Timeline for Given a curve, under which condition is the set of gonal morphisms finite
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10 events
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| Aug 20, 2011 at 20:07 | comment | added | rita | Sorry for the messy notation. $C$ is a curve of genus $g$ which is a double cover of a plane quartic $X$, $D$ is the image of $C$ via the map $f\times g$, where $f\colon C\to X$ is the double cover and $g\colon C\to {\mathbb P}^1$ is the $g^1_d$. The Hodge index theorem is applied to $D$ and $\{pt\}\times X+{\mathbb P}^1\times\{pt\}$. | |
| Aug 20, 2011 at 19:37 | comment | added | Ariyan Javanpeykar | Probably another stupid question, but what is $C$ and what is $D$? How's the birational map $C\longrightarrow D$ defined? Moreover, which index theorem are you using here? Hodge index? | |
| Aug 20, 2011 at 18:29 | comment | added | rita | In your example the gonality is precisely 6 for $g$ large enough. Assume the curve has a $g^1_d$ with $d<6$. Then you get a birational map $C\to D\subset {\mathbb P}^1\times X=:S$. By construction $DK_S=8−2d\le 8$ and by the index theorem $2D^2\le(2+d)^2\le 49$. So $p_a(D)\ge g$ is bounded. | |
| Aug 20, 2011 at 17:08 | vote | accept | Ariyan Javanpeykar | ||
| Aug 20, 2011 at 14:03 | comment | added | Felipe Voloch | There are infinitely many double covers of any curve. I am not fixing the ramification. | |
| Aug 20, 2011 at 13:46 | comment | added | Ariyan Javanpeykar | Ow yes of course. I get it now. Concerning the first paragraph of your answer: there are only finitely many double covers of a given genus 3 curve, right? So to get a negative answer to my questions, one has to consider all genus 3 curves, right? So just to be sure, for each such curve $X_3$ we take a double cover $Y_3$ of $X_3$ of genus $g_{Y}$, where $g_Y$ is a function on $\mathcal{M}_3$ going to infinity. (hmmm...I think I'm writing down things a bit too complicated here.) Anyway, the point is that the genus grows but the gonality is bounded, right? | |
| Aug 20, 2011 at 13:38 | comment | added | Felipe Voloch | @Ariyan: The image of $X$ in $\mathbb{P}^1\times\mathbb{P}^1$, if the map is not injective, which will happen if the genus of $X$ is large enough. If you want to consider all maps of degree $d$ then you need to prove that there is a $Y$ that works simultaneously for all of them. | |
| Aug 20, 2011 at 13:16 | comment | added | Ariyan Javanpeykar | What is your curve $Y$ in the second paragraph? | |
| Aug 20, 2011 at 12:49 | comment | added | naf | A $g^1_d$ is not exactly a map to $\mathbb{P}^1$ of degree $d$ (since it can have basepoints)... But nice answer! | |
| Aug 20, 2011 at 12:39 | history | answered | Felipe Voloch | CC BY-SA 3.0 |