/sci/ - Science & Math

Those following my funeral-posting series will know that recently, for the second time in my life, I've been having a spate of people close to me dying. Sadly this run shows little sign of letting up with today's death of my good friend Stephen Hawking. I'll never forget the joy of teaching him to walk as a small child, or giving him patient hints as to the physical mechanism of Hawking radiation until finally I saw it click in his little machine mind. Sadly Stephen suffered an advanced form of memory loss later in life, first forgetting how to walk, and then even how to talk, as he regressed further and further to the condition of an infant. Today he completed his cycle, being welcomed back into the womb of the cosmological sky-mother to rest and gestate, before he begins anew in some fresh body one day in the future. I for one will miss him, and am saddened at his departing when there was still so much for me to teach him, but gladdened by the thought that I was able to brighten his short life with my weekly rankings of physicists. Rest well, sweet Stephen.

Let [math] \pi: \mathbf{C}[x,y] \to \mathbf{C}[x, y]/(x^2 + y^2 + 1) [/math], and for [math] z = (z_1, z_2) \in \mathbf{C}^2 [/math], define [math] \phi^z: \mathbf{C}[t] \to \mathbf{C}[x, y] [/math] by [math] t \mapsto z_1x + z_2y [/math]. Let [math] \pi_\star [/math] and [math] \phi_\star [/math] denote the induced spectral maps [math] \pi^z_\star: \mathbf{A}^2_{\mathbf{C}} \to \mathbf{A}^1_{\mathbf{C}} [/math], and [math] \phi_\star: V(x^2 + y^2 + 1) \to \mathbf{A}^2_{\mathbf{C}} [/math] . For what values of [math] z [/math] is [math] \psi^z = \pi^z_\star \circ \phi_\star [/math] a finite and surjective morphism?

Pls help

Let π:C[x,y]→C[x,y]/(x2+y2+1), and for z=(z1,z2)∈C2, define ϕz:C[t]→C[x,y] by t↦z1x+z2y. Let π⋆ and ϕ⋆ denote the induced spectral maps πz⋆:A2C→A1C, and ϕ⋆:V(x2+y2+1)→A2C. For what values of z is ψz=πz⋆∘ϕ⋆ a finite and surjective morphism?

Pls help ((