Timeline for K3 surfaces with no −2 curves
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Feb 2, 2023 at 1:20 | comment | added | Cranium Clamp | @YCor yes, take any K3 surface. The second cohomology with the intersection product is isomorphic as a lattice to E8^2 + U(-2)^3 for any K3 surface. | |
| Feb 1, 2023 at 9:21 | comment | added | YCor | What is meant by $H^2(K3,\mathbb{Z})$? do you mean $H^2(S,\mathbb{Z})$ for some K3 surface $S$? | |
| Feb 1, 2023 at 9:08 | history | edited | Misha Verbitsky | CC BY-SA 4.0 |
added 17 characters in body
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| Feb 1, 2023 at 9:06 | comment | added | Misha Verbitsky | Marco: any smooth rational curve on K3 has square -2. Conversely, using Riemann-Roch, you can prove that any line bundle $L$ with $c_1^2=-2$ satisfies $H^2(L)\neq 0$ or $H^2(L^*)'\neq 0$, and the zero divisor of its section contains a smooth rational curve as a component | |
| Feb 1, 2023 at 9:04 | comment | added | Misha Verbitsky | Samir: many thanks, this is what I need I think. | |
| Feb 1, 2023 at 9:04 | comment | added | Misha Verbitsky | Jason: for McMullen's nonprojective K3 surfaces, the Picard lattice is either degenerate or negative definite, and such lattices are easy to construct in any rank. | |
| Feb 1, 2023 at 7:09 | comment | added | Samir Canning | For projective K3s, there are -2 curves whenever the Picard rank is at least 12. See, for example, Huybrechts's Lectures on K3 surfaces, Chapter 14, Corollary 3.8. | |
| Feb 1, 2023 at 1:05 | comment | added | Jason Starr | . . . The rank is zero. | |
| Feb 1, 2023 at 1:01 | comment | added | Jason Starr | Did you look at McMullen’s nonprojective K3 surfaces associated to Salem numbers? I will look now to see their ranks . . . | |
| Jan 31, 2023 at 23:56 | comment | added | Marco Golla | @MishaVerbitsky: pardon my ignorance, but why does the second question imply the first? | |
| Jan 31, 2023 at 22:56 | comment | added | Misha Verbitsky | the question is, for which $d$ we can be sure there are no -2 vectors? What about $d=12$? What about $d=20$? | |
| Jan 31, 2023 at 22:50 | comment | added | LSpice | What is $d$ in "which has rank $\ge d$"? | |
| Jan 31, 2023 at 22:49 | history | edited | LSpice | CC BY-SA 4.0 |
-2 -> $-2$, and consistent hyphenation
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| Jan 31, 2023 at 22:01 | history | asked | Misha Verbitsky | CC BY-SA 4.0 |