Related papers: Elliptic Surfaces

This is a paper in smooth $4$-manifold topology, inspired by the N\'{e}ron-Lang Theorem in number theory. More precisely, we prove that a smooth version $\MW(\pi)$ of Mordell-Weil group of an elliptic fibration $\pi:M\to\Pb^1$ is finitely…

We study threefolds X in a projective space having as hyperplane section a smooth surface with an elliptic fibration. We first give a general theorem about the possible embeddings of such surfaces with Picard number two. More precise…

It has recently become apparent that the elliptic genera of K3 surfaces (and their symmetric products) are intimately related to the Igusa cusp form of weight ten. In this contribution, I survey this connection with an emphasis on string…

Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the…

We study the surface area of an ellipsoid in n-dimensional Euclidean space as the function of the lengths of their major semi-axes. We write down an explicit formula as an integral over the unit sphere, use the formula to derive convexity…

We define general rotational surfaces of elliptic and hyperbolic type in the pseudo-Euclidean 4-space with neutral metric which are analogous to the general rotational surfaces of C. Moore in the Euclidean 4-space. We study Lorentz general…

We describe two geometrically meaningful compactifications of the moduli space of elliptic K3 surfaces via stable slc pairs, for two different choices of a polarizing divisor, and show that their normalizations are two different toroidal…

We consider a rational surface with a relatively minimal fibration. Picard number of a such fibred surface is bounded in terms of the genus of a general fibre. When Picard number is the maximum for any given genus, we characterize a such…

This article is an introduction to Mordell-Weil theorem. In this article, I introduced some basic properties about ellptic curves and proved the theorem in two different ways.

We undertake a study of topological properties of the real Mordell-Weil group $\operatorname{MW}_{\mathbb R}$ of real rational elliptic surfaces $X$ which we accompany by a related study of real lines on $X$ and on the "subordinate" del…

A fibration is said to be isotrivial if all of its smooth fibres are isomorphic to a single fixed variety. We classify the elliptic K3 surfaces that are isotrivial, and use them to construct Lagrangian fibrations that are isotrivial. We…

Given an elliptic quartic of type $Y^2=f(X)$ representing an elliptic curve of positive rank over $\Q$, we investigate the question of when the $Y$-coordinate can be represented by a quadratic form of type $ap^2+bq^2$. In particular, we…

In this survey we report a general and systematic approach to study $[\varphi,\vec{e}_{3}]$-minimal surfaces in $\mathbb{R}^{3}$ from a geometric viewpoint and show some fundamental results obtained in the recent development of this theory.

The aim of the paper is to provide a series of new examples of smooth surfaces in P^4, not of general type, in degrees varying from 12 up to 14, and to describe their geometry. By using mainly syzygies and liaison techniques, we construct…

Let $K$ be a finitely generated field over $\mathbb{Q}$. Let $\mathcal{X}\to \mathcal{B}$ be a family of elliptic surfaces over $K$ such that each elliptic fibration has the same configuration of singular fibers. Let $r$ be the minimum of…

This article wants to show two things: first, that certain problems in Diophantus' Arithmetica lead to equations defining del Pezzo surfaces or other rational surfaces, while certain others lead to K3 surfaces; second, that Diophantus' own…

We classify, up to automorphisms, the elliptic fibrations on the singular K3 surface $X$ whose transcendental lattice is isometric to $\langle 6\rangle\oplus \langle 2\rangle$.

We construct several examples of genus-one fibered K3 surfaces without a global section with type $I_{n}$ fibers, by considering double covers of a special class of rational elliptic surfaces lacking a global section, known as Halphen…

We construct a family of elliptic surfaces with $p_g=q=1$ that arise from base change of the Hesse pencil. We identify explicitly a component of the higher Noether-Lefschetz locus with positive Mordell-Weil rank, and a particular surface…

Let $X$ be a quadratic vector field with a center whose generic orbits are algebraic curves of genus one. To each $X$ we associate an elliptic surface (a smooth complex compact surface which is a genus one fibration). We give the list of…