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 $2$.
181
questions
6
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SYZ mirror symmetry for K3 surfaces
My question is essentially related to this post, but let me formulate it again. Let $f:S \rightarrow \mathbb{P}^1$ be an elliptic fibration, then this can be a SLAG fibration with respect to another ...
- 61
15
votes
1
answer
940
views
Curves on K3 and modular forms
The paper of Bryan and Leung "The enumerative geometry of $K3$ surfaces and modular forms" provides the following formula. Let $S$ be a $K3$ surface and $C$ be a holomorphic curve in $S$ representing ...
- 600
1
vote
1
answer
254
views
Linear system on an abelian surface
On a K3 surface $S$, a linear system $|C|$ is said to be hyperelliptic if the corresponding map is of degree 2 and the image is of degree $g_a(C)-1$ in $\mathbb P^{g_a}$.
For $g_a(C) > 2$, if $|C|...
- 453
3
votes
1
answer
312
views
Spectral sequence associated to elliptic fibration degenerates?
Let $\phi:S\rightarrow \mathbb{CP}^1$ be an elliptic fibration of a K3 surface. When is the Leray spectral sequence associated to the fibration $E_2$-degenerate? Are there any good criteria for the $...
- 31
6
votes
2
answers
395
views
adjacency matrix of a graph and lines on quartic surfaces
Suppose you are given a smooth quartic surface $X$ in $\mathbb P^3$. I would like to find an upper bound for the number of lines on $X$ in the case that there is no plane intersecting the curves in ...
4
votes
1
answer
275
views
Euler number for base change of a K3 surface
Suppose you have a K3 surface $S$ containing a smooth rational curve $C$ and suppose you have an elliptic fibration $S \rightarrow \mathbb P^1$ that restricts to a morphism $C \rightarrow \mathbb P^1$ ...
1
vote
2
answers
529
views
An ample line bundle on a K3 surface
Let $X$ be a K3 surface obtained as a double covering of $\mathbb{P}^1 \times \mathbb{P}^1$ branching along a $(4,4)$-divisor. I think the natural line bundle $\pi^*\mathcal{O}_{\mathbb{P}^1\times \...
- 23
8
votes
2
answers
454
views
Necessary condition on Calabi-Yau manfiold to be a hypersurface in a Fano manifold
Let $X$ be a smooth projective Calabi-Yau threefold. Are there any known obstructions to it
being a member of a base-point-free linear system in a nef-Fano fourfold?
What, in anything, is known ...
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5
votes
1
answer
318
views
Positivity question on K3 surfaces
Let $X$ be a smooth projective complex K3 surface and $L, D$ two effective divisors, $L^2\geq0$ and $D^2\geq0$.
(Q1). do we have $L\cdot D\geq0$ ?
If either one has positive self-intersection, the ...
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5
votes
1
answer
300
views
Does $h^1(D)=0$ imply numerical connectedness on K3 surfaces?
Let $X$ be a complex K3 surface and $D$ an effective divisor on $X$.
We shall say: $D$ is connected if its support is connected. $D$ is numerically connected if for any non-trivial effective ...
- 761
5
votes
1
answer
298
views
K3 surface with $D_{14}$ singular fiber
Let $X$ be an elliptic K3 surface with $D_{14}$ singular fiber. Do you know an explicit equation for such $X$? Also, how many disjoint sections such fibration admits? Any reference would be greatly ...
- 73
2
votes
1
answer
195
views
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
5
votes
0
answers
198
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
5
votes
2
answers
616
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 ...
- 53
0
votes
2
answers
486
views
K3 surface with a non-symplectic involution: a basic question
Let $X$ be a K3 surface (algebraic, complex). An involution $\sigma:X\rightarrow X$ is called non-symplectic if it acts trivially on $H^{2,0}(X)=\Bbb{C}\omega_X$ $\ $ (where $\omega_X$ is any nowhere ...
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votes
0
answers
576
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
310
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
2
votes
2
answers
974
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 ...
- 761
22
votes
1
answer
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views
Monstrous moonshine for $M_{24}$ and K3?
An important piece of Monstrous moonshine is the j-function,
$$j(\tau) = \frac{1}{q}+744+196884q+21493760q^2+\dots\tag{1}$$
In the paper "Umbral Moonshine" (2013), page 5, authors Cheng, Duncan, and ...
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5
votes
2
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views
Action of automorphisms of a $K3$ surface on its $(-2)$-curves
Consider a complex $K3$ surface $X$ and take its group of automorphisms $Aut(X)$. It is a known fact that the action of $Aut(X)$ on the set of rational $-2$ curves of $X$ has only finite number of ...
- 14k
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votes
2
answers
451
views
Reference for Automorphisms of K3 surfaces
I am looking for some introductory reference concerning Automorphisms (of finite order) on K3 surfaces. Any suggestion?
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vote
0
answers
119
views
K3 surface minus finite set
Let $S$ be a complex K3 surface, and $P\subset S$ a finite set of points in $S$. It is known that
$$
H^i(S,\mathbb{Z})\cong H^i(S\setminus P,\mathbb{Z})
$$
for $0\le i \le 2$. Then the Euler ...
- 11
1
vote
0
answers
365
views
Moduli space of K3 surfaces and Bogomolov-Tian-Todorov theorem
The famous Bogomolov-Tian-Todorov theorem says that the moduli space of Calabi-Yau manifold is smooth, that is locally a complex manifold. Doesn't this contradicts to the fact that the moduli space ...
- 1,653
5
votes
1
answer
505
views
Existence of logarithmic structures and d-semistability
I am reading a paper ( Kawamata, Y.; Namikawa, Y. Logarithmic deformations of normal crossing varieties and smoothing of degenerate Calabi-Yau varieties. Invent. math. 1994, 118, 395–409.) I have a ...
