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 $2$.
4
votes
0answers
120 views
Explicit Enriques involutions on the Fermat quartic surface
Let $X$ be the complex Fermat quartic surface defined by the polynomial $x^4+y^4+z^4+w^4$.
By results of Sertöz, we know that the surface $X$ admits at least one Enriques involution, i.e. an ...
5
votes
1answer
317 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-...
3
votes
0answers
161 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. ...
4
votes
0answers
158 views
Vanishing cycles for elliptic fibration on K3 surface?
Let $X$ be an elliptic K3 surface (over $\mathbb{C}$). Assume we have an elliptic fibration on $X$ that only has $I_1$ singular fibers.
If we fix a smooth fiber $F$ of such a fibration and a ...
1
vote
0answers
104 views
Relation between Beauville-Bogomolov form and Intersection Product on Hilbert scheme of K3 surfaces
I am learning about Hilbert scheme of points $S^{[n]}$ on projective K3 surfaces S. Since these are hyperkähler varieties, the second cohomology $H^2(S^{[n]},\mathbb{Z})$ is endowed with the non-...
2
votes
0answers
55 views
The quotient of a superspecial abelian surface by the involution
Let $E_i\!: y_i^2 = f(x_i)$ be two copies of a supersingular elliptic curve over a field of odd characteristics. Consider the involution
$$
i\!: E_1\times E_2 \to E_1\times E_2,\qquad (x_1, y_1, x_2, ...
2
votes
0answers
124 views
Elliptic fibrations on the Fermat quartic surface
Consider the Fermat quartic surface
$$
x^4 + y^4 + z^4 + t^4 = 0
$$
over an algebraically closed field $k$ of characteristics $p$, where $p \equiv 3$ ($\mathrm{mod}$ $4$).
Is there the full ...
2
votes
0answers
173 views
Is the Fermat quartic surface a generalized Zariski surface?
Consider the Fermat quartic surface $$F\!: x^4 + y^4 + z^4 + t^4 = 0$$ over an algebraically closed field $k$ of odd characterstics $p$. Shioda proved that for $p=3$ this surface is a generalized ...
7
votes
1answer
296 views
Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?
Let $E_i\!: y_i^2 = x_i^3 + a_4x_i + a_6$ be two copies ($i = 1$, $2$) of a supersingular elliptic curve over a finite field $\mathbb{F}_{p^2}$, for odd prime $p > 3$. Consider the Kummer surface $...
8
votes
0answers
151 views
Fundamental group of moduli space of K3's
According to Rizov (https://arxiv.org/abs/math/0506120), the moduli stack of primitively polarized K3 surfaces of degree 2d $\mathcal{M}_{d}$ is a Deligne-Mumford stack over $\mathbb{Z}$. I'm looking ...
9
votes
1answer
211 views
Primitivity of subgroups in the Picard groups of anticanonical $K3$ surfaces
Let $X$ be a smooth projective threefold with $h^{0,1}(X) = h^{0,2}(X)=0$ that has a smooth anticanonical section $D$.
Then $D$ is necessarily a $K3$ surface.
Consider a subgroup
$$Pic_X(D) = i^*(Pic(...
2
votes
1answer
250 views
Common gerbes over two K3 surfaces
Let $X$ and $Y$ be K3 surfaces over the complex numbers.
Under what assumptions, do there exist
a finite group $G_X$
a finite group $G_Y$
a $G_X$-gerbe $\mathcal{X}\to X$ (for the fppf topology)
a $...
3
votes
0answers
130 views
Are unirational K3 surfaces defined over finite fields?
Is every supersingular (thus unirational for ${\rm char }\ k = p\geq 5$, from Liedtke) $K3$ surface defined over a finite field? I guess this is true for Kummer surfaces, for example, since ...
7
votes
0answers
390 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^...
6
votes
0answers
174 views
Produce supersingular K3 from rational elliptic surfaces
Given a rational elliptic surface $R \to \Bbb P^1$, is there a way to know if there exists a supersingular K3 surface that arises as a base curve change $S=R\times_{\Bbb P^1} \Bbb P^1 \to \Bbb P^1$, ...
2
votes
0answers
100 views
Is there a hyperkaehler manifold whose mirror is the total space of a tangent/cotangent bundle?
I am looking for an example of a hyperkaehler manifold $Y$ whose mirror is the total space of a tangent bundle $TX$ or a cotangent bundle $T^*X$, where $X$ can be any Riemannian manifold.
Is such a ...
1
vote
0answers
135 views
Is the mirror of a noncompact hyperkaehler manifold also hyperkaehler?
This is essentially a follow-up question from 'Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold?'. Verbitsky's theorem in (https://arxiv.org/pdf/hep-th/9512195.pdf) says that ...
8
votes
2answers
601 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 ...
2
votes
0answers
89 views
Could we construct an inverse transform for the equivalence $D^b(X)\to D^b(M)$ between a K3 surface and its moduli space of semistable sheaves?
Let $X$ be a K3 surface and fix an ample line bundle on $X$. Let $v\in \widetilde{H}(X,\mathbb{Z})$ be a Mukai vector and $M(v)$ be the moduli space of semi-stable coherent sheaves on $X$ with Mukai ...
2
votes
0answers
259 views
Stability notion to smoothing varieties under a flat deformation with a smooth total space
Is there any stability notion that led to an algebraic variety be smoothable in general for Fano varieties or for Calabi-Yau varieties?
