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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20
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1answer
846 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
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).
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 $M_{24}$ ...
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 ...
12
votes
1answer
505 views

Curves on K3 and modular forms

The paper of Bryan and Leung "The enumerative geometry of $K3$ surfaces and modular forms" provides the following formula. Let $S$ be a $K3$ surface and $C$ be a holomorphic curve in $S$ representing ...
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 $x,y,z$?
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 ...
10
votes
1answer
286 views

K3 surfaces that correspond to rational points of elliptic curves

In his work on mirror symmetry (http://arxiv.org/pdf/alg-geom/9502005v2.pdf) Igor Dolgachev has considered families of K3 surfaces of Picard rank at least 19 with the base given by $X_0(n)^+$, the ...
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 ...
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 ...
8
votes
0answers
347 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 ...
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 ...
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 ...
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 ...
7
votes
2answers
285 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
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 ...
7
votes
1answer
197 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 ...
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 ...
6
votes
2answers
187 views

adjacency matrix of a graph and lines on quartic surfaces

Suppose you are given a smooth quartic surface $X$ in $\mathbb P^3$. I would like to find an upper bound for the number of lines on $X$ in the case that there is no plane intersecting the curves in ...
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?
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? ...
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 ...
5
votes
1answer
207 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
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: ...
5
votes
1answer
224 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
0answers
138 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
0answers
157 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
2answers
328 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
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 ...
4
votes
2answers
304 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
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.
4
votes
1answer
348 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 ...
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 ...
4
votes
1answer
137 views

Euler number for base change of a K3 surface

Suppose you have a K3 surface $S$ containing a smooth rational curve $C$ and suppose you have an elliptic fibration $S \rightarrow \mathbb P^1$ that restricts to a morphism $C \rightarrow \mathbb P^1$ ...
4
votes
1answer
387 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 ...
4
votes
2answers
231 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 ...
4
votes
0answers
92 views

On the cohomology ring of the Hilbert scheme of points on k3 or abelian surfaces

There are many results on the cohomology of the Hilbert scheme of points of a surface. Gottsche calcaluted the Betti numbers and Nakajima got the generators of the cohomology. Also there are results ...
4
votes
0answers
282 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
2answers
296 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
1answer
631 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
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 ...
3
votes
2answers
280 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
1answer
439 views

K3 over fields other than C?

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

Spectral sequence associated to elliptic fibration degenerates?

Let $\phi:S\rightarrow \mathbb{CP}^1$ be an elliptic fibration of a K3 surface. When is the Leray spectral sequence associated to the fibration $E_2$-degenerate? Are there any good criteria for the ...
3
votes
1answer
215 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
1answer
189 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
1answer
315 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
1answer
100 views

When will the mirror of a K3 surface be an elliptic K3?

Let $f:Y\rightarrow\mathbb{P}^1$ be an elliptic $K3$ surface, then the holomorphic 2-form $\Omega_Y$ vanishes when restricted to an elliptic fiber $f^{-1}(b)$ with $b\in\mathbb{P}^1$. After a ...
3
votes
1answer
568 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 ...