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 $2$.
180
questions
3
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Intuition behind RDP (Rational Double Points)
Let $S$ be a surface (so a $2$-dimensional proper $k$-scheme) and $s$ a singular point which is a rational double point.
One common characterisation of a RDP is that under sufficient conditions there ...
5
votes
0
answers
165
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 ...
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-...
8
votes
0
answers
339
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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. ...
5
votes
0
answers
224
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 ...
2
votes
0
answers
331
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
0
answers
90
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
$$
i\!: E_1\times E_2 \to E_1\times E_2,\qquad (x_1, y_1, x_2, ...
2
votes
0
answers
268
views
Elliptic fibrations on the Fermat quartic surface
Consider the Fermat quartic surface
$$
x^4 + y^4 + z^4 + t^4 = 0
$$
over an algebraically closed field $k$ of characteristics $p$, where $p \equiv 3$ ($\mathrm{mod}$ $4$).
Is there the full ...
2
votes
0
answers
206
views
Is the Fermat quartic surface a generalized Zariski surface?
Consider the Fermat quartic surface $$F\!: x^4 + y^4 + z^4 + t^4 = 0$$ over an algebraically closed field $k$ of odd characterstics $p$. Shioda proved that for $p=3$ this surface is a generalized ...
7
votes
1
answer
411
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Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?
Let $E_i\!: y_i^2 = x_i^3 + a_4x_i + a_6$ be two copies ($i = 1$, $2$) of a supersingular elliptic curve over a finite field $\mathbb{F}_{p^2}$, for odd prime $p > 3$. Consider the Kummer surface $...
8
votes
0
answers
253
views
Fundamental group of moduli space of K3's
According to Rizov (https://arxiv.org/abs/math/0506120), the moduli stack of primitively polarized K3 surfaces of degree 2d $\mathcal{M}_{d}$ is a Deligne-Mumford stack over $\mathbb{Z}$. I'm looking ...
8
votes
1
answer
253
views
Primitivity of subgroups in the Picard groups of anticanonical $K3$ surfaces
Let $X$ be a smooth projective threefold with $h^{0,1}(X) = h^{0,2}(X)=0$ that has a smooth anticanonical section $D$.
Then $D$ is necessarily a $K3$ surface.
Consider a subgroup
$$Pic_X(D) = i^*(Pic(...
2
votes
1
answer
265
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
0
answers
264
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 ...
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^...
6
votes
0
answers
201
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
0
answers
118
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 ...
2
votes
0
answers
241
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 ...
9
votes
2
answers
763
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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
0
answers
100
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
0
answers
328
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
0
answers
265
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
2
answers
317
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
0
answers
287
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
0
answers
550
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
\...
14
votes
1
answer
950
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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
0
answers
173
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
0
answers
146
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
1
answer
569
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 ...
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 ...
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 ...
66
votes
1
answer
2k
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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 ...
5
votes
1
answer
745
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 ...
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 ...
33
votes
1
answer
1k
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$\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
1
answer
364
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
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 ...
6
votes
0
answers
199
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 $...
3
votes
2
answers
810
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 $\mathbb{P}^3$.
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-...
5
votes
2
answers
1k
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
0
answers
892
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
1
answer
176
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 $d$-...
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/...
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 ...
5
votes
1
answer
692
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
1
answer
429
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
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 ...
4
votes
0
answers
261
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 ...
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 ...