Tagged Questions

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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Rational curves on the Fermat quartic surface

Let $X$ be the Fermat quartic $x^4+y^4+z^4+w^4=0$ in $\mathbb P^3$. It is known that $X$ contains infinitely many $(-2)$-curves, that is, smooth rational curves. (One way to obtain in infinitely many ...
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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-...
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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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Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?

Given an integer solution $s_m$ to the system, $$x_1^2+x_2^2+\dots+x_n^2 = y^2$$ $$x_1^3+x_2^3+\dots+x_n^3 = z^3$$ and define the function, $$F(s_m) = x_1+x_2+\dots+x_n$$ For $n\geq3$, using an ...
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On the cohomology ring of the Hilbert scheme of points on k3 or abelian surfaces

There are many results on the cohomology of the Hilbert scheme of points of a surface. Gottsche calcaluted the Betti numbers and Nakajima got the generators of the cohomology. Also there are results ...
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Is every algebraic $K3$ surface a quartic surface? [closed]

Algebraic $K3$ surface means the $K3$ surface admits an ample line bundle. So the question is equivalent to asking whether every algebraic $K3$ surface can be embedded in $\mathbb{P}^3$.
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Enriques surfaces over $\mathbb Z$

Does there exist a smooth proper morphism $E \to \operatorname{Spec} \mathbb Z$ whose fibers are Enriques surfaces? By a theorem of, independently, Fontaine and Abrashkin, combined with the Enriques-...
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Singular models of K3 surfaces

Let us work over a ground field of characteristic zero. As is well-known, a K3 surface is a smooth projective geometrically integral surface $X$ whose canonical class $\omega_X$ is trivial and for ...
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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). ...
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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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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/...
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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 ...
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When will the mirror of a K3 surface be an elliptic K3?

Let $f:Y\rightarrow\mathbb{P}^1$ be an elliptic $K3$ surface, then the holomorphic 2-form $\Omega_Y$ vanishes when restricted to an elliptic fiber $f^{-1}(b)$ with $b\in\mathbb{P}^1$. After a ...
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K3 surfaces that correspond to rational points of elliptic curves

In his work on mirror symmetry (http://arxiv.org/pdf/alg-geom/9502005v2.pdf) Igor Dolgachev has considered families of K3 surfaces of Picard rank at least 19 with the base given by $X_0(n)^+$, the ...
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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 ...
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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 ...
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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 ...
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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 $C \rightarrow \mathbb P^1$ ...
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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 ...
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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 ...
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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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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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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 ...
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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-...
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construct the elliptic fibration of elliptic k3 surface

Hi all, As we know, every elliptic k3 surface admits an elliptic fibration over $P^1$, but generally how do we construct this fibration? For example, how to get such a fibration for Fermat quartic? ...
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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 ...
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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 ...
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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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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 ...
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 \...