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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3
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0answers
136 views

Topology of K3 as a sum of two abelian fibrations.

Let $E$ be a blow-up of $\mathbb{P}^2$ at 9-points in the bases locus of a pencil of elliptic curves (A $T^2$ fibration over $S^2$). K3 surfaces is obtained by removing a fiber from two copies of $E$ ...
6
votes
1answer
211 views

$K3$ surfaces admitting finite non-symplectic group actions are projective

I have read somewhere that "$K3$ surfaces admitting finite non-symplectic group actions are projective". Could someone remind me of a proof?
9
votes
2answers
658 views

How to compute the Picard rank of a K3 surface?

I'm curious about the following question: Given a K3 surface, how does one proceed to compute its rank? Of course the answer may depend on the form of the input, i.e. how the K3 is "given". So ...
0
votes
0answers
143 views

$T^2$-fibered K3 surface with involution

Let $S$ be a K3 surface and $f:S\rightarrow \mathbb{P}^1$ a $T^2$-fibration (not necessarily holomorphic, I have a special Langrangian fibration in mind). Assume there is a $k$-section, then a fiber ...
2
votes
0answers
260 views

Singular fibers of an elliptic fibered K3 surface.

Let $f:S\rightarrow \mathbb{P}^1$ be an elliptic K3 surface. Assume that $\mathrm{Pic}(S)\cong U$, where $U$ stands for the hyperbolic lattice. I think that the elliptic fibration has only singular ...
5
votes
1answer
424 views

Complex structures on a K3 surface as a hyperkähler manifold

A hyperkähler manifold is a Riemannian manifold of real dimension $4k$ and holonomy group contained in $Sp(k)$. It is known that every hyperkähler manifold has a $2$-sphere $S^{2}$ of complex ...
6
votes
3answers
628 views

2-cycle of K3 surface

Hi there, I want to ask about the 2-cycle of K3 surface. As we know, its betti number $b_2$=22, so there will be 22 2-cycle generators. Is there any topological way to figure out such cycles ...
2
votes
1answer
361 views

Genus two pencil in K3 surface

It is known that smooth $K3$ surface can be obtained as two fold branched cover of rational elliptic surface $E(1) = \mathbb{CP}^2 9 \bar{{\mathbb{{CP}^2}}}$ along the smooth divisor $2F_{E(1)} = 6H - ...
3
votes
2answers
566 views

Question on K3 Surface

Is it possible to realize $K3$ surface as a ramified double cover of rational elliptic surface? If so, is there way to see an elliptic fibration structure on $K3$ from such cover? It seems to me one ...
19
votes
4answers
1k views

Does $(x^2 - 1)(y^2 - 1) = c z^4$ have a rational point, with z non-zero, for any given rational c?

I need this result for something else. It seems fairly hard, but I may be missing something obvious. Just one non-trivial solution for any given $c$ would be fine (for my application).
11
votes
1answer
348 views

Symmetric functions on three parameters being perfect squares

Is it possible for $x+y+z, xy+yz+zx$, and $xyz$ to be perfect squares at the same time for positive integer values of $x,y,z$?
2
votes
1answer
275 views

Picard/cohomology lattice of surfaces of low degree in $\mathbb P^3$

Let $S_{d>3}\subset\mathbb{P}^3_{\mathbb{C}}$ be a smooth surface of degree $d$. What is known (where to read?) about the Picard/cohomology lattice for small d? e.g. for $d=4$ the cohomology ...
1
vote
1answer
300 views

holomorphic sections on elliptic K3 surface

Hi all, I want to ask something about the holomorphic sections on elliptic K3: Is there any obstruction for an ellptic K3 (as an elliptic fibration) to have holomorphic sections? Is that always some ...
5
votes
2answers
1k views

construct the elliptic fibration of elliptic k3 surface

Hi all, As we know, every elliptic k3 surface admits an elliptic fibration over $P^1$, but generally how do we construct this fibration? For example, how to get such a fibration for Fermat quartic? ...
7
votes
2answers
577 views

Polarizations of K3 surfaces over finite fields

Suppose that $X$ is a (projective) K3 surface over a field $k$. A polarization of $X$ is an element $\lambda\in Pic_X(k)$ that is represented over an algebraic closure $\overline{k}$ by an ample line ...
2
votes
1answer
244 views

Sheaves with zero Chern classes on a $K3$ surface.

Let $S$ be a $K3$ surface. Is it true that any sheaf on $S$ with zero Chern classes is isomorphic to $\mathcal{O}_S^{\oplus n}$ for some $n$? If not, do you have any counterexample?
7
votes
1answer
443 views

To what extent does Poincare duality hold on moduli stacks?

