Related papers: Linear systems representing threefolds which are s…
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…
We analyze morphisms from pointed curves to K3 surfaces with a distinguished rational curve, such that the marked points are taken to the rational curve, perhaps with specified cross ratios. This builds on work of Mukai and others…
Some of the 95 families of weighted K3 hypersurfaces have been known to have the isometric lattice polarizations. It is shown that weighted K3 hypersurfaces in such families are to one-to-one correspond by explicitly constructing the…
Let $X$ be an algebraic variety, defined over the rationals. This paper gives upper bounds for the number of rational points on $X$, with height at most $B$, for the case in which $X$ is a curve or a surface. In the latter case one excludes…
Modern extended reality XR systems provide rich analysis of image data and fusion of sensor input and demand AR/VR applications that can reason about 3D scenes in a semantic manner. We present a spatial reasoning framework that bridges…
We construct an explicit K3 surface over the field of rational numbers that has geometric Picard rank one, and for which there is a transcendental Brauer-Manin obstruction to weak approximation. To do so, we exploit the relationship between…
We present a form of algebraic reasoning for computational objects which are expressed as graphs. Edges describe the flow of data between primitive operations which are represented by vertices. These graphs have an interface made of…
The questions of global topological, smooth and holomorphic classifications of the differential systems, defined by covering foliations, are considered. The received results are applied to nonautonomous linear differential systems and…
We classify and construct all line graphs that are $3$-polytopes (planar and $3$-connected). Apart from a few special cases, they are all obtained starting from the medial graphs of cubic (i.e., $3$-regular) $3$-polytopes, by applying two…
In this paper we present algorithms for computing the topology of planar and space rational curves defined by a parametrization. The algorithms given here work directly with the parametrization of the curve, and do not require to compute or…
Links in $S^3$ as well as Legendrian links in the standard tight contact structure on $S^3$ can be encoded by grid diagrams. These consist of a collection of points on a toroidal grid, connected by vertical and horizontal edges. Blackwell,…
Let $\mathbb{P}$ denote the weighted projective space with weights $(1,1,1,3)$ over the rationals, with coordinates $x,y,z,$ and $w$; let $\mathcal{X}$ be the generic element of the family of surfaces in $\mathbb{P}$ given by…
Consider a (non-empty) linear system of surfaces of degree d in P^3 through at most 8 multiple points in general position and let L denote the corresponding complete linear system on the blowing-up X of P^3 along those general points. Then…
A general linear determinantal quartic in $\mathbb{P}^4$ is nodal, non-$\mathbb{Q}$-factorial and rational. We show that the family $\mathcal{F}$ of such quartics also contains rational $\mathbb{Q}$-factorial quartics, and that a generic…
We review recent developments in the arithmetic of K3 surfaces. Our focus lies on aspects of modularity, Picard number and rational points. Throughout we emphasise connections to geometry.
We introduce a simple combinatorial way, which we call a rectangular diagram of a surface, to represent a surface in the three-sphere. It has a particularly nice relation to the standard contact structure on $\mathbb S^3$ and to rectangular…
Let $X\subset \P^5$ be a smooth cubic fourfold. A well known conjecture asserts that $X$ is rational if and only if there an Hodge theoretically associated K3 surface $S$. The surface $S$ can be associated to $X$ in two other different…
We consider certain elliptic threefolds over the projective plane (more generally over certain rational surfaces) with a section in Weierstrass normal form. In particular, over a del Pezzo surface of degree 8, these elliptic threefolds are…
This is a survey on the classification of smooth surfaces in P^4 and smooth 3-folds in P^5. We recall the corresponding results arising from adjunction theory and explain how to construct examples via syzygies. We discuss some examples in…
A canonical branched covering over each sufficiently good simplicial complex is constructed. Its structure depends on the combinatorial type of the complex. In this way, each closed orientable 3-manifold arises as a branched covering over…