Timeline for One-dimensional family of complex algebraic K3 surfaces
Current License: CC BY-SA 4.0
15 events
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| Nov 16, 2022 at 20:50 | comment | added | Ariyan Javanpeykar | Just a little addendum to Jason Starr's comment. Namely, the moduli stack $M_d$ of polarized K3 surfaces of any fixed degree $d$ is hyperbolic (i.e., every holomorphic map $\mathbb{C}\to M_d^{an}$ is isoconstant). This implies that $S$ can't be $\mathbb{A}^1$, unless $X$ is already Picard rank one (take the trivial family). The reason $M_d$ is hyperbolic is because polarized K3 surfaces satisfy the infinitesimal Torelli property, so that hyperbolicity follows from a theorem of Griffiths-Schmid in Hodge Theory. | |
| Nov 16, 2022 at 8:03 | vote | accept | CommunityBot | ||
| Nov 15, 2022 at 22:44 | comment | added | Jason Starr | Welcome new contributor. By Gritsenko-Hulek-Sankaran, the moduli spaces of genus $g$ K3 surfaces are of general type for $g$ sufficiently positive. So you cannot make the base be rational or elliptic. | |
| Nov 15, 2022 at 20:10 | history | edited | user494851 | CC BY-SA 4.0 |
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| Nov 15, 2022 at 19:45 | answer | added | Sasha | timeline score: 3 | |
| Nov 15, 2022 at 19:20 | comment | added | user494851 | @Sasha You are corrected, here I mean the fibre over a general point. | |
| Nov 15, 2022 at 19:19 | history | edited | user494851 | CC BY-SA 4.0 |
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| Nov 15, 2022 at 19:16 | history | edited | LSpice | CC BY-SA 4.0 |
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| Nov 15, 2022 at 18:58 | comment | added | Sasha | The first was complex, i.e., over $\mathbb{C}$, and the second is over the field of functions on $S$ (this would be the standard meaning of $\eta$), so they can't be the same. Or did you mean something else? | |
| Nov 15, 2022 at 18:45 | history | edited | user494851 | CC BY-SA 4.0 |
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| Nov 15, 2022 at 18:45 | comment | added | user494851 | @Sasha They are the same algebraic complex K3 surface. | |
| Nov 15, 2022 at 18:31 | comment | added | Sasha | What is the relation between $X$ in your first and second sentences? | |
| Nov 15, 2022 at 18:17 | history | edited | user494851 | CC BY-SA 4.0 |
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| S Nov 15, 2022 at 18:17 | review | First questions | |||
| Nov 15, 2022 at 22:46 | |||||
| S Nov 15, 2022 at 18:17 | history | asked | user494851 | CC BY-SA 4.0 |