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I have the following question which seems to come up every year in exam papers, but just different numbers...Im really stuck on some parts...

Define $\phi: \mathbb{A}^1_k\rightarrow \mathbb{A}^2_k$ by $\phi(t)=(t^2-1,t^3+2)$

(i) Describe the induced map $\phi$*:k[x,y] $\rightarrow$k[t] and compute $\phi$*$(x^2y-3xy)$

(ii) Describe the ideal $J_\phi$ and explain how to use $J_\phi$to compute the kernel of $\phi$*

(iii) Give the gernerating set for the kernel of $\phi$*

(iv)Is the point (2,-1) in the Zariski closure of the image $\phi$?

Working

I have computed

(i) $\phi$*$(f(x,y))=f(t^2-1,t^3+2)$

$\phi$*$(x^2y-3xy)=(t^2-1)^2(t^3+2)-3(t^2-1)(t^3+2)$

$=t^7-5t^5+2t^4+4t^3-10t^2+8$

.

(ii)

(iii)

(iv)

Please help me!!!

Mathsstudent147
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  • There are a few typos. Is the problem with $\mathbb{R}$ everywhere, or with some arbitrary field $k$? In your work for (i), I know what you mean, but it should say $\phi^*(f(x,y))=f(t^2-1, t^3+2)$ because you input a two-variable polynomial and output a one-variable poly. – Matt May 03 '13 at 23:06
  • There is no mention of $\mathbb{R}$ in the question.. it is with some arbitrary field k.... For my answer to part (ii) I have literally written out what some of my notes suggest it should be.. Although, unfortunately a lot of my lecture notes don't seem to be very inconsistent...Of course it should say that, thank you I will correct it... – Mathsstudent147 May 03 '13 at 23:09
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    Can you define $J_{\phi}$? I am assuming it is the ideal given by the polynomials in $k[x,y]$ which are identically zero on the image of $\phi$, but just to be sure... – Niccolò May 11 '13 at 01:53

1 Answers1

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Write $k[x,y]$ as $k[a,b]$ with $a = x+1$ and $b=y-2$.

In $ab$-coordinates the map $\phi$ is $(t^2, t^3)$ which implicitizes to $a^3=b^2$.

This answers some parts of the question, and for the rest the definition of $J_\phi$ is missing.

zyx
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