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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Picard number of Hilbert modular surfaces

Hilbert modular surfaces are discussed in various papers by Hirzebruch. Following [HZ] (and their notations), one obtains Hilbert modular surfaces by the action of Hilbert modular group on $\mathcal{H}...
4 votes
2 answers
290 views

Is the set of points on an abelian surface which project to rational points on the Kummer surface a subgroup?

Let $C$ be a hyperelliptic curve of genus 2 defined over $\mathbb{Q}$, let $J$ be its Jacobian, and let $X$ be the Kummer surface associated to $J$ (i. e. $X$ is the singular Kummer surface which ...
6 votes
0 answers
264 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-...
2 votes
1 answer
316 views

$K3$ 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 ...
2 votes
0 answers
83 views

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 ...
7 votes
0 answers
228 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 ...
1 vote
0 answers
128 views

complex K3 surfaces with automorphisms of given orders

Concerning complex K3 surfaces, there are various methods to show the non-existence of an automorphism of certain orders. The usually way is to investigate the action of the automorphism on the space $...
7 votes
1 answer
450 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 ...
1 vote
1 answer
228 views

One-dimensional family of complex algebraic K3 surfaces

Let $X$ be an algebraic complex K3 surface, we know that $X$ is deformation equivalent to a smooth quartic surface or more generally a K3 surface with Picard number $1$ (a very general K3 surface in ...
5 votes
0 answers
113 views

Isotopy classes of $CP^1$ in 4-manifolds

Let $S_1$, $S_2$ be homologous embedded 2-spheres in a compact smooth 4-manifold. Under which additional conditions are they smoothly isotopic? I am interested in the state of the art picture when $...
3 votes
0 answers
135 views

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\...
3 votes
0 answers
247 views

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 ...
3 votes
0 answers
235 views

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 ...
6 votes
1 answer
330 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 ...
3 votes
0 answers
91 views

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 ...
2 votes
0 answers
148 views

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 ...
2 votes
0 answers
169 views

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. \...
2 votes
0 answers
245 views

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 vote
0 answers
135 views

Obstruction in construction of some lattices, related with $K3$ surfaces

I am considering a certain $K3$ surface that is lattice-polarized in two ways. This leads to the following simple problem in lattice theory: (Let me borrow notations for lattice from Ch.14 of this ...
2 votes
1 answer
168 views

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,\...
1 vote
1 answer
235 views

$K3$ surfaces in $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$

I am considering $K3$ surfaces in $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$ with an automorphism that preserves an ample divisor class. For an automorphism $\rho$ of a $K3$ surface, let ${\...
6 votes
0 answers
347 views

Quantifying the failure of geometric formality in K3 surfaces

It is known that K3 surfaces are never geometrically formal [1]. That is, the wedge product of two harmonic forms on an arbitrary K3 surface is in general not harmonic, or equivalently, the space $\...
1 vote
1 answer
371 views

Understand the Mukai vector

Let $S$ be a K3 surface and $h:=c_1(i^*\mathcal{O}_{\mathbb{P^3}}(1))$, then we can compute that $c_1(S)=0,c_2(S)=6h^2$. Hence \begin{align} \sqrt{\text{td}(S)}=1+\frac{c_2(S)}{24}=1+\frac{1}{4}h^2 \...
2 votes
0 answers
179 views

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 ...
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 \...
3 votes
1 answer
230 views

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 ...
5 votes
1 answer
307 views

K3 surfaces with small Picard number and symmetry

I am looking for examples of K3 surfaces that have a low Picard rank and at least one holomorphic involution. Here, low is no mathematically precise concept. I want to do computations with Monad ...
5 votes
1 answer
799 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 ...
3 votes
0 answers
188 views

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 ...
1 vote
0 answers
111 views

Does the Mukai's lemma hold for non-algebraic $K3$ surfaces?

In Huybrechts' book Fourier-Mukai Transforms in Algebraic Geometry I found the following result due to Mukai (Page 232, Lemma 10.6) Let $X$ and $Y$ be two $K3$ surfaces. Then the Mukai vector of any ...
1 vote
0 answers
125 views

Global section of unstable vector bundles comparing with (semi)stable vector bundles

Let $X$ be a smooth projective variety, say it is a K3 surface. Fix a Chern character $(ch_0,ch_1,ch_2)$. Then if we consider the global sections of all the possible (semi)stable vector bundles and ...
6 votes
0 answers
176 views

Find an explicit quasi-smooth embedding $X_{38} \subset \mathbb P(5, 6, 8, 19)$

This question is not quite about research-level mathematics, so I apologize for bringing it here. I asked it in Math.SE first, but I got no answers, and only a suggestion to ask it here. Consider the ...
2 votes
0 answers
96 views

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 ...
9 votes
2 answers
727 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 ...
2 votes
2 answers
949 views

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 ...
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 (\...
2 votes
1 answer
257 views

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 votes
0 answers
152 views

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 ...
2 votes
1 answer
117 views

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 $X$...
1 vote
0 answers
133 views

Automorphic representation of weight 3 eigenforms

Let $f$ be a weight 3 eigenform with rational Fourier coefficients. As shown by Elkies and Schutt, $f$ is associated to a singular K3 surface over $\mathbb{Q}$. A construction of Shioda and Inose ...
2 votes
0 answers
191 views

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 $X$ ...
7 votes
0 answers
204 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.
4 votes
1 answer
260 views

Kummer surfaces which are not projective

This is a question from an online note. Let $A$ be a two-dimensional $\mathbb C$-torus. And there is an involution on $A$: $A\to A, x\mapsto -x$. The action has 16 fixed points. Let $Y:=A/\{\pm1\}$, ...
5 votes
2 answers
599 views

Are singular rational curves on K3 surfaces rigid?

Let $S$ be a K3 surface over the complex numbers $\mathbb{C}$. If $C\subset S$ is a smooth rational curve, the normal bundle $N_{C/S}$ is isomorphic to $\mathbb{O}(-2)$ and thus $C$ is rigid. What ...
5 votes
2 answers
522 views

density of singular K3 surfaces

By singular K3 I mean a smooth complex K3 with Néron-Severi rank equal to 20. Are singular K3 surfaces dense in the moduli space of polarized K3 surfaces?
4 votes
0 answers
86 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 vote
0 answers
86 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 ...
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
1 answer
329 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 $. ...