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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.

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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 $\mathbb{C}^2/\Gamma$, ...
WindUpBird's user avatar
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Do elliptic K3 surfaces have line bundle of relative degree $1$?

Let $f:S\to \mathbb P^1$ be an elliptic K3 surface ($S$ is a compact Kähler manifold or projective, if this makes things easier). If $f$ admits a section $C \subset S$, then the line bundle $\mathcal ...
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A line bundle $L$ can be assumed to be ample (after eventually passing to its dual)

I'm reading Huybrechts' Lectures on K3 Surfaces and I got stuck reading example 2.3.9, which shows that any K3 surface $X$ with $\operatorname{Pic}(X)=\mathbb{Z}\cdot L$ and such that $(L)^2=4$ can ...
WindUpBird's user avatar
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What is a monodromy representation?

I'm currently studying the Global Torelli Theorem for K3 surfaces and I encountered the following section in Huybrechts book. If $\mathcal{X}\rightarrow S$ is a smooth proper morphism over a ...
WindUpBird's user avatar
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$R^2f_*\mathbb{Z}$ is isomorphic to the constant sheaf $H^2(X_0,\mathbb{Z})$

Let $f:X \rightarrow S$ be a smooth proper family of K3 surfaces and let $X_t=f^{-1}(t)$. Let's suppose $S$ is a disk in $\mathbb{C}^n$. I know that the Betti/Hodge numbers are constant, so the direct ...
WindUpBird's user avatar
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Isomorphism between Tangent Sheaf and Cotangent Sheaf

I'm studying K3 surfaces and I often encountered the fact that the tangent sheaf $\mathcal{T}$ is isomorphic to the sheaf of differentials $\Omega$. Why is this true? I guess this follows from the ...
WindUpBird's user avatar
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Properties of the Lazarsfeld-Mukai vector bundle - Dualizing an exact sequence of vector bundles

I'm trying to understand a passage from an article by R.Lazarsfeld concerning properties of the Lazarsfeld-Mukai bundle for my reseach. Here is the paper of Lazarsfeld giving the construction (p.3 of ...
NaNoS's user avatar
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The complexification of Hodge class is properly contained in the space of type $V^{k,k}$

I was reading Professor Huybrechts's Lectures on K3 surfaces, there is a statement about Hodge class that I can't figure out. Let's consider the (integral or rational) Hodge structure $V$ with the ...
yi li's user avatar
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K3 surfaces: equivalent definitions

I am just starting to study complex K3 surfaces and I'm trying to understand why the different definitions I found are actually equivalent. Definition 1: a (complex) K3 surface is a compact, connected,...
WindUpBird's user avatar
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Stein manifolds coming from neighbourhood of critical fibers in elliptic fibrations

Let $X$ be a compact, Kähler surface (so $\dim X = 4$) and let $f:X\to \mathbb{CP}^1$ be an elliptic fibration, in particular $f$ is holomorphic. Suppose that $p\in \mathbb{CP}^1$ is a singular value ...
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Reducing a surface

I am supposed to reduce a surface $x^4+y^4+z^4+w^4+12xyzw=0$ to a surface whose equation is a quadratic in $x^2,y^2,z^2,w^2$ by change of coordinates. Is the following a valid reduction: first of all ...
nomadd's user avatar
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Morphism of family of varieties determined on rational points

I am currently studying various Moduli problems and in order to check whether some families have non-trivial automorphisms, I have the strong intuition that the following should hold: Let $k$ be an ...
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Period map for products of K3 surfaces

If $X$ is a K3 surface, it carries a unique (up to scaling with $\mathbb{C}^*$) holomorphic 2-form $\sigma$, determined by the complex structure. Let $\Lambda=3U\oplus-2E_8$ be the K3-lattice. The ...
James's user avatar
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ADE types singularities of $K3$ surfaces

I was reading a paper in which they have an algebraic surface \begin{equation} F(x, y, u) = u^2(1+xy)= (x+y).(x+y-4xy+x^2y+xy^2) \end{equation} After homogenising the surface they get; \begin{equation}...
Mandeep's user avatar
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K3 surface $S \subset \mathbb P^1 \times \mathbb P^2$ of bidegree $(2,3)$ has Picard rank 2?

$\DeclareMathOperator{\Pic}{Pic}$The following example is taken from [1, Example 5.8]. Let $S \subset \mathbb P^1 \times \mathbb P^2$ be a general smooth surface of bidegree $(2,3)$, so $S$ is a K3-...
red_trumpet's user avatar
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Example of a K3 surface where the Picard group is generated by a single elliptic curve.

In a paper by Voisin[1] she considers as an example a K3-surface whose Picard group is generated by the class of an elliptic curve. ... l'exemple d'une surface K3 dont le groupe de Picard est ...
red_trumpet's user avatar
  • 8,951
4 votes
1 answer
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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 ...
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Why the pencil of planes on quadric $Q\subset\mathbb{P}^4$ cuts out on cubic $V\subset\mathbb{P}^4$ a pencil of elliptic curves $|E|$?

I'm reading the proof of Proposition VIII.15 in Beauville's "Complex algebraic surfaces", and I do not understand a part of it: Suppose $g=3k+1$, with $k\ge 1$. Let $Q\subset\mathbb{P}^4$ be ...
Rainbow57's user avatar
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K3 surface is not the blow up of any other smooth surface.

