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 $2$.
178
questions
3
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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 ...
- 2,078
3
votes
1
answer
368
views
Mordell–Weil rank of some elliptic $K3$ surface
Consider a finite field $\mathbb{F}_q$ such that $q \equiv 1 \pmod3$ (i.e., $\omega \mathrel{:=} \sqrt[3]{1} \in \mathbb{F}_q$ for $\omega \neq 1$) and an element $b \in (\mathbb{F}_q^*)^2 \setminus (\...
- 1,801
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votes
1
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420
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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 ...
- 3,055
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votes
1
answer
586
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Kuga-Satake with p-adic methods
Is it possible to construct the Kuga-Satake abelian variety attached to a K3 surfaces (over a local field) only using p-adic methods?
If the K3 surface is defined over a local field, the Kuga-...
3
votes
1
answer
230
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Irrationality of some threefolds
Consider a smooth projective threefold $\overline W$, constructed in section 4 of this paper.
This threefold is a resolution of singularities of the quotient of a product of a K3 surface and $\mathbb ...
- 1,821
3
votes
0
answers
135
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Moduli space with exceptional Mukai vector and tangent spaces at strictly semistable bundles
Assume we work (over $\mathbb{C}$) on a polarized K3 surface $(X,L)$ with a line bundle $M$ on $X$ such that $M^2=-6$ and $ML=0$ as well as $h^0(M)=h^2(M)=0$ and thus $h^1(M)=1$.
Then $E=\mathcal{O}_X\...
- 1,015
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0
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247
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K3 surfaces in Fano threefolds
By K3 surfaces and Fano threefolds, I mean smooth ones.
If a K3 surface $S$ is an anticanonical section of a Fano threefold $V$ of Picard rank one (hence, $Pic(V)=\mathbb Z H_V$ for some ample divisor ...
- 1,821
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0
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235
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A K3 cover over a Del Pezzo surface
Let $V \rightarrow \mathbb P^2$ be the blow-up at two distinct points. ($V$ is a Del Pezzo surface of degree 7.)
Choose a smooth curve $C$ from the linear system $|-2K_V|$ and let $S \rightarrow V$ be ...
- 1,821
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Are supersingular K3 surfaces unirational?
There is a conjecture due to Artin, Rudakov, Shafarevich, Shioda that supersingular K3 surfaces over a finite field are unirational. This paper claims to prove this result but it has had a recent ...
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Toric degeneration of Kummer Surface
I am wondering if there are any explicit examples of a toric degeneration of a Kummer surface (e.g. as a family of projective varieties say), and what the central fibre can look like? (I am working ...
- 205
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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 ...
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0
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262
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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 ...
- 151
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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
\...
- 1,801
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Topology of K3 as a sum of two abelian fibrations.
Let $E$ be a blow-up of $\mathbb{P}^2$ at 9-points in the bases locus of a pencil of elliptic curves (A $T^2$ fibration over $S^2$).
K3 surfaces is obtained by removing a fiber from two copies of $E$ ...
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390
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The surface $ x^2 y^2 + 1 = (x^2 + y^2) z^2 $
Hi, I'm trying to find all rational points on the surface of the title, in connection with the Euler Brick (AKA Rational Box) problem.
This surface is equivalent to $ x^2 z^2 - 1 = (x^2 - z^2) y^2 $, ...
- 1,495
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votes
3
answers
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cuspidal curves in K3 surfaces
Let $S\subset\mathbb{P}^g$ be a smooth polarized K3 surface of genus $g$. I am interested in the existence of certain cuspidal curves in the linear system. We know a general hyperplane section $H\cap ...
- 103
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votes
2
answers
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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 it sufficient in our case to check this for ...
- 761
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answer
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Very general quartic hypersurface in $\mathbb{P}^3$ has Picard group $\mathbb{Z}$
I am looking for a reference from which I can cite the following statement:
The Picard group of a very general quartic hypersurface $X\subset\mathbb{P}^3$ is generated by the class of a hyperplane ...
- 2,863
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votes
1
answer
329
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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 $. ...
- 846
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1
answer
330
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Sheaves with zero Chern classes on a $K3$ surface.
Let $S$ be a $K3$ surface. Is it true that any sheaf on $S$ with zero Chern classes is isomorphic to $\mathcal{O}_S^{\oplus n}$ for some $n$? If not, do you have any counterexample?
- 753
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Picard/cohomology lattice of surfaces of low degree in $\mathbb P^3$
Let $S_{d>3}\subset\mathbb{P}^3_{\mathbb{C}}$ be a smooth surface of degree $d$. What is known (where to read?) about the Picard/cohomology lattice for small d?
e.g. for $d=4$ the cohomology ...
- 2,202
2
votes
1
answer
117
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Fixed locus in the linear system associated to the ramification locus of a K3 double cover of a Del Pezzo surface
Let $X$ be a (smooth) del Pezzo surface over $\mathbb{C}$. Let $\Delta_0$ be a (smooth irreducible) generic curve in the linear system $|-2K_X|$. Let $\rho : S \rightarrow X$ be the double cover of $X$...
- 7,100
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1
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461
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Minimal semistable model for K3-surfaces.
I wonder if a semistalbe K3 surface over a $p$-adic field has a minimal semistable model. I guess yes but I do not find any reference.
Also, if we have a semistable K3 surface with a log structure, ...
2
votes
1
answer
316
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$K3$ surfaces can't be uniruled
Let $S$ be a uniruled surface, ie admits a dominant map $ f:X \times \mathbb{P}^1$. Why then it's canonical divisor $\omega_X$ cannot be trivial? Motivation: I want to understand why $K3$ surfaces ...