Note that Friedman found a nesessary condition that $X$ to be ...
0
votes
0answers
134 views
Classification of Elliptic singularity
For a $K_3$ surface $X$, if there exists a holomorphic surjective map $X\to \mathbb P^1$, with elliptic fibres, i.e. for any generic point on $\mathbb P^1$ whose fiber is diffeomorphic to a torus $\...
2
votes
2answers
253 views
What are sufficient and necessary conditions to be a generalized Zariski surface over a finite field?
Let $X$ be an absolutely irreducible reduced surface over a finite field $k$ of characteristic $p$. What are sufficient and necessary conditions for $X$ to be a generalized Zariski surface over $k$ (...
4
votes
0answers
201 views
What is the Artin invariant of an elliptic supersingular K3 surface?
Let $X$ be a supersingular K3 surface over an algebraically closed field $k$ of positive characteristic $\!p$. Artin proved in the paper https://eudml.org/doc/81948 that the determinant $\mathrm{disc}(...
3
votes
0answers
293 views
The Jacobian surface of an elliptic surface
Let $\mathcal{X}$ be an elliptic surface over $\mathbb{P}^1$ without a section and let $\mathcal{J}$ be an elliptic surface over $\mathbb{P}^1$ with a section. Assume we have the commutative diagram
\...
13
votes
1answer
541 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 ...
5
votes
0answers
147 views
Is a Kummer surface over an finite field $\mathbb{F}_q$ supersingular iff $\mathbb{F}_q$-unirational?
Let $A$ be an abelian surface over an finite field $\mathbb{F}_q$. In particular, I am interested in the case when $A$ is a Jacobian variety. Is the Kummer surface $K_A/\mathbb{F}_q$ Shioda-...
2
votes
0answers
121 views
Open Period Integrals of Elliptically Fibered K3 surfaces
Let M be the period domain for elliptic K3 surfaces $(X,\Omega)$ with a holomorphic two-form. Denote the fiber class $f$. Then $$M=\{\Omega\in f^\perp\otimes \mathbb{C}\,:\, \Omega\cdot \Omega=0, \,\...
5
votes
1answer
387 views
Is the automorphism group of a Calabi-Yau variety an arithmetic group
Let $X$ be a smooth projective variety over the complex numbers with trivial canonical bundle. Suppose that $X$ is Calabi-Yau.
Is the automorphism group of $X$ an arithmetic group?
What if $X$ is a ...
17
votes
1answer
455 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 ...
6
votes
1answer
214 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 ...
58
votes
1answer
1k views
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 ...
5
votes
1answer
497 views
Training towards research on k3 surfaces
I am a graduate student learning basic algebraic geometry (from Hartshorne, Shafarevich). I'm planning to work in k3 surfaces (arithmetic and geometric properties, in my guide's words). I came to know ...
10
votes
1answer
622 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 ...
27
votes
1answer
769 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) ...
1
vote
1answer
266 views
Infinitely many rational nt multisection in elliptic K3 surfaces by deformation theory
I'm trying to read this paper of Bogomolov and Tschinkel http://arxiv.org/pdf/math/9902092.pdf about potential density of rational points on elliptic K3 Surfaces.
I got quite stuck in Corollary 3.27 ...
12
votes
0answers
645 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 ...
6
votes
0answers
185 views
Are all these K3 surfaces supersingular?
Consider all the smooth K3 surfaces given by $X^4+W^2X^2+XW^3 = f(Y,Z,W)$ or $X^4+XW^3 = g(Y,Z,W)$ over $\mathbb F_{2}$ with $f$ or $g$ homogenous of degree 4. There are a lot of choices for $f$ and $...
5
votes
2answers
604 views
Is every algebraic $K3$ surface a quartic surface?
Algebraic $K3$ surface means the $K3$ surface admits an ample line bundle. So the question is equivalent to asking whether every algebraic $K3$ surface can be embedded in $\mathbb{P}^3$.
27
votes
0answers
960 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-...
4
votes
1answer
418 views
Singular models of K3 surfaces
Let us work over a ground field of characteristic zero. As is well-known, a K3 surface is a smooth projective geometrically integral surface $X$ whose canonical class $\omega_X$ is trivial and for ...
2
votes
0answers
721 views
Cubic fourfold and K3 surface: geometric constructions of Hodge isometry
Hodge structure on K3 surface (the middle line of Hodge diamond is 1 20 1) is similar to the Hodge structure of cubic fourfold (the middle line of Hodge diamond of primitive cohomology is 0 1 20 1 0). ...
1
vote
1answer
151 views
Characterization of $d$-gonal curves on a K3 surface
Let $X$ be a K3 surface and $C$ a curve on $X$. We say that $C$ is $d$-gonal if it admits a pencil of degree $d$ (and none of smaller degree).
I am wondering if there exist characterizations of $d$-...
13
votes
0answers
406 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/...
7
votes
0answers
345 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 ...
5
votes
1answer
533 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 ...
3
votes
1answer
287 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 ...
10
votes
1answer
507 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 ...
3
votes
0answers
215 views
Integral cohomology of the hilbert scheme of points on a k3
i'm reading the famous article "Varietes kahleriennes dont la premiere classe de chern est nulle" by Beauville, in particular proposition 6, which characterizes the second cohomology group for the ...
4
votes
1answer
489 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 ...
13
votes
1answer
780 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 ...
