Questions tagged [elliptic-surfaces]
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63
questions
12
votes
3
answers
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A K3 over $P^1$ with six singular $A_1$- fibers?
Hirzebruch, in the paper 'Arrangements of Lines and Algebraic Surfaces'
constructs a special $K3$ surface out of a 'complete quadrilateral' in
$CP^2$. A complete quadritlateral consists of
4 ...
- 8,118
12
votes
1
answer
675
views
Dodecahedral K3?
In pondering
this
MO question and in particularly its 1st answer, and answers to
this one recently posed, I realized there ought to be a dodecahedral K3 surface $X$.
This $X$ would fiber as an ...
- 8,118
9
votes
2
answers
758
views
Do singular fibers determine the elliptic K3 surface, generically?
General elliptic K3 surfaces. Consider K3 surfaces of Picard rank two with Neron-Severi lattice isomorphic to $$\left[\begin{array}{cc}
2d & t \\
t & 0
\end{array}\right]$$ for some positive ...
- 2,260
7
votes
1
answer
568
views
Discriminant locus of elliptic K3 surfaces
Given a complex elliptic K3 surface $\pi\colon X\rightarrow \mathbb P^1$, its discriminant locus is the divisor $$D = \sum_{i = 1}^s n_i P_i$$ on $\mathbb P^1$ such that $n_i$ is equal to the Euler-...
7
votes
1
answer
1k
views
Is there a description of the moduli space of elliptic surfaces?
In this question elliptic surface means a smooth projective complex surface $X$, such that there is an elliptic fibration $\pi \colon X \to C$. (I.e., there is a curve $C$ and a proper map $\pi$, such ...
- 5,444
7
votes
1
answer
223
views
Question on paper of Stewart and Top about ranks of elliptic curves over Q(t)
I'm reading "On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms" by Stewart and Top, and struggling to understand the argument on pg 962 which shows that the rank of a ...
- 585
7
votes
0
answers
668
views
Explicit family of generalized elliptic curves with level n structure
Let $\pi:\mathcal{E}\rightarrow U$ be a family of elliptic curves with level $n$ structure (in the sense of Deligne-Rapoport) where $U\subseteq C$ is some (non-empty) Zariski open set of a smooth ...
- 7,521
6
votes
2
answers
346
views
Mordell-Weil of an elliptic surface after adjoining a nontorsion section: as small as possible?
Let $k$ be an algebraically closed field of characteristic $0$, let $C_{/k}$ be a nice (smooth, projective, geometrically integral curve), let $K = k(C)$, and let $\overline{K}$ be an algebraic ...
- 64.8k
6
votes
3
answers
2k
views
Surfaces ruled over elliptic curves
Ground field $\Bbb{C}$. Algebraic category. Elliptic surfaces are those surfaces endowed with a morphism onto some smooth curve, with generic fiber an elliptic curve.
Suppose $E$ is an elliptic curve ...
- 105
6
votes
1
answer
448
views
Two definitions of the narrow Mordell-Weil group
Let:
$K = k(C)$, where $C/k$ is a projective non-singular curve,
$E/K$ - an elliptic curve,
$\mathcal{E} \to C$ - the minimal elliptic surface associated to $E$.
Consider the "narrow Mordell-Weil ...
- 411
6
votes
1
answer
310
views
Linear systems on bielliptic surfaces
A bielliptic surface is a surface of type $S=E_1 \times E_2/G$ where $E_1, E_2$ are elliptic curves and $G$ is a finite group of translations of $E_1$ acting on $E_2$ such that $E_2/G=\mathbf{P}^1$.
...
- 453
6
votes
0
answers
201
views
Produce supersingular K3 from rational elliptic surfaces
Given a rational elliptic surface $R \to \Bbb P^1$, is there a way to know if there exists a supersingular K3 surface that arises as a base curve change $S=R\times_{\Bbb P^1} \Bbb P^1 \to \Bbb P^1$, ...
