The equation x1 x2 + x2 x3 + x3 x4 + x4 x1 = n

1 It is a subtle question as to when the Diophantine equation of the tittle has solutions in positive integers. Here, we show that the equation in the title does not have solutions in positive integers in the case that [Formula: see text] is of the form [Formula: see text], where [Formula: see text], with [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. We do this by explicitly calculating a Brauer–Manin obstruction to weak approximation on the elliptic surface defined by the title equation.

Elliptic Fibrations and Kodaira Dimensions of Schreieder’s Varieties

We discuss the birational geometry and the Kodaira dimension of certain varieties previously constructed by Schreieder, proving that in any dimension they admit an elliptic fibration and they are not of general type. The $l$-dimensional variety $Y_{(n)}^{(l)}$, which is the quotient of the product of a certain curve $C_{(n)}$ by itself $l$ times by a group $G\simeq \left ({\mathbb{Z}}/n{\mathbb{Z}}\right )^{l-1}$ of automorphisms, was constructed by Schreieder to obtain varieties with prescribed Hodge numbers. If $n=3^c$ Schreieder constructed an explicit smooth birational model of it, and Flapan proved that the Kodaira dimension of this smooth model is 1, if $c>1$; if $l=2$ it is a modular elliptic surface; if $l=3$ it admits a fibration in K3 surfaces. In this paper we generalize these results: without any assumption on $n$ and $l$ we prove that $Y_{(n)}^{(l)}$ admits many elliptic fibrations and its Kodaira dimension is at most 1. Moreover, if $l=2$, its minimal resolution is a modular elliptic surface, obtained by a base change of order $n$ on a specific extremal rational elliptic surface; if $l\geq 3$ it has a birational model that admits a fibration in K3 surfaces and a fibration in $(l-1)$-dimensional varieties of Kodaira dimension at most 0.

Bounding Tangencies of Sections on Elliptic Surfaces

Given an elliptic surface $\mathcal{E}\to \mathcal{C}$ over a field $k$ of characteristic zero equipped with zero section $O$ and another section $P$ of infinite order, we give a simple and explicit upper bound on the number of points where $O$ is tangent to a multiple of $P$.

The Nef Cone of the Hilbert Scheme of Points on Rational Elliptic Surfaces and the Cone Conjecture

We compute the nef cone of the Hilbert scheme of points on a general rational elliptic surface. As a consequence of our computation, we show that the Morrison–Kawamata cone conjecture holds for these nef cones.

UNWEIGHTED DONALDSON–THOMAS THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSES

We further the study of the Donaldson–Thomas theory of the banana 3-folds which were recently discovered and studied by Bryan [3]. These are smooth proper Calabi–Yau 3-folds which are fibred by Abelian surfaces such that the singular locus of a singular fibre is a non-normal toric curve known as a ‘banana configuration’. In [3], the Donaldson–Thomas partition function for the rank 3 sub-lattice generated by the banana configurations is calculated. In this article, we provide calculations with a view towards the rank 4 sub-lattice generated by a section and the banana configurations. We relate the findings to the Pandharipande–Thomas theory for a rational elliptic surface and present new Gopakumar–Vafa invariants for the banana 3-fold.

Explicit moduli spaces for congruences of elliptic curves

We determine explicit birational models over $${{\mathbb {Q}}}$$ Q for the modular surfaces parametrising pairs of N-congruent elliptic curves in all cases where this surface is an elliptic surface. In each case we also determine the rank of the Mordell–Weil lattice and the geometric Picard number.

Quadratic points of surfaces in projective 3-space

Quadratic points of a surface in the projective 3-space are the points which can be exceptionally well approximated by a quadric. They are also singularities of a 3-web in the elliptic part and of a line field in the hyperbolic part of the surface. We show that generically the index of the 3-web at a quadratic point is ±1/3, while the index of the line field is ±1. Moreover, for an elliptic quadratic point whose cubic form is semi-homogeneous, we can use Loewner’s conjecture to show that the index is at most 1. From the above local results, we can conclude some global results: A generic compact elliptic surface has at least 6 quadratic points, a compact elliptic surface with semi-homogeneous cubic forms has at least 2 quadratic points and the number of quadratic points in a hyperbolic disc is odd. By studying the behavior of the cubic form in a neighborhood of the parabolic curve, we also obtain a relation between the indices of the quadratic points of a generic surface with non-empty elliptic and hyperbolic regions.