Highest scored 'k3-surfaces+ag.algebraic-geometry+algebraic-surfaces' questions

30 votes

1 answer

2k views

Enriques surfaces over $\mathbb Z$

Does there exist a smooth proper morphism

$E \to \operatorname{Spec} \mathbb Z$ whose fibers are Enriques surfaces? By a theorem of, independently, Fontaine and Abrashkin, combined with the Enriques-...

Will Sawin

131k

asked Mar 27, 2015 at 22:19

16 votes

4 answers

1k views

K3 surfaces with good reduction away from finitely many places

Let S be a finite set of primes in Q. What, if anything, do we know about K3 surfaces over Q with good reduction away from S? (To be more precise, I suppose I mean schemes over Spec Z[1/S] whose ...

JSE

19.1k

asked Nov 9, 2009 at 0:14

8 votes

3 answers

1k views

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

For me a K3 surface will be a smooth complex projective variety of dimension 2 that is simply-connected and has trivial canonical bundle. Given a K3 surface

$X$ , an elliptic fibration $\pi \colon X \...

John Baez

21k

asked Feb 2, 2022 at 20:53

8 votes

1 answer

664 views

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 ...

user2013

1,633

asked Jun 30, 2013 at 12:29

7 votes

1 answer

406 views

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

Let

$E_i\!: y_i^2 = x_i^3 + a_4x_i + a_6$ be two copies (

$i = 1$ ,

$2$ ) of a supersingular elliptic curve over a finite field

$\mathbb{F}_{p^2}$ , for odd prime

$p > 3$ . Consider the Kummer surface $...

Dimitri Koshelev

1,801

asked Apr 29, 2018 at 11:04

6 votes

0 answers

199 views

Are all these K3 surfaces supersingular?

Consider all the smooth K3 surfaces given by

$X^4+W^2X^2+XW^3 = f(Y,Z,W)$ or

$X^4+XW^3 = g(Y,Z,W)$ over

$\mathbb F_{2}$ with

$f$ or

$g$ homogenous of degree 4. There are a lot of choices for

$f$ and $...

Gabriel Furstenheim

483

asked Apr 13, 2015 at 11:04

5 votes

2 answers

449 views

Reference for Automorphisms of K3 surfaces

I am looking for some introductory reference concerning Automorphisms (of finite order) on K3 surfaces. Any suggestion?

Heitor

761

asked Oct 14, 2013 at 18:09

5 votes

1 answer

307 views

K3 surfaces with small Picard number and symmetry

I am looking for examples of K3 surfaces that have a low Picard rank and at least one holomorphic involution. Here, low is no mathematically precise concept. I want to do computations with Monad ...

user505117

557

asked Dec 13, 2021 at 9:21

5 votes

1 answer

297 views

K3 surface with $D_{14}$ singular fiber

Let

$X$ be an elliptic K3 surface with

$D_{14}$ singular fiber. Do you know an explicit equation for such

$X$ ? Also, how many disjoint sections such fibration admits? Any reference would be greatly ...

guest2014

73

asked Feb 18, 2014 at 5:47

5 votes

1 answer

296 views

Does $h^1(D)=0$ imply numerical connectedness on K3 surfaces?

Let

$X$ be a complex K3 surface and

$D$ an effective divisor on

$X$ . We shall say:

$D$ is connected if its support is connected.

$D$ is numerically connected if for any non-trivial effective ...

Heitor

761

asked Mar 1, 2014 at 15:43

5 votes

0 answers

173 views

Is a Kummer surface over an finite field $\mathbb{F}_q$ supersingular iff $\mathbb{F}_q$ -unirational?

Let

$A$ be an abelian surface over an finite field

$\mathbb{F}_q$ . In particular, I am interested in the case when

$A$ is a Jacobian variety. Is the Kummer surface

$K_A/\mathbb{F}_q$ Shioda-...

