44
votes

Accepted

### Is there an octonionic analog of the K3 surface, with implications for stable homotopy groups of spheres?

Yes, such M exists. The boundary connected sum of 28 copies of the Milnor plumbing has boundary diffeomorphic to $S^7$ so it can be closed off with $D^8$ and you can let $M$ be the connected sum of ...

15
votes

### Seeking concrete examples of "generic" elliptic fibrations of K3 surfaces

Weierstrass equations are probably a good choice. You can try
$$y^2 = x^3 - 3x +2 t^{12},$$ for example.
Here the singular fibers are when $t$ is a $24$th root of unity $\zeta$, and the double point ...

14
votes

Accepted

### Enriques surfaces over $\mathbb Z$

A preprint by Stefan Schröer came out today with the answer to this question: arXiv:2004.07025.
No such Enriques surface exists. In fact, there is no classical Enriques surface over $\mathbb F_2$ ...

13
votes

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### Rational curves on the Fermat quartic surface

I am posting my comments as an answer. I am concerned that I misunderstand the OP, so let me state first the result. There exists a finite field extension $K/\mathbb{Q}$ such that for every closed ...

Community wiki

13
votes

Accepted

### density of singular K3 surfaces

This is a standard argument and there probably exists a reference but it's not hard once you rephrase it in terms of the period domain.
The moduli space of K3 surfaces is locally isomorphic to its ...

12
votes

Accepted

### Do non-projective K3 surfaces have rational curves?

Some of them do, and some don't.
Indeed, by global Torelli theorem, there is a K3 surface $X$ with $\mathrm{Pic}(X) = 0$. Such $X$ has no curves, in particular no rational curves.
On the other hand, ...

11
votes

Accepted

### Reference request: Generic k3 surface has Picard number 1

Welcome new contributor. I am just writing my comment as an answer, and expanding on the observation of Prof. Arapura. For a smooth, projective scheme $X$ over a field $k$, the space of first order ...

Community wiki

10
votes

Accepted

### Is the automorphism group of a Calabi-Yau variety an arithmetic group

The answer for $K3$ surfaces is no. A counterexample, where the group is not even commensurable with an arithmetic group, was given by Totaro in Example 6.3 of this paper.

9
votes

Accepted

### Discriminant locus of elliptic K3 surfaces

The minimal $s$ is $3$.
It is attained by several elliptic K3's,
including $y^2 = x^3 + (t^2-t)^4$ which has IV* fibers at
$t = 0, 1, \infty$ and no other singular fibers.
The comment by Ariyan ...

7
votes

### Vector field on a K3 surface with 24 zeroes

This is probably not as explicit as what you requested, but there are recipes for how to build a Morse function on any smooth hypersurface in $CP^3$ with no index 1 or 3 critical points. There will be ...

7
votes

Accepted

### Training towards research on k3 surfaces

Likely this should only be a comment, but I don't have enough reputation for that...
J.C. Ottem has provided a wonderful reference about the basics of K3 surfaces in his comment. It's my personal ...

7
votes

### Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold?

I think Verbisky proves a refined form of the Mirror Conjecture for hyperkaehler manifolds, not the conjecture in the strict form. This is explained at page 3 of the paper that you link.
In fact, ...

7
votes

### Seeking concrete examples of "generic" elliptic fibrations of K3 surfaces

As far as I know no one has “written down” a hyperkähler structure on a K3 surface. Indeed much of the work in Mirror symmetry revolves around trying to give an asymptotic expansion of such metrics (...

7
votes

### density of singular K3 surfaces

A reference for the density of singular K3 surfaces in the period domain (see Will Sawin's answer) is
Piatetski-Shapiro, Ilya I.; Shafarevich, I. R., Arithmetic of K3 surfaces, Trudy Mezhdunarod. Konf....

6
votes

### K3 surfaces with small Picard number and symmetry

In Section 9 of the paper
I. Shimada: An algorithm to compute automorphism groups of (K3) surfaces and an application to singular (K3) surfaces, Int. Math. Res. Not. 2015, No. 22, 11961-12014 (2015) ...

