Why this $F_*=F(X,Y,1)$

Question

I'm studying Fulton's algebraic curves book.

Someone could help me to prove this phrase highlighted:

I didn't understand why the $F_*$ he defined is the same of the known $F_*=F(X,Y,1)$.

Thanks in advance

Answer 1

After the linear change of coordinates we have $F_*=F/Z^d$ and $F=F(X,Y,Z)$ is homogeneous of degree $d$. Thus $F_*=F(X/Z,Y/Z,1)$. The natural identification mentioned in the text sends $X$ to $X/Z$ and $Y$ to $Y/Z$.