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 ...
user80296's user avatar
  • 751
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 ...
Will Sawin's user avatar
  • 129k
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 $\mathbb F_2$ ...
Davide Cesare Veniani's user avatar
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 ...
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 ...
Will Sawin's user avatar
  • 129k
12 votes
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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, ...
Sasha's user avatar
  • 35.7k
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 ...
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.
Lazzaro Campeotti's user avatar
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 ...
Noam D. Elkies's user avatar
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 ...
Danny Ruberman's user avatar
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 ...
Oli Gregory's user avatar
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, ...
Francesco Polizzi's user avatar
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 (...
Tom Mrowka's user avatar
  • 2,904
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....
Francesco Polizzi's user avatar
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) ...
Francesco Polizzi's user avatar
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 ...
Zach Teitler's user avatar
  • 5,732
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 ...
Jonny Evans's user avatar
  • 6,885
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 $p>2$ ...
Felipe Voloch's user avatar
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 ...
Johannes Nordström's user avatar
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 ...
R.P.'s user avatar
  • 4,630
5 votes
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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 ...
Sasha's user avatar
  • 35.7k
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 ...
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 ...
Misha Verbitsky's user avatar
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 \, (...
Noam D. Elkies's user avatar
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. ...
Sasha's user avatar
  • 35.7k
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 ...
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. ...
Philip Engel's user avatar
  • 1,483
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 ...
Sasha's user avatar
  • 35.7k
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 (...

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