- 151
5
votes
3
answers
488
views
Is there a way to find any non-trivial $\mathbb{F}_p(t)$-point on the given elliptic curve?
Consider a finite field $\mathbb{F}_p$ (where $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$) and the elliptic curve
$$
E\!:y^2 = x^3 + (t^6 + 1)^2
$$
over the univariate ...
- 1,801
5
votes
1
answer
298
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 ...
- 73
5
votes
1
answer
456
views
Properly elliptic surface with no multiple fibers and without a section
I am aware that if an elliptic surface contains multiple fibers, then it has no section. Is the converse false?
In particular, I am looking for an example of a projective, properly elliptic surface (...
- 61
5
votes
0
answers
335
views
Jacobian fibration of an abelian fibration
Let $f \colon S \rightarrow C$ be a minimal elliptic surface and let $g \colon J \rightarrow C$ be its jacobian fibration. In this case, we know that the fibers of $g$ are better behaved that the ones ...
- 625
4
votes
1
answer
467
views
Kodaira classification and the McKay correspondence
Kodaira's table of singular fibers has a singular fiber
for each of $\tilde{A}_n$, $\tilde{D}_n$, $\tilde{E}_6$, $\tilde{E}_7$, and $\tilde{E}_8$; these are chains or cycles of (-2)-curves connected ...
4
votes
1
answer
291
views
Generators of cohomology groups of higher push-forward sheaves
Let $\phi:S\rightarrow \mathbb{P}^1$ be an elliptic fibration of a compact complex surface. Assume that there is a multiple section $s$ of $\phi$. Is it true that $H^0(\mathbb{P}^1,R^2f_*\mathbb{R})$ ...
- 41
4
votes
0
answers
87
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 ...
- 1,801
4
votes
0
answers
299
views
Action of the Picard Scheme of an Elliptic Fibration
Suppose that we have a surface $X$ defined over a field $k$ (I am interested in $k$ being a number field) and an elliptic fibration $f: X \rightarrow \mathbb{P}^1$, i.e. $f$ is proper and almost all ...
- 241
4
votes
0
answers
287
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}(...
- 1,801
4
votes
0
answers
545
views
Singular fibers of an elliptic fibered K3 surface.
Let $f:S\rightarrow \mathbb{P}^1$ be an elliptic K3 surface. Assume that $\mathrm{Pic}(S)\cong U$, where $U$ stands for the hyperbolic lattice. I think that the elliptic fibration has only singular ...
- 41
3
votes
2
answers
1k
views
Singular fibres in the definition of an elliptic surface
I have a question about the definition of an elliptic surface. One defines an elliptic surface $S$ over a base curve $C$ (over some field $k$) as a surjective morphism $f: S \to C$ such that almost ...
- 5,123
3
votes
1
answer
150
views
Foliation by Asymptotic lines
Suppose $(M,g)$ is a compact Riemannian manifold with boundary. I am interested about existence of surfaces $\Gamma$ embedded in $M$ with the following property:
$\Gamma$ is foliated by geodesics (...
- 4,111
3
votes
1
answer
304
views
elliptic fibration over $\mathbb{P}^1$ with exactly two fibres with monodromy of unipotency rank 1
Despite the apparent simplicity of the following question I couldn't find the answer so far.
I am looking to construct an elliptic fibration $X \to \mathbb{P}^1$ with $X$ smooth, and exactly two ...
- 4,060
3
votes
1
answer
312
views
Spectral sequence associated to elliptic fibration degenerates?
Let $\phi:S\rightarrow \mathbb{CP}^1$ be an elliptic fibration of a K3 surface. When is the Leray spectral sequence associated to the fibration $E_2$-degenerate? Are there any good criteria for the $...
- 31
3
votes
1
answer
929
views
Elliptic fibrations with few singular fibers
It is known that non-isotrivial fibrations of genus $g>0$ curves over the projective line have a bunch of singular fibers. There are at least three of them.
