# Questions tagged [elliptic-surfaces]

The elliptic-surfaces tag has no usage guidance.

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### How to link the rank of Elliptic Surfaces and Elliptic Curves over function fields?

I have been investigating certain elliptic surfaces for my research, and when giving a presentation, I was asked why when given an elliptic surface $E$, say given in Weierstrass form
$$E\colon y^2+a_1(...

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1
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### Calculate blowup of a pencil of cubics "by hand"

I have one more question about the Example (I.5.1) on page 7 from
Rick Miranda's the basic theory of elliptic surfaces:
Let $C_1$ be a smooth cubic curve in $\mathbb{P^2}$ and let $C_2$
be any other ...

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### An elliptic threefold and the Mordell–Weil lattices of its reductions

Let $T\!: y^2 = x^3 + a(t, s)x + b(t, s)$ be an elliptic threefold over a finite field $\mathbb{F}_q$ of characteristic $p > 3$. In other words, we have an elliptic curve over the function field $\...

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### 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 ...

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### 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 \...

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1
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237
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### Coefficients of elliptic curves over function fields

Consider the projective plane $\mathbb{P}^2_{\overline{\mathbb{C}(t)}}$ over the algebraic closure of the function field $\mathbb{C}(t)$.
Take the point $p_0 = [0:1:0]\in \mathbb{P}^2_{\overline{\...

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### Elliptic fibrations on some Kummer surface in characteristic $2$

In the question I ask about one elliptic fibration on the surface
$$
K\!: y^2 + x_1x_2y = (x_1x_2)^2(x_1 + x_2 + 1) + (x_1 + x_2)^2.
$$
over a finite field $\mathbb{F}_q$ of characteristic $2$ such ...

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### Classification of genus 1 Lefschetz fibration over a disk with 6 singular fibers

For a genus 1 Lefschetz fibration over a sphere, there is a good classification in B. Moishezon, Complex surfaces and connected sums of complex projective planes. In this case, the number $K$ of ...

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### 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 ...

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### 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 \...

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### 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 (\...

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### 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 ...

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### Vanishing cycles and relative homology of I_n fiber

Suppose $\pi: X \rightarrow D$ is a smooth elliptic fibration with a section over the closed disk without multiple fibers, such that all special fibers are in the interior of $D$. The boundary $\...

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### Is there an infinite order $\mathbb{F}_{p}$-section for a certain elliptic surface $\mathcal{E}_n$?

Consider a natural number $n$, a finite field $\mathbb{F}_{p}$ (such that $p$ is prime, $p \equiv 1 \ (\mathrm{mod} \ 3)$, $p \equiv 3 \ (\mathrm{mod} \ 4)$, and $\sqrt[3]{2} \notin \mathbb{F}_p$), ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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....

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### 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 ...

2
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answers

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### 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/...

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### 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 ...

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### 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 ...

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### 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-...

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### 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 (...

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### 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 ...

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### 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 ...

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### 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 $S=R\times_{\Bbb P^1} \Bbb P^1 \to \Bbb P^1$, ...

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### 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 ...

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### 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 (...

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### 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 ...

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### 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 ...

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### Mordell-Weil groups of Elliptic Surfaces

Consider the moduli space $M_d$ ($d\geq 2$) of relatively minimal Jacobian elliptic surfaces $S\to\mathbb P^1$ with $p_g(S)=d$. This was constructed by Miranda (1981) using Weierstrass form and the ...

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### 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}(...

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### 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
\...

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### 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 $\mathbb{P}^2$. ...

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### 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 ...

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### 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 ...

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### 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 $\mathbb P^1(\mathbb C)$...

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### 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 ...

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### 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 ...

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### 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 ...

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### Understanding a proof of a lemma in elliptic surfaces

In the following paper , The Kahler-Ricci flow on surfaces of positive Kodaira dimension (Sung and Tian) in page 621, I am trying to understand the proof of part 1 of Lemma 3.4.
In the part 1 of ...

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### 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 $E_2/G=\mathbf{P}^1$.
...

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### 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 $H^0(\mathbb{P}^1,R^2f_*\mathbb{R})$ ...

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281
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### 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 $...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...