Poincare duality gives us, for a smooth orientable $n$-manifold, an isomorphism $H^k(M) \to H_{n-k}(M)$ given by $\gamma \mapsto \gamma \frown [M]$ where $[M]$ is the fundamental class of the ...
2
votes
1answer
328 views

Kuga-Satake with p-adic methods

Is it possible to construct the Kuga-Satake abelian variety attached to a K3 surfaces (over a local field) only using p-adic methods? If the K3 surface is defined over a local field, the ...
7
votes
1answer
475 views

Dodecahedral K3?

In pondering this MO question and in particularly its 1st answer, and answers to this one recently posed, I realized there ought to be a dodecahedral K3 surface $X$. This $X$ would fiber as an ...
10
votes
3answers
740 views

A K3 over $P^1$ with six singular $A_1$- fibers?

Hirzebruch, in the paper 'Arrangements of Lines and Algebraic Surfaces' constructs a special $K3$ surface out of a 'complete quadrilateral' in $CP^2$. A complete quadritlateral consists of 4 ...
7
votes
4answers
1k views

Sums of four fourth powers

Apologies in advance if this is a naive question. If I understand correctly, it's well-known that the Fermat quartic surface $X = \lbrace w^4 +x^4+y^4+z^4 =0 \rbrace \subset \mathbf{P}^3$ has ...
2
votes
3answers
676 views

cuspidal curves in K3 surfaces

Let $S\subset\mathbb{P}^g$ be a smooth polarized K3 surface of genus $g$. I am interested in the existence of certain cuspidal curves in the linear system. We know a general hyperplane section $H\cap ...
4
votes
1answer
358 views

Is any K3 surface of degree 8 in P^5 the complete intersection of quadrics?

Here the base field is the field of complex numbers.
0
votes
1answer
538 views

Neron-Severi Lattice of Elliptic K3

I'm trying to compute Neron-Severi lattices of some K3 surfaces. They have elliptic fibrations with multiple sections. Setting one section to be the identity section, I can write down a Weierstrass ...
4
votes
1answer
455 views

Period integrals of the fiber of elliptically fibered K3 manifolds

Suppose I have a smooth elliptically fibered K3 manifold over $\mathbb{P}^1$ defined by the Weierstrass equation, \begin{equation} y^2=x^3+f(z)x+g(z) \end{equation} where $x,y,z$ are local ...
3
votes
0answers
320 views

The surface $ x^2 y^2 + 1 = (x^2 + y^2) z^2 $

Hi, I'm trying to find all rational points on the surface of the title, in connection with the Euler Brick (AKA Rational Box) problem. This surface is equivalent to $ x^2 z^2 - 1 = (x^2 - z^2) y^2 $, ...
14
votes
1answer
1k views

M24 moonshine for K3

There are recent papers suggesting that the elliptic genus of K3 exhibits moonshine for the Mathieu group $M_{24}$ (http://arXiv.org/pdf/1004.0956). Does anyone know of constructions of $M_{24}$ ...
3
votes
1answer
216 views

Spherical objects on Kummer surfaces

Spherical objects E in the derived category of coherent sheaves over a K3 surface satisfy: 1) Hom(E,E)=C 2) Ext^2(E,E)=C 3 Ext^i=0 otherwise. Are the structure sheaf O_X and the sheaves associated ...
4
votes
2answers
582 views

Are any two K3 surfaces over C diffeomorphic?

Let $S$ be a K3 surface over $\mathbb{C}$, that is, $S$ is a simply connected compact smooth complex surface whose canonical bundle is trivial. I recall reading somewhere that any two such surfaces ...
8
votes
2answers
712 views

What classes am I missing in the Picard lattice of a Kummer K3 surface?

Constructing the Kummer K3 of an Abelian surface $A$, we have an obvious 22-dimensional collection of classes in $H^2(K3, \mathbb{Z})$ given by the 16 (-2)-curves (which by construction do not ...
2
votes
3answers
1k views

K3 surface of genus 8

Let $V$ be a complex vector space of dimension 6 and let $G\subset {\mathbb P}^{14}\simeq {\mathbb P}(\Lambda^2V)$ be the image of the Plucker embedding of the Grassmannian $Gr(2, V)$. Why the ...
5
votes
2answers
790 views

Singular K3 — mathematical meaning?

There's a very interesting text by Cumrun Vafa called Geometric Physics. Here I'm particularly interested in Chapter 4, where we take a Calabi-Yau manifold presented as a degenerating fibration: ...
13
votes
4answers
884 views

K3 surfaces with good reduction away from finitely many places

Let S be a finite set of primes in Q. What, if anything, do we know about K3 surfaces over Q with good reduction away from S? (To be more precise, I suppose I mean schemes over Spec Z[1/S] whose ...
3
votes
1answer
439 views

K3 over fields other than C?

How to classify K3 surfaces over an arbitrary field k?
3
votes
1answer
569 views

The existence of primitive and sufficiently ample line bundles on K3 surfaces?

Let S be a surface and L be a line bundle on S. For any zero-dimensional closed subschemes x of S, there is natural map from global sections of L to the global sections of L restricting to x (which is ...