11
votes
0
answers
783
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
1
answer
685
views
A question on an elliptic fibration of the Enriques surface
Let $S$ be an Enriques surface over complex numbers. It is known that $S$ admits an elliptic fibration over $\mathbb{P}^1$ with $12$ nodal singular fibers and $2$ double fibers. How can I see this ...
- 1,653
4
votes
2
answers
317
views
Algebraic cycles on a K3 surface after hyperKahler rotation.
I would like to find a gap in the following observation. I found a suspicious part but cannot prove it wrong. I would appreciate your assistance.
Let $M$ be a lattice of signature $(1,t)$ and $S$ be ...
- 41
6
votes
0
answers
855
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)
- 42.5k
1
vote
1
answer
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views
A question on real surfaces on K3 surfaces.
Let $X$ be a K3 surface and $\omega$ be a nowhere vanishing 2-form on $X$. Suppose $Y\subset X$ be a smooth real surface. How can one see that $\omega|_Y=0$ implies $Y$ is a complex submanifold (a ...
- 11
5
votes
1
answer
613
views
Rookie questions about k3's
Hi everyone,
I am trying to go through parts of Saint-Donat's 1974 paper 'Projective Models of K3-surfaces', and have been stuck on a few claims for a while now - I'd appreciate some help explaining ...
- 349
3
votes
2
answers
658
views
Line bundles on K3 surfaces
Let $L$ be a line bundle on an (algebraic) K3 surface over a field $k$. The Riemann-Roch theorem specializes to
$$
\chi(X, L)=\frac{1}{2}(L\cdot L)+2
$$
which can be rewritten as
$$
h^0(X, L)+h^0(...
- 31
1
vote
1
answer
780
views
Picard group of a K3 surface generated by a curve
In Lazarsfeld's article "Brill Noether Petri without degenerations" he mentions the fact that for any integer $g \geq 2$, one may find a K3 surface $X$ and a curve $C$ of genus $g$ on $X$ such that ...
- 365
1
vote
1
answer
189
views
Is this an embedding of $S^{[2]}$?
The intersection of 3 quadrics in $P^5$ is a K3 surface $S$.
There is a natural map $S^{[2]} \to G(1,5)$ well defined everywhere, because a generic K3 doesn't contain any line and this family is ...
- 453
1
vote
0
answers
211
views
Maximally unipotent monodromy point of a K3 surface
I have a question on maximally unipotent monodromy point (or large complex structure limit) of the family of polarized K3 surfaces $(X,L)$. It is known that the moduli space of such pair is given by ...
- 11
2
votes
1
answer
464
views
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
403
views
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
8
votes
1
answer
2k
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Mirror symmetry for hyperkahler manifold
Hi there,
I have some questions about the mirror symmetry of hyperkahler manifold and K3 surface.
The well-known result said: the mirror symmetry for K3 surface is just given by its hyperkahler ...
- 583
2
votes
1
answer
463
views
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
3
votes
1
answer
336
views
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 ...
- 2,098
1
vote
2
answers
474
views
Isotrivial K3 family and Picard number
Is it true that any family of K3 surfaces over $\mathbb{C}$ whose Picard number is constant is isotrivial? Here isotrivial means locally analytically trivial.
Speculation: Let $\mathcal{M}$ be the ...
- 11
4
votes
1
answer
426
views
(3,3) abelian surface and k3 surfaces
SOrry for the very specific question, but curiosity bites....
So here's the story: an idecomposable principally polarized abelian surface is embedded in $P^8=|3\Theta |^* $ as a deg 18 surface A. ...
- 3,717
3
votes
1
answer
403
views
octic K3s inside cubic 4-folds
From the Thesis of B.Hassett I seem to understand that a smooth cubic 4-fold $X$ containing a $\mathbb{P}^2$ should contain also a octic K3, but I cannot see a natural way by which this K3 octic could ...
- 3,717
1
vote
0
answers
232
views
Moduli Space of an Algebraic K3 surface with singularities.
Suppose that $X$ is an algebraic K3 surface (say polarized). If the singular divisor of $X$ is normal crossing... Do we have a moduli space parametrizing such $K3$ surfaces? If yes do we have a ...
2
votes
0
answers
132
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 ...
- 763
0
votes
1
answer
416
views
Non Smooth K3 surface?
Hi,
My question is related to Algebraic Surfaces. I have seen that we always consider K3 surfaces which are smooth, but I wonder how can we define a non-smooth K3 surface.
The problem I see is on ...
4
votes
1
answer
961
views
Involution of the Fermat quartic
Let $X\subset\mathbb{P}^{3}$ be the Fermat quartic surface given by
$$x^4-y^4-z^4+w^4 = 0$$
and consider the involution
$$i:X\rightarrow X,\: (x,y,z,w)\mapsto (y,x,w,z).$$
The surface $X$ can be seen ...
- 8,852
3
votes
0
answers
154
views
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 $E$ ...
7
votes
1
answer
293
views
$K3$ surfaces admitting finite non-symplectic group actions are projective
I have read somewhere that "$K3$ surfaces admitting finite non-symplectic group actions are projective". Could someone remind me of a proof?
- 71
14
votes
2
answers
2k
views
How to compute the Picard rank of a K3 surface?
I'm curious about the following question:
Given a K3 surface, how does one proceed to compute its rank?
Of course the answer may depend on the form of the input, i.e. how the K3 is "given". So
...
- 3,758
0
votes
0
answers
181
views
$T^2$-fibered K3 surface with involution
Let $S$ be a K3 surface and $f:S\rightarrow \mathbb{P}^1$ a $T^2$-fibration (not necessarily holomorphic, I have a special Langrangian fibration in mind). Assume there is a $k$-section, then a fiber ...
- 1