if $X$ is an irreducible projective variety, its graph under a regular map into projective space is an irreducible subvariety [Harris Ex 2.24]. I tried to modify a proof I gave before for the affine case.
If $X\subset P^n$ is a projective variety, it is the vanishing set of some homogenous polynomials $G_j$ in $n+1$ variables. The regular map $\phi: X \rightarrow P^m$ is of the form $[\phi_0([X_0:\ldots:X_n]):\ldots: \phi_m([X_0:\ldots:X_n])]$, where each $\phi_i$ is regular on $X$. Call the graph of regular map $\phi$,  $\Gamma= \{[x:\phi(x)]:x\in X\} \subset P^n\times P^m$. To show that this graph is a variety, we can exhibit a set of polynomials who zero-locus is $\Gamma$. Consider the set of polynomials $S = \{G_j\} \cup \{ x_{n+i}-\phi_i\}$ where $i=0,\ldots, m$ and its vanishing set $V(S)$. First we show $\Gamma \subset V(S)$. Take some element $[x:\phi(x)]$ in $\Gamma$, by definition $x\in X$ so we know first $n+1$ coordinates of $x$ satisfy all the $G_j$ and the last $m+1$ coordinates satisfy them trivially as they do not appear. We need to see that $x$ satisfy the other $m$ polynomial. The polynomials $x_{n+i}-\phi_i$ are trivially satisfied by the definition of the graph thus $\Gamma \subset V(S)$. So we check $\Gamma \supset V(S)$. Satisfying the $x_{n+i}-\phi_i$ polynomials guarantees we are on the bigger graph $\Gamma' = \{(x,f(x)):x\in P^n\}$ and the other polynomials $G_i$ intersect the first $n$ coordinates with $X$ as a subspace, thus giving $\Gamma$. To see that $\Gamma$ is irreducible if $X$ is, we can look at the ideals of the polynomials. We know the ideal generated by $\{G_j\}$ is prime, and  the ideal generated by $\{-\phi_i+x_{n+i}\}$ are all prime because of the $x_{n+i}$ term being irreducible and degree 1. Since the ideal generators do not intersect, the ideal generated by their union is still prime and therefore the graph is irreducible.  
Two issues I have: the set $S$ needs to be all homogenous polynomials, and the $-\phi_i+x_{n+i}$ are clearly not since $x_{n+i}$ is degree 1 and $\phi_i$ can be any degree. I was thinking of raising $x_{n+1}$ to the degree of $\phi_i$ but I am not sure that would fix anything. Also, then my argument about irreducibility would not work. To be frank, I am not full convinced it works, but I tried several examples and it seems to be the case that if you have generates $S_i$ for a prime ideal and generators $T_i$ for a prime ideal and if $\{S_i\}\cap \{T_i\}=\emptyset$ then $\{S_i\}\cup \{T_i\}$ generates a prime ideal. 
 A: 1) The key to your problem is to remember that a subset  $T\subset \mathbb P^n\times \mathbb P^m$ is closed (=is a subvariety) if it is the zero locus $F_i(z,w)=0 \; (i\in I)$  of a family $F_i(Z,W)$ of bihomogeneous polynomials in the variables $Z=(Z_0,\ldots,Z_n), W=(W_0,\ldots,W_m)$ where $Z$ and $W$ are the homogeneous coordinates of $\mathbb P^n$ and $ \mathbb P^m$.  
2) Suppose  now that $X\subset \mathbb P^n$ is given by the vanishing of the polynomials $G_j(Z)$.
There is a covering of $\mathbb P^n$ by open subsets $U\subset \mathbb P^n$ such that on $U\cap X$ the morphism $\phi$ is given by $\phi(x)=[\phi_0(x):\ldots:\phi_m(x))]$ where the $\phi_k(Z)$ are homogeneous polynomials of the same degree $r$.
Then the graph  graph $\Gamma$ is given in  $U\times \mathbb P^m$ by the vanishing of the family consisting of the $G_j(Z)$ and of the $W_i\phi_k(Z)-W_k\phi_i(Z)$, these last polynomials being of bidegree $(r,1)$ in the variables$(Z,W)$.    
3) So $\Gamma\cap (U \times \mathbb P^m)\subset U \times \mathbb P^m$ is closed and since these $U \times \mathbb P^m$ form an open covering of $\mathbb P^n\times \mathbb P^m,$ conclude that $\Gamma$ is closed in  $\mathbb P^n\times \mathbb P^m$.
4) As usual in Harris's book the exercise is  undoable by a beginner in algebraic geometry, due to the rarity or absence of preliminary explicit calculations .
 Beginners  can by all means consult Harris, a treasure trove of classical algebraic geometry, but should use it in conjunction with another book to learn the basics and find reasonable, encouraging exercises.
