Timeline for Exceptional quartic K3 surfaces
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 20, 2023 at 8:20 | comment | added | Basics | @Cranium Clamp, yes. One can show it by the Torelli theorem for K3 surface. | |
| Sep 20, 2023 at 6:05 | comment | added | Cranium Clamp | I have a question on the side: Can the Picard number of a quartic K3 be any possible integer between 1 and 20 inclusive? | |
| Sep 18, 2023 at 13:15 | comment | added | Basics | I know that there are infinitely many non-isomorphic exceptional K3 surfaces. I guess that there may be infinitely many non-isomorphic quartic exceptional K3 surfaces but I cannot give a clean explanation. I am wondering if somebody can. | |
| Sep 18, 2023 at 12:44 | comment | added | Frank | I realise I perhaps spoke too soon and that you are asking about quartics only. Inose has a paper "Defining equations..." which I cannot get hold of atm but that claims to prove such surfaces are all birationally quartics. Of course these surfaces will have many classes of square 4 (ie the K3 has many quartic models), but whether this class is very ample needs to be tested on -2-curves and one can hope this is a lattice problem one can unravel from Inose's paper. | |
| Sep 18, 2023 at 12:26 | comment | added | Frank | These are more often called "singular K3 surfaces", and a theorem of Shioda-Inose classifies them as resolutions of Kummer quotients of products of isogenous elliptic curves. Indeed there are infinitely many of them (up to isomorphism) even though they do not deform in moduli. See Section 14.3.4 in Huybrechts' "Lectures on K3 surfaces" book. | |
| Sep 18, 2023 at 11:48 | comment | added | Basics | By "non-isomorphic", I mean non-isomorphic as compact complex surfaces. However, if the answer to the original question in the post is not easy, I also would like to know if there are infinitely many quartic exceptional $K3$ surfaces that are not projectively equivalent. | |
| Sep 18, 2023 at 11:32 | comment | added | Jason Starr | Just to clarify, when you say "non-isomorphic", do you mean non-isomorphic as compact complex surfaces, i.e., not biholomorphic, or do you mean non-isomorphic as quartic hypersurfaces, i.e., not projectively equivalent? Some K3 surfaces have infinitely many non-projectively equivalent embeddings in the same projective space. | |
| Sep 17, 2023 at 12:31 | history | asked | Basics | CC BY-SA 4.0 |