Questions tagged [k3-surfaces]

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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Is there an octonionic analog of the K3 surface, with implications for stable homotopy groups of spheres?

The infamous K3 surface has many constructions in many fields ranging from algebraic geometry to algebraic topology. Its many properties are well known. For this question I am really interested in the ...
Chris Schommer-Pries's user avatar
33 votes
1 answer
1k views

$\int_0^\infty x \, [J_0(x)]^5 \, dx$: source and context, if any?

QUESTION Numerical calculation with gp (first to the default 38-digit precision, then tripled) supports the conjecture that $$ \int_0^\infty x \, [J_0(x)]^5 \, dx = \frac{\Gamma(1/15) \, \Gamma(2/15) ...
Noam D. Elkies's user avatar
30 votes
1 answer
2k views

Enriques surfaces over $\mathbb Z$

Does there exist a smooth proper morphism $E \to \operatorname{Spec} \mathbb Z$ whose fibers are Enriques surfaces? By a theorem of, independently, Fontaine and Abrashkin, combined with the Enriques-...
Will Sawin's user avatar
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23 votes
2 answers
4k 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? ...
Jay's user avatar
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22 votes
1 answer
2k 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 ...
Tito Piezas III's user avatar
20 votes
4 answers
3k 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).
John R Ramsden's user avatar
19 votes
1 answer
770 views

Vector field on a K3 surface with 24 zeroes

In https://mathoverflow.net/a/44885/4177, Tilman points out that one can use a $K3$ surface minus the zeroes of a generic vector field to build a nullcobordism for $24[SU(2)]$. Given that a) this is a ...
David Roberts's user avatar
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19 votes
1 answer
2k 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 $M_{24}$ ...
Jeff Harvey's user avatar
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16 votes
4 answers
1k 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 ...
JSE's user avatar
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15 votes
1 answer
939 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 ...
IBazhov's user avatar
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14 votes
2 answers
2k 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 ...
36min's user avatar
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14 votes
1 answer
950 views

Rational curves on the Fermat quartic surface

Let $X$ be the Fermat quartic $x^4+y^4+z^4+w^4=0$ in $\mathbb P^3$. It is known that $X$ contains infinitely many $(-2)$-curves, that is, smooth rational curves. (One way to obtain in infinitely many ...
byu's user avatar
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14 votes
0 answers
530 views

Am I missing something about this notion of Mirror Symmetry for abelian varieties?

This is a continuation of my recent question: Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s. In the comments of the question, I was directed to the paper http://arxiv.org/abs/...
Simon Rose's user avatar
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12 votes
3 answers
1k 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 ...
Richard Montgomery's user avatar
12 votes
2 answers
2k 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 ...
Simon Rose's user avatar
  • 6,240
12 votes
1 answer
675 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 ...
Richard Montgomery's user avatar
12 votes
0 answers
698 views

Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system, $$x_1^2+x_2^2+\dots+x_n^2 = y^2$$ $$x_1^3+x_2^3+\dots+x_n^3 = z^3$$ and define the function, $$F(s_m) = x_1+x_2+\dots+x_n$$ For $n\geq3$, using an ...
Tito Piezas III's user avatar
11 votes
1 answer
400 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 $x,y,z$?
Hej's user avatar
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11 votes
1 answer
873 views

Non-algebraic K3 surfaces in characteristic $p$

I have a very naive question. Recall that over the field of complex numbers, there exist non-algebraic K3 surfaces. Namely, smooth non-projective simply connected compact complex surfaces with ...
Daniel Loughran's user avatar
11 votes
0 answers
782 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 \...
Rogelio Yoyontzin's user avatar
10 votes
1 answer
604 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 ...
Lev Borisov's user avatar
  • 5,166
9 votes
2 answers
763 views

Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold?

Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold? What I know so far is as follows: In this paper (https://arxiv.org/pdf/hep-th/9512195.pdf) by Verbitsky, it is claimed that ...
Mtheorist's user avatar
  • 1,135
9 votes
2 answers
751 views

Do singular fibers determine the elliptic K3 surface, generically?

General elliptic K3 surfaces. Consider K3 surfaces of Picard rank two with Neron-Severi lattice isomorphic to $$\left[\begin{array}{cc} 2d & t \\ t & 0 \end{array}\right]$$ for some positive ...
Evgeny Shinder's user avatar
8 votes
4 answers
2k 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 ...
user avatar
8 votes
3 answers
1k views

Seeking concrete examples of "generic" elliptic fibrations of K3 surfaces

For me a K3 surface will be a smooth complex projective variety of dimension 2 that is simply-connected and has trivial canonical bundle. Given a K3 surface $X$, an elliptic fibration $\pi \colon X \...
John Baez's user avatar
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8 votes
1 answer
2k 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 ...
Jay's user avatar
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8 votes
2 answers
453 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 ...
Lev Borisov's user avatar
  • 5,166
8 votes
1 answer
772 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 manifold,...
Simon Rose's user avatar
  • 6,240
8 votes
1 answer
253 views

