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Questions tagged [k3-surfaces]

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

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

79 questions
18
votes
1 answer

Why is the rank of the Picard group of a K3 surface bounded above by 22?

I understand that, over $\mathbb{C}$, the rank of the Picard group of a K3 surface $X$ is bounded above by $20$ because we can use the exponential sheaf sequence: $0 \to 2\pi i \mathbb{Z} \to \mathcal{O}_X \to \mathcal{O}_X^\times \to 0$, and…
Will
  • 1,850
12
votes
1 answer

K3 surfaces as complete intersections

I'm following Beauville's book "Complex Algebraic Surfaces". If $S$ is a K3 surface and $C$ is a smooth not hyperelliptic curve of genus g, then we have a birational morphism $\phi : S\rightarrow\mathbb{P}^g$, whose restriction to $C$ is the…
idioteca
  • 571
12
votes
2 answers

Why are K3 surfaces minimal?

I need to prove that all K3 surfaces are minimal surfaces, so that every birational map between K3 surfaces is an isomorphism. I've started to read Beauville's book on complex algebraic surfaces: there it says that the fact that K3 surfaces are…
ciccio
  • 523
12
votes
1 answer

Where does this elliptic curve come from?

In Zeta functions of an infinite family of K3 surfaces, Scott Alhgren, Ken Ono and David Penniston compute the zeta functions (given a good reduction restriction mentioned below) of the K3 surfaces $X_\lambda$ that are the smooth complete model of…
Alex
  • 4,178
11
votes
2 answers

Why the Picard group of a K3 surface is torsion-free

Let $X$ be a K3 surface. I want to prove that $Pic(X)\simeq H^1(X,\mathcal{O}^*_X)$ is torsion-free. From D.Huybrechts' lectures on K3 surfaces I read that if $L$ is torsion then the Riemann-Roch formula would imply that $L$ is effective. But then…
ciccio
  • 523
10
votes
2 answers

References to understand $K3$ surface as a double cover of $\mathbb{P}^2$ ramified along a sextic

My goal is to understand that $2:1$ cover of $\mathbb{P}^2(\mathbb{C})$ ramified along a sextic is a $K3$ surface. My main problem is in understanding the theory of ramified covering of $\mathbb{P}^2.$ Since $\mathbb{P}^2$ is compact, there are…
user166467
8
votes
1 answer

Varieties with the property that the cotangent bundle restricted to a complete nonsingular curve is free

Let $X$ be a $d$-dimensional smooth projective connected variety with cotangent sheaf $\Omega^1_X$ over $\mathbb C$. Suppose that for any nonsingular complete curve $C$ and non-constant morphism $\gamma:C\to X$, we have that $$\gamma^\ast \Omega^1_X…
Tom
  • 647
8
votes
1 answer

Can a complex manifold that is not a Calabi-Yau manifold be homeomorphic to a Calabi-Yau manifold?

This is a kind of a follow up to this question, which actually already had an answer here, in which it is asserted that Hodge numbers in general are not topological invariants. Could it be so extreme that a Calabi-Yau $n$-manifold (compact Kähler…
doetoe
  • 4,082
6
votes
2 answers

K3 surface criteria

Suppose I have an affine equation $f(x, y) = 0$ which after homogenizing becomes $f(X, Y, Z) = 0$ in $\mathbb{P}^{3}$. Are there ways to check that $f$ represents a K3 surface?
ADF
  • 1,745
5
votes
1 answer

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 equivalents and I know that $K3$ surfaces are…
Nutella Warrior
  • 265
5
votes
2 answers

Why is every smooth quartic in $\Bbb{P}^3$ a K3 surface?

Usually the first example of a K3 surface presented to us is the Fermat quartic $x_0^4+x_1^4+x_2^4+x_3^4=0$ in $\Bbb{P}_\Bbb{C}^3$. But I've just found out that actually any smooth quartic in $\Bbb{P}^3$ is K3, and I'm trying to understand why. I…
rmdmc89
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  • 94
5
votes
1 answer

elliptic k3 surface and Shioda Inose structure

We know that suppose given two elliptic curves $E$ and $E'$, there is a Kummer surface $km(E,E')$. And I'm curious suppose we know a $K3$ surface is kummer, how do we recover the pair $(E,E')$? For example, in the note …
Leo D
  • 347
4
votes
1 answer

Find an explicit quasi-smooth embedding $X_{38} \subset \mathbb P(5, 6, 8, 19)$

Consider the weighted projective space $\mathbb P(5,6,8,19)$ with weighted homogeneous coordinates $x,y,z,w$, in this order. I want to construct an explicit quasi-smooth embedding of the weighted K3 surface $X_{38} \subset \mathbb P(5,6,8,19)$, in…
isekaijin
  • 1,795
4
votes
1 answer

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 appears in the Euler characteristic…
red_trumpet
  • 11,169
4
votes
1 answer

cohomology of $T_X$ on K3 surfaces

I'm starting to study K3 surfaces, do you know some books on the subject which I could read? In particular I need the proofs that for a complex K3 surface $X$ it is $H^0(X,T_X)=H^2(X,T_X)=0$ and $h^1(X,T_X)=20$ (I consider the Dolbeault cohomology…
ciccio
  • 523
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