This is the exercise 2.5.5 in the book 'Complex Geometry' by Huybrechts: Let $X$ be a K3 surface, i.e. $X$ is a compact complex surface with $K_X\cong\mathscr{O}_X$ and $h^1(X,\mathscr{O}_X)=0$. Show ...
WakeUp-X.Liu's user avatar
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complete intersection in $\mathbb{P}^{n+2}$ as K3 surface

I am reading D. Huybrechts' lecture "Lectures on K3 surfaces", http://www.math.uni-bonn.de/people/huybrech/K3Global.pdf He claims that a smooth complete intersection of type $(d_1,\cdots, ...
Invariance's user avatar
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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 ...
Harry Richman's user avatar
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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 ...
user avatar
1 vote
1 answer
144 views

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 ...
anon1432's user avatar
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233 views

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 ...
rmdmc89's user avatar
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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 ...
Mee98's user avatar
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5 votes
1 answer
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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 ...
Nutella Warrior's user avatar
4 votes
1 answer
103 views

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 ...
red_trumpet's user avatar
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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 ...
Ian Gershon Teixeira's user avatar
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1 answer
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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 $|C|$. ...
red_trumpet's user avatar
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2 votes
1 answer
259 views

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 ...
Bernie's user avatar
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1 answer
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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 ...
rmdmc89's user avatar
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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 $H^1(X,\mathcal{O}_X)=0$ ...
rmdmc89's user avatar
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5 votes
2 answers
574 views

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{...
rmdmc89's user avatar
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1 vote
1 answer
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Canonical bundle and canonical divisor in a $K3$ surface

The algebraic definition of a $K3$ surface is this: A smooth algebraic suface $X$ is called $K3$ if: i) $X$ has trivial canonical bundle; ii) $h^1(X,\mathcal{O}_X)=0$. I know that i) means that the ...
rmdmc89's user avatar
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1 vote
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Theorem on normal singularities from Badescu's Algebraic Surfaces

Reading Lucian Badescu's Algebraic Surfaces I have encountered a proof I can't understand. That's Theorem 3.28 (M. Artin) at pages 41/42: Theorem 3.28 (M. Artin). Let $(Y, y)$ be a two-dimensional ...
user267839's user avatar
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Kummer suface ; cohomology of the resolution

I have questions regarded to the resolution of Kummer surface. You can see the other 2 ones here At first, I am describing a resolution of Kummer surface: Get a lattice of rank 4 ; $\Gamma$ on $ \...
Amin Ebrahimi's user avatar
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0 answers
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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 ...
Amin Ebrahimi's user avatar
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0 answers
132 views

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 ...
user avatar
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1 answer
311 views

Fibered surface has connected fibers

I throwed a look into C. Liedtke's paper SEMI-STABLE REDUCTION FOR CURVES and have some problems to understand an argument (page 1): Let $S$ be a connected Dedekind scheme. We denote by $K=...
user avatar
3 votes
0 answers
104 views

Conics in a projective $K3$ surface

Given a complex $K3$ surface $S$ in a projective space $\mathbb{P}^N$, $N>3$, is there some cohomological characterization that allows us to see if $S$ contains some conic $C$? I am particularly ...
Nutella Warrior's user avatar
2 votes
0 answers
134 views

Global sections of a line bundle associated to a smooth irreducible curve on a K3 surface.

I would like to know it the following proof is correct: Lemma: Let $C$ be a smooth, irreducible curve of genus $g$ over a complex projective K3 surface $X$. Let $L:=\mathcal{O}_X(C)$. Then $C^2=2g-2$...
Tnarf's user avatar
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Completions of Stalks of a Surface

Let $S$ be a surface (so a $2$-dimensional proper $k$-scheme). My question arises from following another thread of mine. I'm looking for conditions for $S$ which garantee that the completion of every ...
user267839's user avatar
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1 vote
1 answer
253 views

Kähler Potential on Blowup of $\mathbb{C}/\{\pm 1\}$

The book Joyce: Riemannian Holonomy Groups and Calibrated Geometry contains on page 205 the Eguchi-Hanson space as an example: Consider $\mathbb{C}^2$ with complex coordinates $(z_1,z_2)$, acted ...
user505117's user avatar
2 votes
1 answer
254 views

blowing-up preserves the first Betti number?

Let $G$ be a finite group and $S$ be a K3 surface. $G$ acts effectively and symplectically(fix the nowhere vanishing 2-form of $S$) on $S$. Since the action is symplectic,quotient surface $S/G$ has ...
nero's user avatar
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2 votes
0 answers
223 views

K3 coming from Enriques surface

We know every Enriques surfaces has a K3 surface as its universal cover. My question is: what kind of K3 surface is the double cover of an Enriques surface? Or equivalently, what kind of K3 surface ...
Rust Z's user avatar
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Any concrete example(s) of $K3$ surfaces with Picand number 18 and does not admit Shioda-Inose structure?

I am looking for some explicit examples of (elliptic) $K3$-families defined over a number field (better to be over $\mathbb{Q}$) with Picard number $18$ but does not admit Shioda-Inose structure, i.e. ...
Leo D's user avatar
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Is the Calabi-Yau property of a smooth manifold a differential invariant?

In this question I asked if it could happen that two complex manifolds are homeomorphic, and one of them is a Calabi-Yau manifold but the other isn't. It turns out that there are complex surfaces that ...
doetoe's user avatar
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8 votes
1 answer
344 views

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 ...
doetoe's user avatar
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3 votes
2 answers
848 views

Let $X$ be a K3 surface, show that $H_1(X,\mathbb Z)=0$

Let $X$ be a(n algebraic) K3 surface, i.e., $X$ is a smooth algebraic surface with trivial canonical bundle and $H^1(X,\mathcal{O})=0$. This assumption directly implies that $H^1(X,\mathbb C)=0$, so $...
AG learner's user avatar
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1 vote
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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 ...
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