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Questions tagged [elliptic-surfaces]

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2
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0answers
112 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
167 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
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0answers
118 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 ...
5
votes
1answer
336 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
191 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
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0answers
148 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
324 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
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0answers
176 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
145 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
120 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
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1answer
108 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
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0answers
118 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 ...
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0answers
156 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
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0answers
203 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
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0answers
305 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
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0answers
112 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
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0answers
146 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
471 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
158 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
195 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 ...
8
votes
1answer
692 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
569 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
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0answers
238 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 ...
5
votes
1answer
230 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
258 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
227 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
95 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 ...
1
vote
0answers
321 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
275 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
350 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 ...
2
votes
1answer
282 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
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0answers
532 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
308 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
291 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
335 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
935 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 ...
3
votes
0answers
144 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 $E$ ...
4
votes
0answers
470 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 ...
3
votes
2answers
738 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 ...
10
votes
1answer
555 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 ...
11
votes
3answers
1k views

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 ...