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 ...

28
votes

Accepted

### $\int_0^\infty x \, [J_0(x)]^5 \, dx$: source and context, if any?

See Densities of Short Uniform Random Walks (with an appendix by Don Zagier) by
Jonathan M. Borwein, Armin Straub, James Wan, and Wadim Zudilin,
Canad. J. Math. Vol. 64 (5), 2012 pp. 961–990. http://...

20
votes

### Is every algebraic $K3$ surface a quartic surface?

More generally, the moduli space of algebraic K3 surfaces is a countable union on 19 dimensional subvarieties of the 20 dimensional moduli space of complex K3 surfaces, and exactly one of those ...

19
votes

Accepted

### Curves on K3 and modular forms

The answer to your first question: "What is the relationship between $G_2$ and $\Delta$?" is
$$q\frac{d}{dq} \log \Delta = -24G_2 $$
where
$$\Delta = q\prod_{m=1}^\infty (1-q^m)^{24}$$ and
$$...

15
votes

Accepted

### Non-algebraic K3 surfaces in characteristic $p$

Let me briefly expand on Jason's comment.
Actually, "formal scheme" is the right word here.
For any K3 surface $X$ over an algebraically closed field $k$ of any characteristic one has $$h^0(X, T_X)=...

15
votes

Accepted

### Is every algebraic $K3$ surface a quartic surface?

No. Consider a K3 surface with a polarization of degree 2 and with Picard rank 1. Since the tautological line bundle on $\mathbb{P}^3$ pulls back to a degree 4 line bundle, it follows that such a K3 ...

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 ...

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 ...

13
votes

Accepted

### 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

12
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 $\mathbb F_2$ ...

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 ...

8
votes

Accepted

### Singular models of K3 surfaces

Yes, this is true: the smooth minimal model is a $K3$ surface.
In fact, let $\bar{X}$ be the resolution of the singularities of $X$. Then the following holds.
(1) Rational double points impose no ...

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

### 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

### 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

### 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....

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 (...

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 ...

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

### 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) ...

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

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

### 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 ...

4
votes

### K3 surfaces that correspond to rational points of elliptic curves

You can ask similar questions for elliptic curves, and as far as I know, the answer to your question of "What is special..." is not much. Thus let $\mu:X_0(n)\to E$ be a modular parametrization of an ...

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