6
votes

Accepted

### Very general quartic hypersurface in $\mathbb{P}^3$ has Picard group $\mathbb{Z}$

That is the Noether-Lefschetz theorem. Searching online should find plenty of results in web pages and lecture notes. If you want a published source, how about: Mark Green, A new proof of the explicit ...

6
votes

### (1/2) K3 surface or half-K3 surface: Ways to think about it?

Of course what you've written is too vague to be a definition, but I can guess what they're talking about. In low-dimensional topology there's a 4-manifold called $E(1)$; this is a rational complex ...

5
votes

### Is there a purely inseparable covering $\mathbb{A}^2 \to K$ of a Kummer surface $K$ over $\mathbb{F}_{p^2}$?

See Proposition 4.5 of my paper with D. Abramovich, "Lang’s Conjectures, Fibered Powers, and Uniformity", New York J. Math. 2 (1996) 20–34.
A supersingular elliptic curve in characteristic $p>2$ ...

5
votes

Accepted

### Primitivity of subgroups in the Picard groups of anticanonical $K3$ surfaces

I think the following may give an example $X$ where the image of the map in question is not primitive.
Let $S$ be the blow-up of $\mathbb{P}^2$ in 9 points that are the intersection of two cubics (so ...

5
votes

### Singular models of K3 surfaces

For what it's worth, I wrote up a proof (pretty detailed) following the hints in Francesco Polizzi's answer. It's in an unpublished preprint found here (p. 38 onwards). I am not a geometer, so the ...

5
votes

Accepted

### Fixed part of a line bundle on a K3 surface

(1) I guess, as a scheme, the base locus might have embedded points. But the fixed part is defined as the pure 1-dimensional part of the base locus scheme.
(2) If $F$ is the fixed part, it means that ...

5
votes

### Do singular fibers determine the elliptic K3 surface, generically?

I am expanding naf's comments to make a self-contained community wiki answer. By an elliptic fibration we mean a smooth projective relatively minimal surface $f: X \to C$ with general fiber given by ...

Community wiki

5
votes

### automorphism group of K3 surfaces

Calabi-Yau theorem implies that any diffeomorphism of a Calabi-Yau manifold which preserves
the complex structure and the Kahler class also preserves the Calabi-Yau metric. However, the group of ...

4
votes

Accepted

### Mordell–Weil rank of some elliptic $K3$ surface

Let $E_0$ be the elliptic curve $y^2 = x^3 + 1$,
and choose $\beta$ in $k = {\bf F}_q$ so that $b = \beta^2$.
Then $W = W_b$ has $\rho=18$ unless the elliptic curve
$$
E_\beta : Y^2 = X^3 + \beta \, (...

3
votes

Accepted

### Fixed locus in the linear system associated to the ramification locus of a K3 double cover of a Del Pezzo surface

One has
$$
\rho_*\mathcal{O}_S \cong \mathcal{O}_X \oplus \omega_X
$$
and the involution of $S$ induces the involution of this sheaf that acts by 1 on the first summand and by $-1$ on the second. ...

3
votes

Accepted

### Irrationality of some threefolds

I am just posting my comment as an answer. All such threefolds are rational.
By the hypotheses on the involution of the K3 surface, the quotient surface is a rational surface. The projection from ...

Community wiki

3
votes

Accepted

### Loci in the moduli space of K3 surfaces associated to lattices

I won't completely answer your question, but will try to just rephrase it in a certain way. You are asking when two given moduli spaces of lattice-polarized K3 surfaces $M_L$ and $M_{L'}$ intersect. ...

3
votes

Accepted

### One-dimensional family of complex algebraic K3 surfaces

Choose any ample class in $\mathrm{Pic}(X)$, assume its degree is $d$. Let $M_d$ be the moduli space of polarized K3 surfaces of degree $d$ with appropriate level structure so that it has a universal ...

2
votes

Accepted

### Common gerbes over two K3 surfaces

As Jason Starr says in his comments, such data exists if and only if $X$ is isomorphic to $Y$.
Indeed, let $\mathcal{X}\to X$ be a $G_X$-gerbe, and let $\mathcal{Y}\to Y$ be a $G_Y$-gerbe. As the (...

Community wiki

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