Primitivity of subgroups in the Picard groups of anticanonical $K3$ surfaces

Let $X$ be a smooth projective threefold with $h^{0,1}(X) = h^{0,2}(X)=0$ that has a smooth anticanonical section $D$. Then $D$ is necessarily a $K3$ surface. Consider a subgroup $$Pic_X(D) = i^*(Pic(...
Basics's user avatar
  • 1,821
8 votes
1 answer
681 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 ...
user2013's user avatar
  • 1,653
8 votes
0 answers
339 views

Concrete example of $K3$ surfaces with Picard 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
  • 451
8 votes
0 answers
253 views

Fundamental group of moduli space of K3's

According to Rizov (https://arxiv.org/abs/math/0506120), the moduli stack of primitively polarized K3 surfaces of degree 2d $\mathcal{M}_{d}$ is a Deligne-Mumford stack over $\mathbb{Z}$. I'm looking ...
CKlevdal's user avatar
  • 111
8 votes
0 answers
751 views

Hirzebruch $\chi_y$ genus of a K3 surface

I would like to compute the $\chi_y$ genus of an elliptically fibered K3 surface. For $X$ a compact algebraic manifold, Hirzebruch's $\chi_y$ genus is defined as $\chi_y (X) = \sum_{p,q} (-1)^{p+q} h^...
michele's user avatar
  • 81
8 votes
0 answers
395 views

Mirror symmetry for polarized abelian surfaces and Shioda-Inose K3s

It is well known (cf. Dolgachev) that there is a beautiful notion of mirror symmetry for lattice-polarized K3 surfaces. That is, if we are given a rank $r$ lattice $M$ of signature $(1, r - 1)$ and a ...
Simon Rose's user avatar
  • 6,240
7 votes
3 answers
923 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 direct?...
Jay's user avatar
  • 583
7 votes
2 answers
854 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 ...
Keerthi Madapusi's user avatar
7 votes
1 answer
466 views

Do non-projective K3 surfaces have rational curves?

Define a compact Kähler surface $X$ to be a K3 surface if $X$ is simply connected, $K_X \simeq \mathcal{O}_X$, and $h^{0,1}=0$. If $X$ is projective, then a theorem typically attributed to Bogomolov ...
AmorFati's user avatar
  • 1,349
7 votes
1 answer
567 views

Discriminant locus of elliptic K3 surfaces

Given a complex elliptic K3 surface $\pi\colon X\rightarrow \mathbb P^1$, its discriminant locus is the divisor $$D = \sum_{i = 1}^s n_i P_i$$ on $\mathbb P^1$ such that $n_i$ is equal to the Euler-...
Davide Cesare Veniani's user avatar
7 votes
1 answer
292 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?
Anne's user avatar
  • 71
7 votes
1 answer
411 views

Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?

Let $E_i\!: y_i^2 = x_i^3 + a_4x_i + a_6$ be two copies ($i = 1$, $2$) of a supersingular elliptic curve over a finite field $\mathbb{F}_{p^2}$, for odd prime $p > 3$. Consider the Kummer surface $...
Dimitri Koshelev's user avatar
7 votes
0 answers
236 views

K3 surfaces with no −2 curves

I seem to remember that a K3 surface with big Picard rank always has smooth rational curves. This question is equivalent to the following question about integral quadratic lattices. Let us call a ...
Misha Verbitsky's user avatar
7 votes
0 answers
214 views

Does there exists a compact Ricci-flat K3 surface whose metric tensor is expressed in explicit formula?

Yau's theorem showed the existence. But I had difficulty finding examples other than complex tori. Any information will be appreciated.
clvolkov's user avatar
  • 193
6 votes
1 answer
889 views

Reference request: Generic k3 surface has Picard number 1

I keep running into the statement that "the generic k3 surface has Picard rank 1". For instance the answer of this question (end) and this paper (following Example 1.1) or this paper (proof ...
user147163's user avatar
6 votes
1 answer
748 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.
user15720's user avatar
6 votes
1 answer
924 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 ...
Michel's user avatar
  • 375
6 votes
2 answers
395 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 ...
Davide Cesare Veniani's user avatar
6 votes
1 answer
378 views

automorphism group of K3 surfaces

It is known that smooth complex hypersurfaces with degree bigger than 2 and dimension bigger than 1 have finite automorphism groups, except for K3 surfaces. But the group of polarised automorphisms ...
user avatar
6 votes
1 answer
255 views

Loci in the moduli space of K3 surfaces associated to lattices

The moduli space of K3 surfaces forms a 20-dimensional family with countably many 19-dimensional components $M_d$ corresponding to the polarized K3s $(X,L)$ with $L^2=d$. The moduli space $M_d$ has a ...
gsvr's user avatar
  • 235
6 votes
1 answer
833 views

SYZ mirror symmetry for K3 surfaces

My question is essentially related to this post, but let me formulate it again. Let $f:S \rightarrow \mathbb{P}^1$ be an elliptic fibration, then this can be a SLAG fibration with respect to another ...
Vladhagen's user avatar
6 votes
0 answers
268 views

Exceptional quartic K3 surfaces

Recall that a $K3$ surface is called exceptional if its Picard number is 20. The Fermat quartic $K3$ surface in $\mathbb P^3$ is exceptional. My question is, Are there infinitely many non-...
Basics's user avatar
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