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 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 ...
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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 ...
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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 ...
user494851
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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 \...
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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 ...
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$K3$ 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 ${\...
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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
\...
user479269
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Characterization of $d$-gonal curves on a K3 surface
Let $X$ be a K3 surface and $C$ a curve on $X$. We say that $C$ is $d$-gonal if it admits a pencil of degree $d$ (and none of smaller degree).
I am wondering if there exist characterizations of $d$-...
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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 ...
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Infinitely many rational nt multisection in elliptic K3 surfaces by deformation theory
I'm trying to read this paper of Bogomolov and Tschinkel http://arxiv.org/pdf/math/9902092.pdf about potential density of rational points on elliptic K3 Surfaces.
I got quite stuck in Corollary 3.27 ...
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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|...
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holomorphic sections on elliptic K3 surface
Hi all,
I want to ask something about the holomorphic sections on elliptic K3:
Is there any obstruction for an ellptic K3 (as an elliptic fibration) to have holomorphic sections? Is that always some ...
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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}...
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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
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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 ...
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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 ...
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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
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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 ...
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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 ...
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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 ...
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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 ...
- 1,633
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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 ...
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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 ...
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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 ...
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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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Neron-Severi Lattice of Elliptic K3
I'm trying to compute Neron-Severi lattices of some K3 surfaces. They have elliptic fibrations with multiple sections. Setting one section to be the identity section, I can write down a Weierstrass ...
- 309
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Classification of Elliptic singularity
For a $K_3$ surface $X$, if there exists a holomorphic surjective map $X\to \mathbb P^1$, with elliptic fibres, i.e. for any generic point on $\mathbb P^1$ whose fiber is diffeomorphic to a torus $\...
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$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 ...
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