# Questions tagged [k3-surfaces]

For all questions about *K3 surfaces*, that is complex or algebraic smooth surfaces which are regular with trivial canonical bundle.

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### Canonical divisor of Kummer Surface

I want to prove that a Kummer surface is K3 so, first of all, I want to focus on the canonical divisor $K_X$. Just to fix some notations: $A$ is a complex torus of the form $\mathbb{C}^2/\Gamma$, ...
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### Examples of a K3 surface in a product of $\mathbb P^1$'s

On the Wikipeida page for K3 surfaces, there are several examples listed of how to produce a K3 surface as a subvariety of projective space by taking polynomials of specified degrees. Namely, a K3 ...
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### Image of Kähler class of an octic K3 surface is primitive and satisfies $p(\kappa) \cdot p(\kappa) = 8$

I am trying to understand the following sentence from a paper by Kovalev called Constructions of Compact $G_2$ Holonomy Manifolds. Unfortunately I can't find a copy of the paper publicly available so ...
1 vote
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### Morphisms between K3 surfaces

Let $X$ and $Y$ be K3 surfaces over an algebraically closed field of characteristic 0. Is it possible to have a dominant morphism $X\to Y$ which is not an isomorphism? In positive characteristic, the ...
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### Explicit elliptic fibration for $x_0^4+x_1^4+x_2^4+x_3^4=0$ (Fermat's quartic)

In Huybrecht's Lectures on K3 surfaces, there's an explicit description of an elliptic fibration for the K3 surface $X:=\{(x_0:x_1:x_2:x_3)\in\Bbb{P}_\Bbb{C}^3\mid x_0^4+x_1^4+x_2^4+x_3^4=0\}$ (Fermat ...
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### Adjunction formula projective hypersurface interpretation

I was reading a proof that all degree 4 hypersurfaces $X$ in $P^3(\mathbb{C})$ are K3 surfaces. When they were showing that the canonical bundle $\omega_X$ of $X$ is trivial, they suddenly used the ...
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### Deformations of K3 surface is again a K3 surface

I define a $K3$ surface as a smooth complex manifold of dimension two which is simply-connected and such that the canonical bundle is trivial. I know that two $K3$ surfaces are always deformation ...
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### Dimension of linear system $|C|$ is the genus of $C$?

Let $C \subset S$ be a smooth curve in a K3 surface $S$. Why is the dimension of the linear system $|C|$ the genus of $C$? Here is what I tried: $\dim |C| = \dim H^0(\mathcal{O}(C)) - 1$, and this ...
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### Is a $K3$ surface a spherical variety?

Do there exist any $K3$ surfaces that admit the structure of a spherical variety? That is, does there exists a $K3$ surface $X$, a reductive algebraic group $G$, and a Borel subgroup $B$ of $G$, such ...
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### How to define a family of curves from a linear system?

Let $S$ be a projective K3 surface, and let $C \subset S$ be an integral curve. Assume also, that all other curves in the linear system $|C|$ are integral, and let $n$ be the dimension of $|C|$. ...
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### Globally generated sheaf is globally generated by fewer sections away from finitely many points?

Assume $S$ is a smooth projective surface and $E$ is a nontrivial torsion free sheaf of rank $r$ on $S$ that comes with a surjective morphism $\mathcal{O}_S^{\oplus t}\rightarrow E\rightarrow 0$ for ...
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### If $X$ is a K3 surface and $X\to C$ is an elliptic fibration, then $C\simeq \Bbb{P}^1$

Sometime ago I heard someone say in a talk that if an algebraic surface $X$ is K3 and has an elliptic fibration $\pi:X\to C$ (where $C$ is a curve), then $C$ must be rational. The speaker just said ...
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### Complete intersection of a quadric and a cubic in $\Bbb{P}^4$ is a K3 surface

I'm familiar with the fact that a smooth quartic $X=Z(F)\subset\Bbb{P}^3$ is a K3 surface. The condition that $K_X\sim 0$ can be shown directly by the adjunction formula, and $H^1(X,\mathcal{O}_X)=0$ ...
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### Kummer suface ; the resolution has a holomorhic (2,0)-form .

At first, I am describing a resolution of Kummer surface: Get a lattice of rank 4 ; $\Gamma$ on $\mathbb{C}^2$. The quotient $\mathbb{T}^4:\mathbb{C}^2 /\Gamma$ would be a complex tori . Now ...
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### Ramanujan and Other Fields

Ramanujan's works also included K-3 Surfaces (although it was discovered much later as mentioned here and here) which in the recent years have found their way into physics as one can see here. I ...
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### References for properties of a $K3$ surface
I have read that the surface $K3$ in $\mathbb{CP}^3$ defined by the equation $x^4+y^4+z^4+w^4=0$ is a spin manifold whose $\hat{A}$-genus is non-vanishing. However, not knowing algebraic geometry, I ...