- 5,716
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votes
1
answer
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Concrete descriptions of $S^1$-bundles over smooth manifold $Y$ underying a K3 surface
Let $Y$ be the smooth manifold underlying a K3 surface. As a manifold, $Y$ is diffeomorphic to $\{[x_0:x_1:x_2:x_3]\in\mathbb{C}P^3\colon X_0^4+x_1^4+X_2^4+X_3^4=1\}$. It is well known that $H^2(Y,\...
- 133
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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 $...
- 107
2
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1
answer
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Weyl group of a K3 surface
I am wondering wether the action of the Weyl group $W_X$ of a K3 surface $X$ is transitive on the sets of curves of fixed genus.
Suppose $W_X$ is non-trivial. Given two curves $C,C'$ of genus $g\...
- 761
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votes
1
answer
397
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Elliptic fibration of K3 surface
Let $X$ be a K3 surface with $U(k) \subset NS(X)$ for some $k$. Here $U$ is the hyperbolic lattice. Can one conclude that $X$ admits an elliptic fibration? If my memory serves, I read somewhere that ...
- 21
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answer
460
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Isometry of K3 surface.
Let $S$ be a K3 surface and $\iota$ be anti-symplectic involution of $S$. Suppose that $g$ is a Kahler-Einstein metric on $S$. My question is;
Why $\iota$ is an isometry of $S$ with respect to $g$?...
- 21
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0
answers
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Explicit Lagrangian fibrations of a K3 surface
I would like to look at the behaviour of the fibres of a Lagrangian fibration (such that at least some fibres are not special Lagrangian) $X\to\mathbb{CP}^1$ under the mean curvature flow (in relation ...
- 355
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0
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Automorphisms of finite order on $K3$ surfaces
Is there a $K3$ surface (algebraic, complex) that has infinitely many automorphisms of finite order?
Many K3 surfaces have infinite automorphism groups.
In particular, all K3 surfaces of Picard ...
- 1,821
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0
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169
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Automorphisms of a K3 surface
I was studying the following algebraic surface in $\mathbb{P}^5$ defined by the following three quadrics:
\begin{cases}
x^2 + xy + y^2=w^2\\
x^2 + 3xz + z^2=t^2\\
y^2 + 5yz + z^2=s^2.
\...
- 585
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0
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245
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Example of a K3 surface with two non-symplectic involutions
$\DeclareMathOperator\Pic{Pic}$Let $X$ be a K3 surface (algebraic, complex). An involution $\sigma:X\rightarrow X$ is called non-symplectic if it acts as multiplication by $-1$ on $H^{2,0}(X)=\Bbb{C}\...
- 1,821
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rational curves over K3 surfaces over $\mathbb{Q}$
There are many partial results towards the following conjecture:
Every projective K3 surface over an algebraically closed field contains infinitely many integral rational curves.
My question is: is ...
- 585
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0
answers
96
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Amoeba for a K3 surface in $\mathbb {CP}^3$
Let $X=X_\Delta$ be the toric variety associated to a reflexive polyhedron $\Delta$.
Consider a Calabi-Yau hypersurface $Y\subset X$, and the image of $Y$ under the moment map $\mu:X\to \Delta$ has ...
- 2,709
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0
answers
152
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Degree $4$ curves on K3 double covers of Del-Pezzo surfaces
Let $S$ be a smooth del-Pezzo surface and $\pi : X \longrightarrow S$ be the double cover of $S$ ramified in a smooth section of $-2K_S$. Going through the classification of del-Pezzo surfaces, one ...
- 7,100
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0
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191
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2 K3s and cubic fourfolds containing a plane
Two K3 surfaces show up when talking about cubic fourfolds containing a plane. Let $P\subset X\subset \mathbb{P}^5$ be the plane inside the cubic. Since $P$ is cut out by 3 linear equations then $X$ ...
- 3,697
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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 $0$)
$$
E\!:y_1^2 = ...
- 1,801
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0
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318
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Relation between Beauville-Bogomolov form and Intersection Product on Hilbert scheme of K3 surfaces
I am learning about Hilbert scheme of points $S^{[n]}$ on projective K3 surfaces S. Since these are hyperkähler varieties, the second cohomology $H^2(S^{[n]},\mathbb{Z})$ is endowed with the non-...
- 175
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0
answers
85
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The quotient of a superspecial abelian surface by the involution
Let $E_i\!: y_i^2 = f(x_i)$ be two copies of a supersingular elliptic curve over a field of odd characteristics. Consider the involution
$$
i\!: E_1\times E_2 \to E_1\times E_2,\qquad (x_1, y_1, x_2, ...
- 1,801
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votes
0
answers
259
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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 ...
- 1,801
2
votes
0
answers
204
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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 ...
- 1,801
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0
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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 ...
- 1,105
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votes
0
answers
236
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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 ...
- 1,105
2
votes
0
answers
100
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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 ...
- 8,637
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votes
0
answers
321
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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 ...
user21574
2
votes
0
answers
146
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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, \,\...
- 1,493
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votes
0
answers
884
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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). ...
- 600
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0
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131
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Coherent systems on K3 surfaces
Does anyone know whether the theory of coherent systems on $K3$ surfaces has been studied and, if yes, can you give me a reference? In particular, is there an analogue of Gieseker stability and of ...
- 753
2
votes
2
answers
317
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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 $k$ (...
- 1,801