All Questions
Tagged with k3-surfaces ag.algebraic-geometry
54
questions with no upvoted or accepted answers
14
votes
0
answers
520
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/...
- 6,242
11
votes
0
answers
774
views
Torelli-like theorem for K3 surfaces on terms of its étale cohomology
Is there a proof of a Torelli-like Theorem for a K3-surface over any field (non complex) in terms of its etale or crystalline cohomology?
For example: If $K\ne \mathbb{C} $ and $X\rightarrow \...
8
votes
0
answers
336
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. ...
- 451
8
votes
0
answers
715
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^...
- 81
8
votes
0
answers
392
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 ...
- 6,242
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 ...
- 9,045
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-...
- 1,821
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 ...
- 183
6
votes
0
answers
198
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 $S=R\times_{\Bbb P^1} \Bbb P^1 \to \Bbb P^1$, ...
- 151
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 $...
6
votes
0
answers
570
views
Semistable minimal model of a $K3$-surface and the special fibre
Suppose that $K$ is a $p$-adic field, that is a field of characteristic $0$ whose ring of integers is a complete discrete valuation ring $\mathcal O_K$ and with residue field $k$ (algebraic closed) of ...
6
votes
0
answers
299
views
Non minimal K3 surfaces as hypersurfaces of weighted projective spaces
I recently learnt that the hypersurface
$$
S:=(x^2+y^3+z^{11}+w^{66}=0) \subset \mathbb{P}(33,22,6,1)
$$
is birational to a K3 surface. This is surprising because the surface is quasi-smooth, well-...
- 933
6
votes
0
answers
850
views
Possible automorphism groups of a K3 surface
Which finite groups are automorphism groups of polarized K3 surfaces (let's say over ℂ)?
...and does the answer change is I remove "polarized"?
(polarized = equipped with an ample line bundle)
- 42.2k
5
votes
0
answers
164
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 ...
5
votes
0
answers
223
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 ...
user74900
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-...
- 1,801
5
votes
0
answers
196
views
on the automorphisms of the transcendental Hodge structure of a K3 surface
Let $S$ be a complex projective K3 surface and consider the sub-Hodge structure
$$
T(S) \subset H^2(S, \mathbb{Q})
$$ consisting of transcendental cycles. Let $\varphi$ be an automorphism of Hodge ...
- 51
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 ...
- 1,801
4
votes
0
answers
279
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}(...
- 1,801
4
votes
0
answers
260
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 ...
4
votes
0
answers
540
views
Singular fibers of an elliptic fibered K3 surface.
Let $f:S\rightarrow \mathbb{P}^1$ be an elliptic K3 surface. Assume that $\mathrm{Pic}(S)\cong U$, where $U$ stands for the hyperbolic lattice. I think that the elliptic fibration has only singular ...
- 41
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\...
- 1,015
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 ...
- 1,821
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 ...
- 1,821
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 ...
- 205
3
votes
0
answers
609
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 ...
- 5,716
3
votes
0
answers
262
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 ...
- 151
3
votes
0
answers
541
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
\...
- 1,801
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 ...
- 1,821
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.
\...
- 585
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}\...
- 1,821
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 ...
- 585
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 ...
- 7,100
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 $X$ ...
- 3,697
2
votes
0
answers
141
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 $0$)
$$
E\!:y_1^2 = ...
- 1,801
2
votes
0
answers
318
views
Relation between Beauville-Bogomolov form and Intersection Product on Hilbert scheme of K3 surfaces
I am learning about Hilbert scheme of points $S^{[n]}$ on projective K3 surfaces S. Since these are hyperkähler varieties, the second cohomology $H^2(S^{[n]},\mathbb{Z})$ is endowed with the non-...
- 175
2
votes
0
answers
85
views
The quotient of a superspecial abelian surface by the involution
Let $E_i\!: y_i^2 = f(x_i)$ be two copies of a supersingular elliptic curve over a field of odd characteristics. Consider the involution
$$
i\!: E_1\times E_2 \to E_1\times E_2,\qquad (x_1, y_1, x_2, ...
- 1,801
2
votes
0
answers
259
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 ...
- 1,801
2
votes
0
answers
204
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 ...
- 1,801
2
votes
0
answers
115
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 ...
- 1,105
2
votes
0
answers
236
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 ...
- 1,105
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 ...
- 8,637
2
votes
0
answers
321
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 ...
user21574
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, \,\...
- 1,493
2
votes
0
answers
884
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). ...
- 600
2
votes
0
answers
131
views
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
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 $...
user498585
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 ...
- 1,821
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 ...
- 8,637
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 ...
- 253