The following is from David Mumford's book Algebraic geometry I, page 29-30.
Let $X$ be a projective variety of the form $X=V[\mathscr{\beta}]$, with $\mathscr{\beta}\subset \mathbb{C}[X_{1},...,X_{n}]$. Let $f,g\in \mathbb{C}[X_{1},...,X_{n}]$ are homogeneous of the same degree where $g\not \in \mathscr{\beta}$. Let $Y_{0},Y_{1}$ be homogeneous coordinates in $\mathbb{P}_{1}$. Then consider the closed algebraic set $Z=V(Y_{1}g-Y_{0}f)\subset X\times \mathbb{P}^{1}$
Mumford claim that if we decompose $Z$ into irreducibles, then we will have $Z=Z^{*}\cup Y_{1}\cup Y_{2}...\cup Y_{k}$ where $Y_{i}\subset \text{proper subvariety of X}\times \mathbb{P}^{1}$
Further he asserts that $p_{1}:Z^{*}\rightarrow X$ is surjective by Zariski's main theorem. Why is this true? Mumford says "..Thus $Z^{*}$ is the Zariski-closure of the graph of the map $x\rightarrow \frac{f(x)}{g(x)}$ and is a rational map from $X$ to $\mathbb{P}^{1}$". Since $Z^{*}$ is not given explicitly I feel I am a bit lost why this must be true.