It is not difficult to prove that an ...
- 492
3
votes
1
answer
382
views
Mordell–Weil rank of some elliptic $K3$ surface
Consider a finite field $\mathbb{F}_q$ such that $q \equiv 1 \pmod3$ (i.e., $\omega \mathrel{:=} \sqrt[3]{1} \in \mathbb{F}_q$ for $\omega \neq 1$) and an element $b \in (\mathbb{F}_q^*)^2 \setminus (\...
- 1,801
3
votes
1
answer
235
views
Existence of elliptic surface on Riemann surface with marked points
Is there any proof for the following statement? It has been used as trivial fact in the one of papers of Edward Witten
Let $\Sigma$ be a compact connected Riemann surface as orbifold, with marked ...
user21574
3
votes
1
answer
360
views
Specializing p-torsion in a family of elliptic surfaces
Let $R$ be a DVR of mixed characteristic, with algebraically closed residue field of characteristic $p$ and fraction field $K$. Let $Y\longrightarrow \operatorname{Spec} R$ be a smooth projective ...
- 599
3
votes
1
answer
522
views
A question on existence of degeneration of Enriques surface.
Let $S$ be an Enriques surface, i.e. a quotient of a K3 surface by a free involution. Enriques surfaces arise as elliptic fibrations $S\rightarrow \mathbb{P}^1$ with 12 singular fibers and 2 double ...
- 1,653
3
votes
0
answers
242
views
Ample divisor of degree two on a blow-up of $\mathbb P^2$ at nine points
Let $\pi:S \rightarrow \mathbb P^2$ be a blow-up at nine points in general position.
I am finding an ample divisor $L$ on $S$ of degree two ($L^2=2$).
Since $Pic(S) = \mathbb Z h \oplus \mathbb Z e_1 \...
- 1,821
3
votes
0
answers
165
views
Reference Request: CM Motives over Function Fields
Inspired by Schutt and Shioda's lovely "An interesting elliptic surface over an elliptic curve", I have been investigating the following surface:
$$
\mathcal{E} : y^2 = x^3 - 27ux - 54v \...
3
votes
0
answers
142
views
Are there three ordinary elliptic curves $E$, $E_1$, $E_2$ such that $E^2 \cong E_1 \!\times\! E_2$?
Consider the elliptic curve $E\!: y^2 = x^3 + 1$ of $j$-invariant $0$ over an algebraically closed field $k$ of characteristics $p$. Let me remind that $E$ is ordinary (i.e., non-supersingular) iff $p ...
- 1,801
3
votes
0
answers
553
views
The Jacobian surface of an elliptic surface
Let $\mathcal{X}$ be an elliptic surface over $\mathbb{P}^1$ without a section and let $\mathcal{J}$ be an elliptic surface over $\mathbb{P}^1$ with a section. Assume we have the commutative diagram
\...
- 1,801
3
votes
0
answers
165
views
Can a toric surface be an elliptic surface?
It is known that a rational elliptic surface is a blow-up of $\mathbb{P}^2$ at 9 points. More precisely it is obtained as the blow-up of the base locus of a pencil of cubic curves in $\mathbb{P}^2$. ...
- 31
3
votes
0
answers
172
views
Elliptic fibration arising from a higher genus linear system
Let $H$ be a very ample linear system on a smooth compact complex surface $X$ whose Kodaira dimension is $\geq 0$. A general element of $H$ is smooth and has genus $\geq 2$.
Let $L\subset H$ be a ...
- 492
3
votes
0
answers
154
views
Topology of K3 as a sum of two abelian fibrations.
Let $E$ be a blow-up of $\mathbb{P}^2$ at 9-points in the bases locus of a pencil of elliptic curves (A $T^2$ fibration over $S^2$).
K3 surfaces is obtained by removing a fiber from two copies of $E$ ...
2
votes
1
answer
424
views
Example of non-modular elliptic surface?
