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 ...
- 148k
15
votes
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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 $\mathbb F_2$ ...
14
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 ...
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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, ...
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12
votes
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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 ...
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 ...
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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 (...
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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....
- 66.3k
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, ...
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6
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 ...
- 39.3k
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 ...
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6
votes
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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 ...
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6
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 ...
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) ...
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6
votes
Why do symmetries of K3 surfaces lie in the Mathieu group $M_{24}$?
This is clearly not the best reference on the subject — I would recommend Mukai's original paper: Finite groups of automorphisms of K3 surfaces and the Mathieu group, Invent. Math. 94 (1988), no. 1, ...
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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 ...
- 9,237
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 $p>2$ ...
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5
votes
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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 ...
- 1,936
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 ...
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4
votes
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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
$$
E_\beta : Y^2 = X^3 + \beta \, (...
- 79.9k
4
votes
Is the set of points on an abelian surface which project to rational points on the Kummer surface a subgroup?
The answer to your second question is no. If $x\notin J(\mathbb Q)$, then the condition $\pi(x)\in X(\mathbb Q)$ implies that $\mathbb Q(x)/\mathbb Q$ is a quadratic extension. If you take a point $y$ ...
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4
votes
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Is the set of points on an abelian surface which project to rational points on the Kummer surface a subgroup?
Varieties over a field $k$ can be understood by their $\bar k$-points with the Zariski topology, together with the $\operatorname{Gal}(\bar k/k)$-action. Any morphism $f \colon X \to Y$ of $k$-...
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4
votes
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K3 surfaces and density of rational curves
The OP requested that I add my comment as an answer:
Your argument in the last paragraph seems to assume that any dense subset of $S$ contains a nonempty open set. That is not true.
Specifically in ...
3
votes
Accepted
Fixed locus in the linear system associated to the ramification locus of a K3 double cover of a Del Pezzo surface
One has
$$
\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. ...
- 39.3k
3
votes
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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
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 ...
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2
votes
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$K3$ surfaces can't be uniruled
The comment by user @naf is completely correct: there is a simpler argument than the argument I sketched. However, the argument I sketched gives a stronger result, which Mumford conjectured is sharp.
...
2
votes
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Concrete descriptions of $S^1$-bundles over smooth manifold $Y$ underying a K3 surface
You can say a fair amount about the topology of the total spaces of the different bundles, although I suspect none of them is a particularly well-known manifold that has a `name'. (Except of course ...
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2
votes
$K3$ surfaces in $\mathbb P^1 \times \mathbb P^1 \times \mathbb P^1$
In the generic case you'll have $\text{NS}(S)\otimes\mathbb R\cong\mathbb R^3$ and you can compute the $3$-by-$3$ matrix for the action of the three involutions $i_1,i_2,i_3$ on, say, a basis ...
- 47.4k
Only top scored, non community-wiki answers of a minimum length are eligible