Take the tour ×  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 t