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 $2$.

Filter by
Sorted by
Tagged with
1 vote
0 answers
57 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,\...
user avatar
  • 121
1 vote
0 answers
108 views

Automorphism group of a K3-surface

I am interested to know more about the automorphism group of a K3 surface, more specifically: is there any easy way to determine if it is infinite or not? The specific case I am looking at is the ...
user avatar
  • 143
2 votes
1 answer
181 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 ${\...
user avatar
  • 1,051
6 votes
0 answers
317 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 $\...
user avatar
1 vote
1 answer
113 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 \...
user avatar
2 votes
0 answers
149 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 ...
user avatar
  • 143
8 votes
3 answers
735 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 \...
user avatar
  • 19.3k
4 votes
1 answer
215 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 ...
user avatar
  • 1,051
5 votes
1 answer
234 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 ...
user avatar
2 votes
0 answers
61 views

Some questions about purely non-symplectic automorphisms of K3 surfaces and eigenspaces

I am reading this paper. Let $S$ be a (algebraic) K3 surface, an automorphism $\alpha_S\in \text{Aut}(S)$ of finite order $n:= |\alpha_S|$ is purely non-symplectic (of order n) if $\alpha^*_S(\...
user avatar
  • 165
5 votes
1 answer
433 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 ...
user avatar
3 votes
0 answers
138 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 ...
user avatar
  • 205
1 vote
0 answers
99 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 ...
user avatar
  • 8,079
1 vote
0 answers
80 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 ...
user avatar
  • 253
6 votes
0 answers
151 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 ...
user avatar
  • 183
2 votes
0 answers
83 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 ...
user avatar
  • 2,611
9 votes
2 answers
529 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 ...
user avatar
3 votes
1 answer
273 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 (\...
user avatar
2 votes
1 answer
180 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 ...
user avatar
  • 2,277
2 votes
0 answers
133 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 ...
user avatar
  • 6,169
2 votes
1 answer
103 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 $X$...
user avatar
  • 6,169
1 vote
0 answers
116 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 ...
user avatar
  • 167
2 votes
0 answers
161 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 $X$ ...
user avatar
  • 3,665
7 votes
0 answers
134 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.
user avatar
  • 193
4 votes
1 answer
190 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 $Y:=A/\{\pm1\}$, ...
user avatar
  • 301
5 votes
2 answers
484 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?
user avatar
  • 3,665
4 votes
0 answers
78 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 ...
user avatar
1 vote
0 answers
75 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 ...
user avatar
1 vote
1 answer
199 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 $. ...
user avatar
3 votes
1 answer
290 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 \...
user avatar
  • 9,966
2 votes
0 answers
137 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 = ...
user avatar
3 votes
0 answers
426 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 ...
user avatar
  • 679
5 votes
0 answers
158 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 ...
user avatar
7 votes
1 answer
487 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-...
user avatar
8 votes
0 answers
314 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. ...
user avatar
  • 433
5 votes
0 answers
207 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 ...
user avatar
2 votes
0 answers
255 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-...
user avatar
2 votes
0 answers
81 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, ...
user avatar
2 votes
0 answers
214 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 ...
user avatar
2 votes
0 answers
195 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 ...
user avatar
7 votes
1 answer
374 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 $...
user avatar
8 votes
0 answers
209 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 ...
user avatar
  • 111
9 votes
1 answer
233 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(...
user avatar
  • 769
2 votes
1 answer
262 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 $...
user avatar
3 votes
0 answers
234 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 ...
user avatar
8 votes
0 answers
626 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^...
user avatar
  • 81
6 votes
0 answers
189 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 $S=R\times_{\Bbb P^1} \Bbb P^1 \to \Bbb P^1$, ...
user avatar
2 votes
0 answers
112 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 ...
user avatar
  • 1,033
2 votes
0 answers
196 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 ...
user avatar
  • 1,033
9 votes
2 answers
678 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 ...
user avatar
  • 1,033