Questions tagged [elliptic-surfaces]

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Is there an isotrivial elliptic surface of positive rank having a section of order $3$?

Let $k$ be a field of characteristic $p > 3$. I cannot find any example of ordinary isotrivial elliptic $k$-surface $E$ (i.e., elliptic $k(t)$-curve, where $t$ is a variable) whose Mordell-Weil ...
Dimitri Koshelev's user avatar
1 vote
0 answers
89 views

Is it possible to lift a pair of points on an elliptic $\mathbb{F}_{\!q}$-curve to a pair of short points on an elliptic $\mathbb{F}_{\!q}(t)$-curve?

Let $E$ be an (ordinary) elliptic curve over a finite field $\mathbb{F}_{\!q}$ of a (quite large) characteristic. For simplicity, suppose that $E(\mathbb{F}_{\!q})$ is a prime group. In addition, let $...
Dimitri Koshelev's user avatar
0 votes
1 answer
104 views

Is there a method to check if two sections of an elliptic surface are dependent over the endomorphism ring or not?

Let $E\!: y^2 = x^3 + a(t)x + b(t)$ be an elliptic curve over the function field $\mathbb{F}_{q}(t)$ over a finite field $\mathbb{F}_{q}$ of characteristic $5$ or greater. For simplicity, let $E$ be ...
Dimitri Koshelev's user avatar
1 vote
0 answers
152 views

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(...
Sam Gue's user avatar
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1 answer
238 views

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 ...
user267839's user avatar
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1 vote
0 answers
92 views

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 $\...
Dimitri Koshelev's user avatar
1 vote
1 answer
274 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 ...
user267839's user avatar
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3 votes
0 answers
238 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 \...
Basics's user avatar
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1 answer
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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{\...
Puzzled's user avatar
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1 vote
0 answers
75 views

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 ...
Dimitri Koshelev's user avatar
1 vote
0 answers
64 views

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 ...
Faniel's user avatar
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9 votes
2 answers
739 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 ...
Evgeny Shinder's user avatar
3 votes
0 answers
161 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 \...
Angus McAndrew's user avatar
3 votes
1 answer
378 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 (\...
Dimitri Koshelev's user avatar
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 ...
pozio's user avatar
  • 599
1 vote
0 answers
70 views

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 $\...
moep's user avatar
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1 vote
0 answers
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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 $\sqrt[3]{2} \notin \mathbb{F}_p$), ...
Dimitri Koshelev's user avatar
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 ...
Dimitri Koshelev's user avatar
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's user avatar
5 votes
0 answers
327 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 ...
Stefano's user avatar
  • 625
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....
Dimitri Koshelev's user avatar
5 votes
3 answers
484 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 ...
Dimitri Koshelev's user avatar
2 votes
0 answers
135 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/...
Ron's user avatar
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3 votes
1 answer
296 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 ...
Dima Sustretov's user avatar
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 ...
Dimitri Koshelev's user avatar
7 votes
1 answer
565 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-...
Davide Cesare Veniani's user avatar
5 votes
1 answer
439 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 (...
user564401's user avatar
4 votes
0 answers
297 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 ...
Sam Streeter's user avatar
6 votes
1 answer
446 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 ...
Jędrzej Garnek's user avatar
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 $S=R\times_{\Bbb P^1} \Bbb P^1 \to \Bbb P^1$, ...
Vinicius M.'s user avatar
3 votes
1 answer
232 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 ...
user avatar
3 votes
1 answer
149 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 (...
Ali's user avatar
  • 3,995
2 votes
1 answer
249 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 ...
Andrei Halanay's user avatar
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 ...
Dimitri Koshelev's user avatar
1 vote
0 answers
216 views

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 ...
François's user avatar
  • 464
4 votes
0 answers
285 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's user avatar
3 votes
0 answers
547 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's user avatar
3 votes
0 answers
163 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 $\mathbb{P}^2$. ...
Fan's user avatar
  • 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 ...
Alan Muniz's user avatar
3 votes
1 answer
909 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 ...
Alan Muniz's user avatar
2 votes
1 answer
257 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 $\mathbb P^1(\mathbb C)$...
Lucien's user avatar
  • 828
7 votes
1 answer
222 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 ...
stl's user avatar
  • 585
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 ...
jmc's user avatar
  • 5,434
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 ...
ozheidi's user avatar
  • 229
0 votes
0 answers
248 views

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 ...
user avatar
6 votes
1 answer
306 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 $E_2/G=\mathbf{P}^1$. ...
sqrt2sqrt2's user avatar
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 $H^0(\mathbb{P}^1,R^2f_*\mathbb{R})$ ...
Hua's user avatar
  • 41
3 votes
1 answer
309 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 $...
Hua's user avatar
  • 31
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 ...
Ste3an's user avatar
  • 133
2 votes
0 answers
413 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 ...
Pig's user avatar
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