# 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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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 $g\... 0answers 159 views ### on the automorphisms of the transcendental Hodge structure of a K3 surface Let$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 ... 1answer 195 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 ... 2answers 208 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$is any nowhere ... 0answers 360 views ### Semistable minimal model of a$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 ... 0answers 196 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-... 2answers 405 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 ... 1answer 991 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 ... 2answers 404 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 ... 2answers 355 views ### Reference for Automorphisms of K3 surfaces I am looking for some introductory reference concerning Automorphisms (of finite order) on K3 surfaces. Any suggestion? 0answers 110 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 ... 0answers 195 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 ... 1answer 215 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 ... 0answers 481 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 \...
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 ...
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 ...