Questions tagged [k3-surfaces]
Questions about K3 surfaces, which are smooth complex surfaces $X$ with trivial canonical bundle and vanishing $H^1(O_X)$. They are examples of Calabi-Yau varieties of dimension $2$.
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The definition of complex multiplication on K3 surfaces
I am reading this paper on the complex multiplication of K3 surfaces. It seems that this is only defined for complex K3 surfaces, or K3 surfaces over number fields. Is there a more general defintion ...
- 131
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Isomorphism between two K3 surfaces in characteristic $11$ and the action of $\operatorname{PSL}(2, \, \mathbb{F}_{11}) $
We work over a field $k$ with $\operatorname{char}(k)=11$.
In the paper [1], Lemma 3.5, it is shown that the K3 surface $X_0$ defined as the weighted projective hypersurface of degree $12$ $$X_0=V(t_0^...
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Why do symmetries of K3 surfaces lie in the Mathieu group $M_{24}$?
I'm having trouble following some steps of this argument from the appendix of Eguchi, Ooguri and Tachikawa's paper Notes on the K3 surface and the Mathieu group M24:
Now let us recall that the ...
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Formula for Pushforward of structure sheaf for branched coverings
I have some questions of same flavour about two following constructions in Daniel Huybrechts's notes on K3 surfaces.
Construction 1: Kummer surface (Example 1.3 (iii), page 8) Let $k$ be a field of $...
- 5,780
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Action on Enriques surface by sections of Jacobian fibration
A question about a statement in Shigeyuki Kondo's paper Enriques surfaces with finite automorphism groups:
The setup: Let $\pi: Y \to \mathbb{P}^1$ be a special elliptic pencil of complex Enriques ...
- 5,780
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Picard number of Hilbert modular surfaces
Hilbert modular surfaces are discussed in various papers by Hirzebruch. Following [HZ] (and their notations), one obtains Hilbert modular surfaces by the action of Hilbert modular group on $\mathcal{H}...
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Is the set of points on an abelian surface which project to rational points on the Kummer surface a subgroup?
Let $C$ be a hyperelliptic curve of genus 2 defined over $\mathbb{Q}$, let $J$ be its Jacobian, and let $X$ be the Kummer surface associated to $J$ (i. e. $X$ is the singular Kummer surface which ...
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Exceptional quartic K3 surfaces
Recall that a $K3$ surface is called exceptional if its Picard number is 20.
The Fermat quartic $K3$ surface in $\mathbb P^3$ is exceptional.
My question is,
Are there infinitely many non-...
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$K3$ surfaces can't be uniruled
Let $S$ be a uniruled surface, ie admits a dominant map $ f:X \times \mathbb{P}^1$. Why then it's canonical divisor $\omega_X$ cannot be trivial? Motivation: I want to understand why $K3$ surfaces ...
- 5,780
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Explicit Lagrangian fibrations of a K3 surface
I would like to look at the behaviour of the fibres of a Lagrangian fibration (such that at least some fibres are not special Lagrangian) $X\to\mathbb{CP}^1$ under the mean curvature flow (in relation ...
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K3 surfaces with no −2 curves
I seem to remember that a K3 surface with big Picard rank always
has smooth rational curves.
This question is equivalent to the following question about integral quadratic lattices. Let us call a ...
- 9,157
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complex K3 surfaces with automorphisms of given orders
Concerning complex K3 surfaces, there are various methods to show the non-existence of an automorphism of certain orders. The usually way is to investigate the action of the automorphism on the space $...
user498585
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Do non-projective K3 surfaces have rational curves?
Define a compact Kähler surface $X$ to be a K3 surface if $X$ is simply connected, $K_X \simeq \mathcal{O}_X$, and $h^{0,1}=0$. If $X$ is projective, then a theorem typically attributed to Bogomolov ...
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One-dimensional family of complex algebraic K3 surfaces
Let $X$ be an algebraic complex K3 surface, we know that $X$ is deformation equivalent to a smooth quartic surface or more generally a K3 surface with Picard number $1$ (a very general K3 surface in ...
user494851
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Isotopy classes of $CP^1$ in 4-manifolds
Let $S_1$, $S_2$ be homologous embedded 2-spheres
in a compact smooth 4-manifold. Under which additional
conditions are they smoothly isotopic? I am interested
in the state of the art picture when $...
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