**20**

votes

**1**answer

791 views

### Monstrous moonshine for $M_{24}$ and K3?

An important piece of Monstrous moonshine is the j-function,
$$j(\tau) = \frac{1}{q}+744+196884q+21493760q^2+\dots\tag{1}$$
In the paper "Umbral Moonshine" (2013), page 5, authors Cheng, Duncan, and ...

**19**

votes

**4**answers

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).

**14**

votes

**1**answer

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 $M_{24}$ ...

**13**

votes

**4**answers

857 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 ...

**11**

votes

**1**answer

342 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 $x,y,z$?

**10**

votes

**3**answers

699 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 ...

**9**

votes

**2**answers

610 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
...

**7**

votes

**4**answers

959 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 ...

**7**

votes

**2**answers

679 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 ...

**7**

votes

**1**answer

461 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 ...

**7**

votes

**2**answers

548 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 ...

**7**

votes

**2**answers

260 views

### Necessary condition on Calabi-Yau manfiold to be a hypersurface in a Fano manifold

Let $X$ be a smooth projective Calabi-Yau threefold. Are there any known obstructions to it
being a member of a base-point-free linear system in a nef-Fano fourfold?
What, in anything, is known ...

**7**

votes

**1**answer

187 views

### A question on an elliptic fibration of the Enriques surface

Let $S$ be an Enriques surface over complex numbers. It is known that $S$ admits an elliptic fibration over $\mathbb{P}^1$ with $12$ nodal singular fibers and $2$ double fibers. How can I see this ...

**7**

votes

**0**answers

308 views

### Torelli-like theorem for K3 surfaces on terms of its étale cohomology

Is there a proof of a Torelli-like Theorem for a K3-surface over any field (non complex) in terms of its etale or crystalline cohomology?
For example: If $K\ne \mathbb{C} $ and $X\rightarrow ...

**6**

votes

**3**answers

610 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 ...

**6**

votes

**1**answer

202 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?

**6**

votes

**1**answer

427 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 ...

**5**

votes

**2**answers

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?
...

**5**

votes

**1**answer

393 views

### Complex structures on a K3 surface as a hyperkähler manifold

A hyperkÃ¤hler manifold is a Riemannian manifold of real dimension $4k$ and holonomy group contained in $Sp(k)$. It is known that every hyperkÃ¤hler manifold has a $2$-sphere $S^{2}$ of complex ...

**5**

votes

**1**answer

181 views

### Does $h^1(D)=0$ imply numerical connectedness on K3 surfaces?

Let $X$ be a complex K3 surface and $D$ an effective divisor on $X$.
We shall say: $D$ is connected if its support is connected. $D$ is numerically connected if for any non-trivial effective ...

**5**

votes

**2**answers

782 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:
...

**5**

votes

**1**answer

321 views

### Involution of the Fermat quartic

Let $X\subset\mathbb{P}^{3}$ be the Fermat quartic surface given by
$$x^4-y^4-z^4+w^4 = 0$$
and consider the involution
$$i:X\rightarrow X,\: (x,y,z,w)\mapsto (y,x,w,z).$$
The surface $X$ can be seen ...

**5**

votes

**1**answer

209 views

### Positivity question on K3 surfaces

Let $X$ be a smooth projective complex K3 surface and $L, D$ two effective divisors, $L^2\geq0$ and $D^2\geq0$.
(Q1). do we have $L\cdot D\geq0$ ?
If either one has positive self-intersection, the ...

**5**

votes

**0**answers

128 views

### on the automorphisms of the transcendental Hodge structure of a K3 surface

Let $S$ be a complex projective K3 surface and consider the sub-Hodge structure
$$
T(S) \subset H^2(S, \mathbb{Q})
$$ consisting of transcendental cycles. Let $\varphi$ be an automorphism of Hodge ...

**5**

votes

**0**answers

141 views

### Non minimal K3 surfaces as hypersurfaces of weighted projective spaces

I recently learnt that the hypersurface
$$
S:=(x^2+y^3+z^{11}+w^{66}=0) \subset \mathbb{P}(33,22,6,1)
$$
is birational to a K3 surface. This is surprising because the surface is quasi-smooth, ...

**4**

votes

**2**answers

290 views

### Action of automorphisms of a $K3$ surface on its $(-2)$-curves

Consider a complex $K3$ surface $X$ and take its group of automorphisms $Aut(X)$. It is a known fact that the action of $Aut(X)$ on the set of rational $-2$ curves of $X$ has only finite number of ...

