The Stacks project

Exercise 111.34.3. Let $A, B, C \in \mathbf{C}[t]$ be nonzero polynomials. Consider the morphism of schemes

Show there does exist a nonempty open $U \subset S$ and a morphism $\sigma : U \to X$ such that $\pi \circ \sigma = \text{id}_ U$. (Hint: Symbolically, write $x = X/Z$, $y = Y/Z$ for some $X, Y, Z \in \mathbf{C}[t]$ of degree $\leq d$ for some $d$, and work out the condition that this solves the equation. Then show, using dimension theory, that if $d >> 0$ you can find nonzero $X, Y, Z$ solving the equation.)