birational map P1

Question

I have the following problem to prove that $V(x_0^3-x_2x_1^2)\subset\mathbb P_k^2 $, where $k$ algebraically closed field, is birational to $\mathbb P_k^1$.

I'm a beginner at this stuff, so someone tell me please if I'm following the right track. I thought to prove that the coordinate rings of $V(x_0^3-x_2)$ and $\mathbb A^1_k$ are isomorphic. Is this correct and enough?

I think you meant $\Bbb A^1_k$ – Vinicius M. Dec 18 '13 at 23:55 — Vinicius M., Dec 18 '13 at 23:55
Yes, I did indeed. Thank you, now I edited. – guest Dec 19 '13 at 00:10 — guest, Dec 19 '13 at 00:10

score 0 · Answer 1 · answered Dec 18 '13 at 23:53

0

Yes: As you suggested, $V(x_0^3-x_2) = V(x_0^3 - x_2x_1^2) \cap \{x_1\neq 0\}$ is an open set of $V(x_0^3-x_2)$. Now use that two varieties are birrational if you can find isomorphic open sets in them.

answered Dec 18 '13 at 23:53

Vinicius M.

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birational map P1

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