**4**

votes

**2**answers

571 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 ...

**4**

votes

**2**answers

285 views

### Reference for Automorphisms of K3 surfaces

I am looking for some introductory reference concerning Automorphisms (of finite order) on K3 surfaces. Any suggestion?

**4**

votes

**1**answer

342 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.

**4**

votes

**1**answer

432 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 ...

**4**

votes

**1**answer

366 views

### Rookie questions about k3's

Hi everyone,
I am trying to go through parts of Saint-Donat's 1974 paper 'Projective Models of K3-surfaces', and have been stuck on a few claims for a while now - I'd appreciate some help explaining ...

**3**

votes

**2**answers

276 views

### Line bundles on K3 surfaces

Let $L$ be a line bundle on an (algebraic) K3 surface over a field $k$. The Riemann-Roch theorem specializes to
$$
\chi(X, L)=\frac{1}{2}(L\cdot L)+2
$$
which can be rewritten as
$$
h^0(X, ...

**3**

votes

**1**answer

535 views

### Mirror symmetry for hyperkahler manifold

Hi there,
I have some questions about the mirror symmetry of hyperkahler manifold and K3 surface.
The well-known result said: the mirror symmetry for K3 surface is just given by its hyperkahler ...

**3**

votes

**2**answers

547 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 ...

**3**

votes

**2**answers

256 views

### Nefness on a K3 surface

Let $D$ be a divisor on a (complex) K3 surface.
Suppose $D^2\geq0$. In general, $D$ is nef if $D\cdot C\geq0$ for all irreducible curves on the surface.
Is in our case sufficient to check this for ...

**3**

votes

**1**answer

437 views

**3**

votes

**1**answer

211 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 ...

**3**

votes

**1**answer

202 views

### Euler characteristic of nodal K3 surfaces (as in singular)

This is probably easy, but I was just wondering if there is a nice and easy formula for the topological Euler characteristic of a K3 surface $X$ with say $k$ nodes. If there is no general formula, is ...

**3**

votes

**1**answer

173 views

### K3 surface with $D_{14}$ singular fiber

Let $X$ be an elliptic K3 surface with $D_{14}$ singular fiber. Do you know an explicit equation for such $X$? Also, how many disjoint sections such fibration admits? Any reference would be greatly ...

**3**

votes

**1**answer

303 views

### (3,3) abelian surface and k3 surfaces

SOrry for the very specific question, but curiosity bites....
So here's the story: an idecomposable principally polarized abelian surface is embedded in $P^8=|3\Theta |^* $ as a deg 18 surface A. ...

**3**

votes

**2**answers

225 views

### Algebraic cycles on a K3 surface after hyperKahler rotation.

I would like to find a gap in the following observation. I found a suspicious part but cannot prove it wrong. I would appreciate your assistance.
Let $M$ be a lattice of signature $(1,t)$ and $S$ be ...

**3**

votes

**1**answer

550 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 ...

**3**

votes

**0**answers

153 views

### Semistable minimal model of a $K3$-surface and the special fibre

Suppose that $K$ is a $p$-adic field, that is a field of characteristic $0$ whose ring of integers is a complete discrete valuation ring $\mathcal O_K$ and with residue field $k$ (algebraic closed) of ...

**3**

votes

**0**answers

236 views

### Possible automorphism groups of a K3 surface

Which finite groups are automorphism groups of polarized K3 surfaces (let's say over ℂ)?
...and does the answer change is I remove "polarized"?
(polarized = equipped with an ample line bundle)

**3**

votes

**0**answers

134 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 $E$ ...

**3**

votes

**0**answers

312 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 $, ...

**2**

votes

**3**answers

907 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 ...

**2**

votes

**3**answers

655 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 ...

**2**

votes

**1**answer

222 views

### octic K3s inside cubic 4-folds

From the Thesis of B.Hassett I seem to understand that a smooth cubic 4-fold $X$ containing a $\mathbb{P}^2$ should contain also a octic K3, but I cannot see a natural way by which this K3 octic could ...

**2**

votes

**1**answer

145 views

### Are singular rational curves on K3 surfaces rigid?

Let $S$ be a K3 surface over the complex numbers $\mathbb{C}$. If $C\subset S$ is a smooth rational curve, the normal bundle $N_{C/S}$ is isomorphic to $\mathbb{O}(-2)$ and thus $C$ is rigid. What ...