In "On elliptic modular surfaces", Shioda proves some interesting theorems on smooth elliptic surfaces (admitting a section); he then focuses on "modular elliptic surfaces" and proves some more ...
2
votes
1
answer
183
views
Is there a way to find any $\mathbb{F}_2(t)$-point on the elliptic curve $\mathcal{E}$?
Consider the ordinary elliptic curves
$$
E\!:y_1^2 + x_1y_1 = x_1^3 + 1,\qquad E^\prime\!: y_2^2 + x_2y_2 = x_2^3 + x_2^2 + 1
$$
over the field $\mathbb{F}_2$. They are quadratic twists to each other....
- 1,801
2
votes
1
answer
254
views
Analogue of Kodaira surfaces
Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{C}$. A (primary) Kodaira surface is a principal bundle $X \to E_1$ with fibre $E_2$. $X$ is a compact complex surface with trivial canonical bundle ...
- 830
2
votes
1
answer
259
views
Looking for Schmickler-Hirzebruch' monograph on elliptic surfaces
I wonder if it is possible to find (and if yes, where?) an electronic copy of the following monograph:
Author: Schmickler-Hirzebruch, Ulrike
Title: Elliptische Flächen über $\mathbb P^1(\mathbb C)$...
- 828
2
votes
1
answer
445
views
Specialization of sections in an elliptic fibration
Let $\pi: X \rightarrow S$ be the Neron model of an elliptic curve over a dedekind domain (but probably any minimal elliptic fibration will suffice).
Let $\eta$ be the generic point of $S$, $K = S(\...
- 33
2
votes
0
answers
65
views
What conditions are sufficient for two points to be independent in the Mordell-Weil group?
Consider a finite field $\mathbb{F}_{q}$ and an elliptic surface
$$
\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(t),
$$
where $a_i(t) \in \mathbb{F}_{q}[t]$. I am mainly ...
- 1,801
2
votes
0
answers
136
views
Global sections of ample line bundles over (rational) elliptic fibration
Let $S$ be a smooth, complex elliptic fibration over $\mathbb{P}^1$ and $L$ be an ample invertible sheaf on $S$. I am looking for criterion under which $L$ has a non-trivial global section. Any idea/...
- 2,126
2
votes
0
answers
137
views
Is a supersingular Kummer surface $k$-unirational in characteristic 2?
Let $k$ be a perfect field of even characteristic. Consider the simplest example of a supersingular genus 2 curve, i.e.,
$$
C\!: y^2 + y = x^5.
$$
By the article of J. S. Müller "Explicit Kummer ...
- 1,801
2
votes
0
answers
115
views
Elliptic surfaces with different Kodaira symbols
Are there examples of surfaces $E$ of Kodaira dimension one that have two elliptic fibrations $p,q:E\to C$ over some curve $C$ such that $p$ has semi-stable fibres but $q$ has an additive fibre?
Can ...
- 133
2
votes
0
answers
418
views
Average rank of elliptic curves over function fields
De Jong showed in 2002 if the finite field $\mathbb{F}_q$ has characteristic not equal to 3, then the limsup of the average of 3-Selmer rank is bounded above, where the average is taken over the ...
- 809
1
vote
1
answer
303
views
Blow-up of a pencil of cubic curves (from Miranda's basic theory of elliptic surfaces)
Rick Miranda's "The basic theory of elliptic surfaces" the Example (I.5.1) see page 7 on a pencil of plane curves contains an argument that I do not understand yet.
Let $C_1$ be a smooth ...
- 6,000
1
vote
1
answer
589
views
Can monodromy be described by the same matrix for chosen generators in case of the same singularity type?
Let $X$ be a surface in $\mathbb{P}^3$. We have a fibration $f: X \longrightarrow \mathbb{P}^1$, and $f^{-1}(s_1)$ and $f^{-1}(s_2)$ have the same singularity type. Let $\gamma_1$ and $\gamma_2$ be ...
- 319