Related papers: Arrangements of rational sections over curves and …
We study the surface $\bar{S}$ parametrizing cuboids: it is defined by the equations relating the sides, face diagonals and long diagonal of a rectangular box. It is an open problem whether a `rational box' exists, i.e., a rectangular box…
This article introduces the theory of Veronese polytopes, a broad generalisation of cyclic polytopes. These arise as convex hulls of points on curves with one or more connected components, obtained as the image of the rational normal curve…
We study intersection theory and Chern classes of reflexive sheaves on normal varieties. In particular, we define generalization of Mumford's intersection theory on normal surfaces to higher dimensions. We also define and study the second…
We construct a linearly normal smooth rational surface S of degree 11 and sectional genus 8 in the projective fivespace. Surfaces satisfying these numerical invariants are special, in the sense that $h^1(\mathscr{O}_S(1))>0$. Our…
We study linear relations between face numbers of levels in arrangements. Let $V = \{ v_1, \ldots, v_n \} \subset \mathbf{R}^{r}$ be a vector configuration in general position, and let $\mathcal{A}(V)$ be polar dual arrangement of…
We study various generalisations of rationally connected varieties, allowing the connecting curves to be of higher genus. The main focus will be on free curves $f:C\to X$ with large unobstructed deformation space as originally defined by…
We obtain a class of examples of non-rational adjoint classical groups of type $^2A_n$ and a group of type $^2D_3$ over the function field $F$ of a smooth geometrically integral curve over a $p$-adic field with $p \neq 2$. We also show that…
We study properties of rational curves on complete intersections in positive characteristic. It has long been known that in characteristic 0, smooth Calabi-Yau and general type varieties are not uniruled. In positive characteristic,…
We construct a good compactification of the variety of irreducible projective plane curves of degree n with d nodes and no other singularities.
This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, which are called irreducible triangulations. The structure has been introduced by Xin He under…
We study the compactification of the locus parametrizing lines with a fixed intersection to a given line, inside the moduli space of line arrangements in the projective plane constructed for weight one by Hacking-Keel-Tevelev and Alexeev…
This paper continues the authors' work on the question of unitary equivalence of matrices with entries in the complex-valued functions of a topological space (matrices over spaces). Specifically, we here consider the question of unitary…
A rational triangle is a triangle with rational side lengths. We consider three different families of rational triangles having a fixed side and whose vertices are rational points in the plane. We display a one-to-one correspondence between…
Consider a simple algebraic group G of adjoint type, and its wonderful compactification X. We show that X admits a unique family of minimal rational curves, and we explicitly describe the subfamily consisting of curves through a general…
Linear topological spaces with partial ordering (linear kinematics) are studied. They are defined by a set of 8 axioms implying that topology, linear structure and ordering are compatible with each other. Most of the results are valid for…
A compact classification of the projective lines defined over (commutative) rings (with unity) of all orders up to thirty-one is given. There are altogether sixty-five different types of them. For each type we introduce the total number of…
Surprisingly, the order-$k$ Voronoi diagram of line segments had received no attention in the computational-geometry literature. It illustrates properties surprisingly different from its counterpart for points; for example, a single…
Let $X \subset \mathbb{P}^{n+1}$ be a smooth Fano hypersurface of dimension $n$ and degree $d$. The derived category of coherent sheaves on $X$ contains an interesting subcategory called the Kuznetsov component $\mathcal{A}_X$. We show that…
Periodic frameworks with crystallographic symmetry are investigated from the perspective of a general deformation theory of periodic bar-and-joint structures in $R^d$. It is shown that natural parametrizations provide affine section…
In this paper, we study unirational differential curves and the corresponding differential rational parametrizations. We first investigate basic properties of proper differential rational parametrizations for unirational differential…