Timeline for Non-dominant polynomial maps in the plane
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 27, 2010 at 17:46 | comment | added | Terry Tao | Ah, normalisation was the key ingredient I was missing. Thanks! | |
| Oct 27, 2010 at 17:46 | vote | accept | Terry Tao | ||
| Oct 27, 2010 at 16:37 | comment | added | Qing Liu | Just to add some details on the genus of C (image of $f: k^2\to k^2$): it is unirational (i.e. dominated by a rational variety), and it is known that unirational curves are rational. The proof is simple: if for all $a\in k$, $x\to f(x, a)$ is constant, then the map $a \to f(0,a)$ is not constant (otherwise $f$ would be constant) and we get a non-constant map from $k$ to $C$. By Luroth, $C$ is rational. If for some $a\in k$, $x\to f(x,a)$ is not constant, then we again have a non-constant map from $k$ to $C$. | |
| Oct 27, 2010 at 8:54 | history | answered | Angelo | CC BY-SA 2.5 |