Dimitri Koshelev

1,801

asked Jun 2, 2016 at 18:46

4 votes

1 answer

630 views

Genus two pencil in K3 surface

It is known that smooth

$K3$ surface can be obtained as two fold branched cover of rational elliptic surface

$E(1) = \mathbb{CP}^2 9 \bar{{\mathbb{{CP}^2}}}$ along the smooth divisor $2F_{E(1)} = 6H - ...

user24328

87

asked Jun 11, 2012 at 11:51

4 votes

1 answer

1k views

The existence of primitive and sufficiently ample line bundles on K3 surfaces?

Let S be a surface and L be a line bundle on S. For any zero-dimensional closed subschemes x of S, there is natural map from global sections of L to the global sections of L restricting to x (which is ...

user761

41

asked Oct 19, 2009 at 19:20

4 votes

0 answers

86 views

Is there a way to calculate the Picard $\mathbb{F}_q$ -number of an (rational or K3) elliptic surface?

Consider a finite field

$\mathbb{F}_{q}$ and an elliptic surface

$\mathcal{E}\!: y^2 + a_1(t)xy + a_3(t)y = x^3 + a_2(t)x^2 + a_4(t)x + a_6,$ where

$a_i(t) \in \mathbb{F}_{q}[t]$ . Is there a way ...

Dimitri Koshelev

1,801

asked May 3, 2020 at 9:39

4 votes

0 answers

279 views

What is the Artin invariant of an elliptic supersingular K3 surface?

Let

$X$ be a supersingular K3 surface over an algebraically closed field

$k$ of positive characteristic

$\!p$ . Artin proved in the paper https://eudml.org/doc/81948 that the determinant $\mathrm{disc}(...

Dimitri Koshelev

1,801

asked Aug 7, 2016 at 15:05

3 votes

1 answer

402 views

octic K3s inside cubic 4-folds

From the Thesis of B.Hassett I seem to understand that a smooth cubic 4-fold

$X$ containing a

$\mathbb{P}^2$ should contain also a octic K3, but I cannot see a natural way by which this K3 octic could ...

IMeasy

3,697

asked Oct 22, 2012 at 15:00

3 votes

2 answers

813 views

Question on K3 Surface

Is it possible to realize

$K3$ surface as a ramified double cover of rational elliptic surface? If so, is there way to see an elliptic fibration structure on

$K3$ from such cover? It seems to me one ...

user24328

87

asked Jun 9, 2012 at 22:33

3 votes

1 answer

490 views

K3 over fields other than C?

How to classify K3 surfaces over an arbitrary field k?

Ilya Nikokoshev

14.8k

asked Oct 19, 2009 at 19:28

3 votes

0 answers

235 views

A K3 cover over a Del Pezzo surface

Let

$V \rightarrow \mathbb P^2$ be the blow-up at two distinct points. (

$V$ is a Del Pezzo surface of degree 7.) Choose a smooth curve

$C$ from the linear system

$|-2K_V|$ and let

$S \rightarrow V$ be ...

Basics

1,821

asked Jul 1, 2022 at 23:05

3 votes

0 answers

262 views

Are unirational K3 surfaces defined over finite fields?

Is every supersingular (thus unirational for

${\rm char }\ k = p\geq 5$ , from Liedtke)

$K3$ surface defined over a finite field? I guess this is true for Kummer surfaces, for example, since ...

Vinicius M.

151

asked Oct 31, 2017 at 11:52

3 votes

0 answers

541 views

The Jacobian surface of an elliptic surface

Let

$\mathcal{X}$ be an elliptic surface over

$\mathbb{P}^1$ without a section and let

$\mathcal{J}$ be an elliptic surface over

$\mathbb{P}^1$ with a section. Assume we have the commutative diagram \...

Dimitri Koshelev

1,801

asked Aug 1, 2016 at 15:05

2 votes

1 answer

330 views

Fixed part of a line bundle on a K3 surface

This question comes from Huybrechts' lecture notes on K3 surfaces, more specifically, chapter 2. Let

$X$ be a K3 surface (over an algebraically closed field

$k$ ) and

$L$ a line bundle on

$X$ . ...

