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Suppose $\mathbb{A}^1$ and $\mathbb{P}^1$ are affine space and projective space respectively. I'm not sure if it matters, but I don't mind if we assume that we're working over algebraically closed fields.

I'm curious, is it possible to find a surjective, regular mapping $\mathbb{A}^1\to\mathbb{P}^1$?

ankit
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Clara
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1 Answers1

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Yes it is possible to find a surjective, regular mapping $\mathbb{A}^1\to\mathbb{P}^1$!

By algebra
$\mathbb A^1\to \mathbb P^1:z\mapsto[z:z^2+1] $
By geometry (better !)
Take a ramified $2$-covering $f:\mathbb P^1\to \mathbb P^1$, consider a non-critical point $a\in \mathbb P^1$ (easy: there are only two critical points for $f$ !) and the required surjective morphism is the restriction $$\text{res}(f):\mathbb P^1\setminus \{a\}=\mathbb A^1\to \mathbb P^1$$ By concrete geometry (best !)
Consider the projection $p$ of a smooth conic $C\subset \mathbb P^2$ from a point $Q$ outside $C$ onto a line $L\subset \mathbb P^2$, take a point $a\in C$ such that the tangent line $\Theta_aC\subset \mathbb P^2$ at $a$ does not pass through $Q$ (easy: there are only two such undesirable points!) and restrict the projection $p$ to the complement of $a$ to obtain the required surjective morphism $$\text{res}(p):C\setminus \{a\}=\mathbb A^1 \to L=\mathbb P^1$$Confused? Make a drawing and just LOOK!

Georges Elencwajg
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    I had seen this answer before, and the last sentence makes me happy every time I see it. – pjs36 Jun 10 '15 at 20:33
  • Dear @pjs36, I'm happy that you too have a visual relation to algebraic geometry. May I ask where you saw this answer ? I don't know any reference mentioning it. – Georges Elencwajg Jun 10 '15 at 20:37
  • I most likely found it following this answer. But since many of your answers are enlightening even when I do not know the subject (as is the case with algebraic geometry, although I intend to rectify that!), I browse through them occasionally. – pjs36 Jun 10 '15 at 20:55
  • Ah, but that was an answer by me referring to the above. I thought you had seen a reference by somebody else. Anyway, thanks for the kind words, @pjs36. – Georges Elencwajg Jun 11 '15 at 07:35