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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4
votes
0answers
69 views

Is there a way to calculate the Picard $\mathbb{F}_q$-number of an (rational or K3) elliptic surface?

Consider a finite field $\mathbb{F}_{q}$ and an elliptic surface $$ \mathcal{E}\!: y^2 + a_1(t)xy + a_3(t)y = x^3 + a_2(t)x^2 + a_4(t)x + a_6, $$ where $a_i(t) \in \mathbb{F}_{q}[t]$. Is there a way ...
1
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0answers
67 views

Picard numbers of isogenous K3 surfaces over a non-closed field

Let $S_1, S_2$ be K3 surfaces defined over a field $k$ and $\phi\!: S_1 \dashrightarrow S_2$ a dominant rational $k$-map (so-called isogeny). It is known that $\rho(S_1) = \rho(S_2)$ for the complex ...
1
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1answer
105 views

Fixed part of a line bundle on a K3 surface

This question comes from Huybrechts' lecture notes on K3 surfaces, more specifically, chapter 2. Let $ X $ be a K3 surface (over an algebraically closed field $ k $) and $ L $ a line bundle on $ X $. ...
2
votes
1answer
214 views

(1/2) K3 surface or half-K3 surface: Ways to think about it?

I heard from string theorists thinking of the so-called "(1/2) K3 surface" or "half-K3 surface" as follows: Let $T^2 \times S^1$ be a 3-torus with spin structure periodic in all directions. $T^2 \...
0
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45 views

What are other elliptic $\mathbb{F}_2$-fibrations on the Kummer surface with an $\mathbb{F}_2$-section?

Consider the ordinary elliptic curves $$ E\!:y_1^2 + x_1y_1 = x_1^3 + 1,\qquad E^\prime\!: y_2^2 + x_2y_2 = x_2^3 + x_2^2 + 1 $$ over the field $\mathbb{F}_2$. They are quadratic twists to each other....
2
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0answers
133 views

Is there a way to explicitly find any rational $\mathbb{F}_p$-curve on the Kummer surface?

Consider a finite field $\mathbb{F}_p$ (where $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$), $\mathbb{F}_{p^2}$-isomorphic elliptic curves (of $j$-invariant $0$) $$ E\!:y_1^2 = ...
4
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225 views

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
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144 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 ...
6
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1answer
393 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
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305 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. ...
5
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181 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 ...
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0answers
151 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
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0answers
68 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
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159 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
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0answers
188 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
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1answer
337 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
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0answers
180 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
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1answer
224 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
256 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
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0answers
183 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
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501 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
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0answers
184 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
106 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
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0answers
161 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
625 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
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0answers
92 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
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0answers
282 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
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0answers
170 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
281 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
219 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
362 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
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1answer
648 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
150 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
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0answers
138 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
451 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
485 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
229 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 ...
61
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
554 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
709 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 ...
30
votes
1answer
895 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
287 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
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0answers
668 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
188 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 $...
6
votes
2answers
687 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 $\mathbb{P}^3$.
29
votes
1answer
1k 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
2answers
903 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
778 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
159 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
0answers
447 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/...