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 $2$.
5
votes
1answer
149 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 ...
5
votes
1answer
173 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 ...
3
votes
1answer
166 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 ...
2
votes
1answer
113 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 ...
5
votes
0answers
126 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 ...
2
votes
1answer
139 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 ...
1
vote
2answers
102 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 ...
3
votes
0answers
137 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 ...
5
votes
0answers
136 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, ...
3
votes
2answers
251 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 in our case sufficient to check this for ...
20
votes
1answer
777 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 ...
4
votes
2answers
281 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 ...
4
votes
2answers
281 views
Reference for Automorphisms of K3 surfaces
I am looking for some introductory reference concerning Automorphisms (of finite order) on K3 surfaces. Any suggestion?
0
votes
0answers
98 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 ...
1
vote
0answers
96 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 ...
2
votes
1answer
113 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 ...
7
votes
0answers
300 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 ...
7
votes
1answer
183 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 ...
3
votes
2answers
224 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 ...
3
votes
0answers
222 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)
1
vote
1answer
98 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 ...
4
votes
1answer
362 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 ...
3
votes
2answers
273 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, ...
1
vote
1answer
190 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 ...
1
vote
1answer
155 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 ...
1
vote
0answers
94 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 ...
2
votes
1answer
255 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, ...
1
vote
1answer
211 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 ...
3
votes
1answer
502 views
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 ...
2
votes
1answer
343 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 ...
3
votes
1answer
196 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 ...
1
vote
2answers
264 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 ...
3
votes
1answer
300 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. ...
2
votes
1answer
220 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 ...
1
vote
0answers
141 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
0answers
99 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 ...
0
votes
1answer
257 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 ...
5
votes
1answer
310 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 ...
3
votes
0answers
132 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 $E$ ...
6
votes
1answer
199 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?
9
votes
2answers
590 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
...
0
votes
0answers
139 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 ...
2
votes
0answers
233 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 ...
5
votes
1answer
384 views
Complex structures on a K3 surface as a hyperkähler manifold
A hyperkähler manifold is a Riemannian manifold of real dimension $4k$ and holonomy group contained in $Sp(k)$. It is known that every hyperkähler manifold has a $2$-sphere $S^{2}$ of complex ...
6
votes
3answers
607 views
2-cycle of K3 surface
Hi there,
I want to ask about the 2-cycle of K3 surface.
As we know, its betti number $b_2$=22, so there will be 22 2-cycle generators.
Is there any topological way to figure out such cycles ...
2
votes
1answer
290 views
Genus two pencil in K3 surface
It is known that smooth $K3$ surface can be obtained as two fold branched cover of rational elliptic surface $E(1) = \mathbb{CP}^2 9 \bar{{\mathbb{{CP}^2}}}$ along the smooth divisor $2F_{E(1)} = 6H - ...
3
votes
2answers
539 views
Question on K3 Surface
Is it possible to realize $K3$ surface as a ramified double cover of rational elliptic surface? If so, is there way to see an elliptic fibration structure on $K3$ from such cover? It seems to me one ...
19
votes
4answers
987 views
Does $(x^2 - 1)(y^2 - 1) = c z^4$ have a rational point, with z non-zero, for any given rational c?
I need this result for something else. It seems fairly hard, but I may be missing something obvious.
Just one non-trivial solution for any given $c$ would be fine (for my application).
11
votes
1answer
342 views
Symmetric functions on three parameters being perfect squares
Is it possible for $x+y+z, xy+yz+zx$, and $xyz$ to be perfect squares at the same time for positive integer values of $x,y,z$?
2
votes
1answer
267 views
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 ...
