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I'd like to solve the following problem: >Let $\pi: \mathbf{C}[x,y] \to \mathbf{C}[x, y]/(x^2 + y^2 + 1)$, and for $z = (z_1, z_2) \in \mathbf{C}^2$, define $\phi^z: \mathbf{C}[t] \to \mathbf{C}[x, y]$ by $t \mapsto z_1x + z_2y$. Let $\pi_\star$ and $\phi_\star$ denote the induced spectral maps $\pi^z_\star: \mathbf{A}^2_{\mathbf{C}} \to \mathbf{A}^1_{\mathbf{C}}$, and $\phi_\star: V(x^2 + y^2 + 1) \to \mathbf{A}^2_{\mathbf{C}}$. For what values of $z$ is $\psi^z = \pi^z_\star \circ \phi_\star$ a finite and surjective morphism? Since everything here is affine and $(x^2 + y^2 + 1)$ is a prime ideal (since it is generated by an irreducible poly), I know that for the finiteness part, it suffices to find all $z$ such that $\pi \circ \phi^z$ is an integral morphism. I was really struggling to make progress here and would love a few hints!
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Fri, 16 Feb 2018 03:47 GMT