# Tagged Questions

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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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### 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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### 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 ...
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### 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 ...
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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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### 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 ...
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### 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 ...
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### 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)
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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 ...
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### $K3$ 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?
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### 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 ...
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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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### 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 ...
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### 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 ...
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 ...