Questions tagged [elliptic-surfaces]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1
vote
1answer
223 views

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{\...
1
vote
0answers
54 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 ...
1
vote
0answers
46 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 ...
9
votes
2answers
441 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 ...
3
votes
0answers
127 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
1answer
249 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 (\...
3
votes
1answer
279 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 ...
1
vote
0answers
43 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 $\...
1
vote
0answers
90 views

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 $\sqrt[3]{2} \notin \mathbb{F}_p$), ...
2
votes
0answers
62 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 ...
4
votes
0answers
77 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 ...
5
votes
0answers
193 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 ...
2
votes
1answer
162 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....
5
votes
3answers
435 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 ...
2
votes
0answers
129 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/...
3
votes
1answer
222 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 ...
3
votes
0answers
137 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 ...
7
votes
1answer
475 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-...
5
votes
1answer
300 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 (...
4
votes
0answers
220 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 ...
6
votes
1answer
395 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 ...
6
votes
0answers
187 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 $S=R\times_{\Bbb P^1} \Bbb P^1 \to \Bbb P^1$, ...
3
votes
1answer
182 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 ...
3
votes
1answer
139 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 (...
2
votes
1answer
164 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 ...
2
votes
0answers
130 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
vote
0answers
189 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 ...
4
votes
0answers
232 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}(...
3
votes
0answers
436 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 \...
3
votes
0answers
137 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 $\mathbb{P}^2$. ...
3
votes
0answers
159 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 ...
3
votes
1answer
715 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 ...
2
votes
1answer
191 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 $\mathbb P^1(\mathbb C)$...
7
votes
1answer
208 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 ...
7
votes
1answer
881 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 ...
1
vote
1answer
584 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 ...
0
votes
0answers
244 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 ...
6
votes
1answer
260 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 $E_2/G=\mathbf{P}^1$. ...
4
votes
1answer
276 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 $H^0(\mathbb{P}^1,R^2f_*\mathbb{R})$ ...
3
votes
1answer
271 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 $...
2
votes
0answers
103 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 ...
2
votes
0answers
375 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 ...
5
votes
1answer
289 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 ...
4
votes
1answer
405 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 ...
1
vote
1answer
360 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 ...
7
votes
0answers
606 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 ...
6
votes
2answers
332 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 ...
2
votes
1answer
364 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(\...
3
votes
1answer
419 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 ...
6
votes
3answers
1k 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 ...