Cranium Clamp

846

asked Mar 2, 2020 at 19:54

2 votes

1 answer

330 views

Sheaves with zero Chern classes on a $K3$ surface.

Let

$S$ be a

$K3$ surface. Is it true that any sheaf on

$S$ with zero Chern classes is isomorphic to

$\mathcal{O}_S^{\oplus n}$ for some

$n$ ? If not, do you have any counterexample?

ginevra86

753

asked Dec 6, 2011 at 13:28

2 votes

1 answer

400 views

Picard/cohomology lattice of surfaces of low degree in $\mathbb P^3$

Let

$S_{d>3}\subset\mathbb{P}^3_{\mathbb{C}}$ be a smooth surface of degree

$d$ . What is known (where to read?) about the Picard/cohomology lattice for small d? e.g. for

$d=4$ the cohomology ...

Dmitry Kerner

2,202

asked Mar 24, 2012 at 11:19

2 votes

1 answer

316 views

$K3$ surfaces can't be uniruled

Let

$S$ be a uniruled surface, ie admits a dominant map

$f:X \times \mathbb{P}^1$ . Why then it's canonical divisor

$\omega_X$ cannot be trivial? Motivation: I want to understand why

$K3$ surfaces ...

user267839

5,716

asked Apr 5 at 21:05

2 votes

0 answers

148 views

Automorphisms of finite order on $K3$ surfaces

Is there a

$K3$ surface (algebraic, complex) that has infinitely many automorphisms of finite order? Many K3 surfaces have infinite automorphism groups. In particular, all K3 surfaces of Picard ...

Basics

1,821

asked Jun 2, 2022 at 15:14

2 votes

0 answers

169 views

Automorphisms of a K3 surface

I was studying the following algebraic surface in

$\mathbb{P}^5$ defined by the following three quadrics: \begin{cases} x^2 + xy + y^2=w^2\\ x^2 + 3xz + z^2=t^2\\ y^2 + 5yz + z^2=s^2. \...

did

585

asked May 28, 2022 at 17:50

2 votes

0 answers

245 views

Example of a K3 surface with two non-symplectic involutions

$\DeclareMathOperator\Pic{Pic}$ Let

$X$ be a K3 surface (algebraic, complex). An involution

$\sigma:X\rightarrow X$ is called non-symplectic if it acts as multiplication by

$-1$ on $H^{2,0}(X)=\Bbb{C}\...

Basics

1,821

asked May 24, 2022 at 5:47

2 votes

0 answers

179 views

rational curves over K3 surfaces over $\mathbb{Q}$

There are many partial results towards the following conjecture: Every projective K3 surface over an algebraically closed field contains infinitely many integral rational curves. My question is: is ...

did

585

asked Mar 10, 2022 at 17:50

2 votes

0 answers

141 views

Is there a way to explicitly find any rational $\mathbb{F}_p$ -curve on the Kummer surface?

Consider a finite field

$\mathbb{F}_p$ (where

$p \equiv 1 \ (\mathrm{mod} \ 3)$ ,

$p \equiv 3 \ (\mathrm{mod} \ 4)$ ),

$\mathbb{F}_{p^2}$ -isomorphic elliptic curves (of

$j$ -invariant

$0$ ) $$ E\!:y_1^2 = ...

Dimitri Koshelev

1,801

asked Jul 26, 2019 at 5:46

2 votes

0 answers

85 views

The quotient of a superspecial abelian surface by the involution

Let

$E_i\!: y_i^2 = f(x_i)$ be two copies of a supersingular elliptic curve over a field of odd characteristics. Consider the involution $$ i\!: E_1\times E_2 \to E_1\times E_2,\qquad (x_1, y_1, x_2, ...

