Could anyone verify whether the following is a typo? I'm studying William Fulton's "Algebraic Curves" and he's in the process of studying the relationship between affine space and projective space. With $K$ as an algebraically closed field, he defines the following:
"Let $V$ be an algebraic set in $\mathbb{A}^n$, $I=I(V) \subset K[x_1,\cdots,x_n]$. Let $I^*$ be the ideal in $K[x_1,\cdots,x_{n+1}]$ generated by $\{F^*:F \in I\}$. This $I^*$ is a homogenous ideal; we define $V^*$ to be $V(I^*) \subset \mathbb{P}^n$.
Conversely, let $V$ be an algebraic set in $\mathbb{P}^n$, $I=I(V) \subset K[x_1,\cdots,x_n]$..."
That last part, on the last sentence, is a typo, correct? It should read $I(V) \subset K[x_1,\cdots, x_{n+1}]$, right?
It continues: "Let $I_*$ be the ideal in $K[x_1,\cdots,x_n]$ generated by $\{F_*:F \in I\}$." What does this last sentence mean? Does it imply we plug in an arbitrary non-zero constant into the $x_{n+1}$ and consider the resultant polynomial?
$\textbf{Edit:}$ Fulton defines the notation $F_*$ in a previous section. It'd just been so long since he defined it (and then never used it), that I forgot about it. Indeed, if $F \in K[x_1,\cdots,x_{n+1}]$, then $F_*=F(x_1,\cdots,x_n,1)$.