Timeline for Fixed-point free holomorphic involutions
Current License: CC BY-SA 4.0
13 events
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| Jul 13, 2022 at 19:28 | comment | added | Francesco Polizzi | @JasonStarr: very interesting, thank you. | |
| Jul 13, 2022 at 18:24 | comment | added | Jason Starr | @FrancescoPolizzi. There do exist some such Enriques involutions on quartic hypersurfaces. But none of these Enriques involutions are induced by (regular) automorphisms of the projective space. There is a nice model of all such (complex) K3 surfaces as $(2,2,2)$ complete intersections in $\mathbb{P}^5$ that are (setwise) fixed by an involution of $\mathbb{P}^5$ with fixed locus a disjoint union of two $2$-planes (both disjoint from the K3 surface). I used this model in one article with Graber, Harris, and Mazur (and a similar model in another article I wrote about char $p$). | |
| Jul 12, 2022 at 19:17 | comment | added | Mohammad Farajzadeh-Tehrani | Your point is valid. I was not completely sure what I want. I understand the last comment; do you know of any tool that helps decide which ones do admit such an involution. The answer to your question is positive. I think Fermat quartic admits such an involution (I asked this a couple of years ago on MathoverFlow). | |
| Jul 12, 2022 at 19:14 | comment | added | Francesco Polizzi | By the way, I do not remember now whether there exists an Enriques involution over a K3 which is a quartic hypersurface of $\mathbb{P}^3$. | |
| Jul 12, 2022 at 18:56 | comment | added | Francesco Polizzi | Ok. But, in your question, you write "I am looking for a larger pool of such varieties in complex dimensions 2 and 3.", and then you ask "(2) Are all examples in dim 2 and 3 abelian or K3 fibered varieties?". Indeed, my answer provides lots of examples in every dimension (since there are plenty of irregular varieties in every dimension) and answers negatively question (2). If you wanted to know specifically about hypersurfaces in projective space, in my opinion you should have asked it explicitly. Please note that I do not want to make any controversy, I am just explaining my point. | |
| Jul 12, 2022 at 18:40 | comment | added | Mohammad Farajzadeh-Tehrani | Do you have answer for my comment above about hypersurfaces in projective space. For the application I have in mind, it would be more convenient to decide whether a given $X$ might have this property or not. So what I actually meant in my deleted comment was given $X$ it is probably equally hard to understand whether it is double cover as you described or not. | |
| Jul 12, 2022 at 18:35 | comment | added | Mohammad Farajzadeh-Tehrani | I deleted my comment. I guess I understand your argument better now. I did not mean to ignore your answer, I was just having a conversation. | |
| Jul 12, 2022 at 16:12 | comment | added | Francesco Polizzi | Furthermore, it seems that you are ignoring the fact that I answered (negatively) your second question. Anyway, it does not matter... | |
| Jul 12, 2022 at 16:10 | comment | added | Francesco Polizzi | Honestly, I do not understand your comment. This is the general recipe, so all the examples (including Jason's) must arise in this way. I also do not understand the "equally hard": for instance, take any irregular surface of general type (there is plenty of examples, for instance, product of curves with genus $\geq 2$, or complete intersections in Abelian varieties), choose any two-torsion divisor and you have your new variety endowed with the holomorphic involution. This is completely explicit. | |
| Jul 11, 2022 at 21:05 | history | edited | Francesco Polizzi | CC BY-SA 4.0 |
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| Jul 11, 2022 at 20:56 | history | edited | Francesco Polizzi | CC BY-SA 4.0 |
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| Jul 11, 2022 at 20:50 | history | edited | Francesco Polizzi | CC BY-SA 4.0 |
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| Jul 11, 2022 at 20:45 | history | answered | Francesco Polizzi | CC BY-SA 4.0 |