Timeline for Existence of orientation preserving, finite order self homeomorphism on a genus 2 surface without fixed point
Current License: CC BY-SA 3.0
14 events
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| Sep 9, 2014 at 18:19 | comment | added | Jim Bryan | @BoyuZhang: everything is orientation preserving (my automorphism is even algebraic and hence preserves the complex structure and orientation). You have to interpret the description the automorphism of the graph correctly: the swapping of the two vertices is done by rotation, not reflection. | |
| Sep 8, 2014 at 19:15 | comment | added | Boyu Zhang | I have just read Daniele Zuddas's answer. It seems that his construction is very similar to yours but he added another reflection to make it orientation-preserving. Thank you so much for your help! I have accepted his answer, but I will vote yours up as well. | |
| Sep 8, 2014 at 19:11 | comment | added | Boyu Zhang | Thank you so much for your help. But it seems that your example is not orientation-preserving. If we look at the third model for the example, when we swamp the two vertices, the orientation seems to be reversed. Am I correct? | |
| Sep 4, 2014 at 16:37 | comment | added | Daniele Zuddas | I see! actually, the construction of my answer is the topological counterpart of this. | |
| Sep 4, 2014 at 15:22 | history | edited | Jim Bryan | CC BY-SA 3.0 |
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| Sep 4, 2014 at 15:20 | comment | added | Jim Bryan | @abx yes you are correct about needing a third root of unity in this example. I'm going to edit the answer thanks. | |
| Sep 4, 2014 at 15:19 | comment | added | Jim Bryan | @DanieleZuddas: the projectivization is singular so one has to normalize to get the "smooth projective model". | |
| Sep 4, 2014 at 9:26 | comment | added | Francesco Polizzi | In fact, the answer to the question is yes. I give another construction in my answer below. If I read correctly Broughton's table, $n=6$ is actually the only possibility, so basically there is only one example. | |
| Sep 4, 2014 at 8:57 | comment | added | Francesco Polizzi | @abx: ok, you are right. | |
| Sep 4, 2014 at 8:54 | comment | added | abx | @Francesco Polizzi: the Euler number excludes a free action of a finite group, but your automorphism may have no fixed points while some power has fixed points. | |
| Sep 4, 2014 at 8:52 | comment | added | abx | The curve is smooth but, as Francesco observes, the automorphism has 2 fixed points at infinity. One should take e.g. $(x,y)\mapsto (\omega x,-y)$ for the same curve, with $\omega $ a nontrivial third root of unity. | |
| Sep 4, 2014 at 8:38 | comment | added | Francesco Polizzi | Are you sure of this example? The Euler number of a genus $2$ surface is $-2$, it seems to me that this forbids free cyclic actions on it. Am I missing something? | |
| Sep 4, 2014 at 4:57 | comment | added | Daniele Zuddas | The projectivization of this curve seems to be singular at $[0,1, 0]$. | |
| Sep 4, 2014 at 4:15 | history | answered | Jim Bryan | CC BY-SA 3.0 |