Dimitri Koshelev

1,801

asked May 21, 2018 at 7:46

2 votes

0 answers

259 views

Elliptic fibrations on the Fermat quartic surface

Consider the Fermat quartic surface

$x^4 + y^4 + z^4 + t^4 = 0$ over an algebraically closed field

$k$ of characteristics

$p$ , where

$p \equiv 3$ (

$\mathrm{mod}$

$4$ ). Is there the full ...

Dimitri Koshelev

1,801

asked May 20, 2018 at 12:53

2 votes

0 answers

204 views

Is the Fermat quartic surface a generalized Zariski surface?

Consider the Fermat quartic surface

$F\!: x^4 + y^4 + z^4 + t^4 = 0$ over an algebraically closed field

$k$ of odd characterstics

$p$ . Shioda proved that for

$p=3$ this surface is a generalized ...

Dimitri Koshelev

1,801

asked May 18, 2018 at 11:04

2 votes

2 answers

317 views

What are sufficient and necessary conditions to be a generalized Zariski surface over a finite field?

Let

$X$ be an absolutely irreducible reduced surface over a finite field

$k$ of characteristic

$p$ . What are sufficient and necessary conditions for

$X$ to be a generalized Zariski surface over

$k$ (...

Dimitri Koshelev

1,801

asked Sep 11, 2016 at 21:37

1 vote

1 answer

245 views

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 $|C|...

sqrt2sqrt2

453

asked Jun 1, 2014 at 12:24

1 vote

0 answers

86 views

Picard numbers of isogenous K3 surfaces over a non-closed field

Let

$S_1, S_2$ be K3 surfaces defined over a field

$k$ and

$\phi\!: S_1 \dashrightarrow S_2$ a dominant rational

$k$ -map (so-called isogeny). It is known that

$\rho(S_1) = \rho(S_2)$ for the complex ...

Dimitri Koshelev

1,801

asked May 2, 2020 at 16:18

Stack Exchange Network

All Questions

Enriques surfaces over $\mathbb Z$

K3 surfaces with good reduction away from finitely many places

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

A question on an elliptic fibration of the Enriques surface

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

Are all these K3 surfaces supersingular?

Reference for Automorphisms of K3 surfaces

K3 surfaces with small Picard number and symmetry

K3 surface with $D_{14}$ singular fiber

Does $h^1(D)=0$ imply numerical connectedness on K3 surfaces?

Is a Kummer surface over an finite field $\mathbb{F}_q$ supersingular iff $\mathbb{F}_q$ -unirational?

Genus two pencil in K3 surface

The existence of primitive and sufficiently ample line bundles on K3 surfaces?

Is there a way to calculate the Picard $\mathbb{F}_q$ -number of an (rational or K3) elliptic surface?

What is the Artin invariant of an elliptic supersingular K3 surface?

octic K3s inside cubic 4-folds

Question on K3 Surface

K3 over fields other than C?

A K3 cover over a Del Pezzo surface

Are unirational K3 surfaces defined over finite fields?

The Jacobian surface of an elliptic surface

Fixed part of a line bundle on a K3 surface

Sheaves with zero Chern classes on a $K3$ surface.

Picard/cohomology lattice of surfaces of low degree in $\mathbb P^3$

$K3$ surfaces can't be uniruled

Automorphisms of finite order on $K3$ surfaces

Automorphisms of a K3 surface

Example of a K3 surface with two non-symplectic involutions

rational curves over K3 surfaces over $\mathbb{Q}$

Is there a way to explicitly find any rational $\mathbb{F}_p$ -curve on the Kummer surface?

The quotient of a superspecial abelian surface by the involution

Elliptic fibrations on the Fermat quartic surface

Is the Fermat quartic surface a generalized Zariski surface?

What are sufficient and necessary conditions to be a generalized Zariski surface over a finite field?

Linear system on an abelian surface

Picard numbers of isogenous K3 surfaces over a non-closed field

All Questions

Related Tags