相关论文: Pencilled regular parallelisms
New examples of Cameron-Liebler line classes in $\mathrm{PG}(3,q)$ are given with parameter $\frac{1}{2}(q^2 -1)$. These examples have been constructed for many odd values of $q$ using a computer search, by forming a union of line orbits…
We define strict confluent drawing, a form of confluent drawing in which the existence of an edge is indicated by the presence of a smooth path through a system of arcs and junctions (without crossings), and in which such a path, if it…
We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to…
The sparse analogue of Szemer\'edi's regularity method has played a central role in the development of extremal results for random graphs. While the sparse embedding lemma (the KLR conjecture) has been resolved, the corresponding sparse…
We prove that for a generic Lefschetz pencil of plane curves of degree $d\geq 3$ there exists a curve $H$ (called the Hesse curve of the pencil) of degree $6(d-1)$ and genus $3(4d^2-13d+8)+1$, and such that: $(i)$ $H$ has $d^2$ singular…
A long standing conjecture, known to us as the Eisenbud Goto conjecture, states that an n-dimensional variety embedded with degree $d$ in the $N$- dimensional projective space is $(d-(N-n)+1)$-regular in the sense of Castelnuovo-Mumford. In…
We give a description of the relative Hilbert scheme of lines in the Dwork pencil of quintic threefolds. We describe the corresponding relative Hilbert scheme associated to the mirror family of quintic threefolds.
The classical Clifford correspondence for normal subgroups is considered in the more general setting of semisimple Hopf algebras. We prove that this correspondence still holds if the extension determined by the normal Hopf subalgebra is…
A \emph{geometric graph} is a graph whose vertex set is a set of points in general position in the plane, and its edges are straight line segments joining these points. We show that for every integer $k \ge 2$, there exists a constat $c>0$…
In semialgebraic geometry, projections play a prominent role. A definable choice is a semialgebraic selection of one point in every fiber of a projection. Definable choices exist by semialgebraic triviality, but their complexity depends…
The square-peg problem asks if every Jordan curve in the plane has four points which are the vertices of a square. The problem is open for continuous Jordan curves, but it has been resolved for various regularity classes of curves between…
We show that a generic real projective n-dimensional hypersurface of degree 2n-1 contains "many" real lines, namely, not less than (2n-1)!!, which is approximately the square root of the number of complex lines. This estimate is based on…
In 2021, Brock, Elkies, and Jordan generalized the theory of periodic continued fractions (PCFs) over $\mathbb{Z}$ to the ring of integers in a number field. In particular, they considered the case where the number field is an intermediate…
The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and…
We compute the equivariant homology and cohomology of projective spaces with integer coefficients. More precisely, in the case of cyclic groups, we show that the cellular filtration of the projective space $P(k\rho )$, of lines inside…
We give a geometric description of singular pencils of quadrics of constant rank, relating them to the splitting type of some naturally associated vector bundles on $\mathbb{P}^1$. Then we study their orbits in the Grassmannian of lines,…
Series-parallel (SP) graphs are binary edge-labeled graphs with a designated source and target vertex, built using serial and parallel composition. A set of graphs is recognizable if membership depends only on its image under a homomorphism…
This paper demonstrates that parallel vector curves are piecewise cubic rational curves in 3D piecewise linear vector fields. Parallel vector curves -- loci of points where two vector fields are parallel -- have been widely used to extract…
We generalize the classical theory of periodic continued fractions (PCFs) over ${\mathbf Z}$ to rings ${\mathcal O}$ of $S$-integers in a number field. Let ${\mathcal B}=\{\beta, {\beta^*}\}$ be the multi-set of roots of a quadratic…
We consider the greatest common divisor (GCD) of all sums of $k$ consecutive terms of a sequence $(S_n)_{n\geq 0}$ where the terms $S_n$ come from exactly one of following six well-known sequences' terms: Pell $P_n$, associated Pell $Q_n$,…