Related papers: Non-connected toric Hilbert schemes

We show that, for planar point sets, the number of non-crossing Hamiltonian paths is polynomially bounded in the number of non-crossing paths, and the number of non-crossing Hamiltonian cycles (polygonalizations) is polynomially bounded in…

We give a classification of ordered five points in $\mathbb P^3$ under the diagonal action of $GL_4$ over an algebraically closed field of characteristic $0$, by an explicit description of the diagonal action of $GL_4$ on the quintuple of…

We consider tight contact structures on plumbed 3-manifolds with no bad vertices. We discuss how one can count the number of tight contact structures with zero Giroux torsion on such 3-manifolds and explore conditions under which Giroux…

Any simple planar graph can be triangulated, i.e., we can add edges to it, without adding multi-edges, such that the result is planar and all faces are triangles. In this paper, we study the problem of triangulating a planar graph without…

Given a finite connected bipartite graph, finite-dimensional indecomposable semisimple Leibniz algebras are constructed. Furthermore, any finite-dimensional indecomposable semisimple Leibniz algebra admits a similar construction.

The purpose of this document is to provide data about known Betti numbers of unordered configuration spaces of small graphs in order to guide research and avoid duplicated effort. It contains information for connected multigraphs having at…

A hypergraph is called an r by r grid if it is isomorphic to a pattern of r horizontal and r vertical lines. Three sets form a triangle if they pairwise intersect in three distinct singletons. A hypergraph is linear if every pair of edges…

Let $X_{2k}$ be a set of $2k$ labeled points in convex position in the plane. We consider geometric non-intersecting straight-line perfect matchings of $X_{2k}$. Two such matchings, $M$ and $M'$, are disjoint compatible if they do not have…

We construct, for any positive integer n, a family of n congruent convex polyhedra in R^3, such that every pair intersects in a common facet. Previously, the largest such family contained only eight polytopes. Our polyhedra are Voronoi…

Let $G$ be a cyclically $5$-connected cubic graph with a $5$-edge-cut separating $G$ into two cyclic components $G_1$ and $G_2$. We prove that each component $G_i$ can be completed to a cyclically $5$-connected cubic graph by adding three…

We exhibit generically nonreduced components of the Hilbert scheme of at least $21$ points on a smooth variety of dimension at least four. The result was announced in~[Jelisiejew__open_problems] and answers a question~[Problem~3.8, AIMPL].…

Flip graphs of combinatorial and geometric objects are at the heart of many deep structural insights and connections between different branches of discrete mathematics and computer science. They also provide a natural framework for the…

We prove that 3-connected triangulations with at most one separating triangle are hamiltonian-connected. In order to show bounds on the strongest form of this theorem, we proved that for any $s\geq4$ there are 3-connected triangulation with…

In [5] I solved the Thom's conjecture that a proper Thom map is triangulable. In this paper I drop the properness condition in the semialgebraic case and, moreover, in the definable case in an o-minimal structure.

A complete structural characterization of graphs with no $K_{3,4}$ minor is obtained, and the following consequences are established. Every $4$-connected non-planar graph with at least seven vertices and minimum degree at least five…

In this paper we prove two results which further classify smoothness properties of Hilbert schemes of points. This is done by counting classes of arrows on Young diagrams corresponding to monomial ideals, building on the approach taken by…

We study certain top intersection products on the Hilbert scheme of points on a nonsingular surface relative to an effective smooth divisor. We find a formula relating these numbers to the corresponding intersection numbers on the…

We construct the defining data of two-dimensional topological field theories (TFTs) enriched by non-invertible symmetries/topological defect lines. Simple formulae for the three-point functions and the lasso two-point functions are derived,…

We introduce a new technique that is used to show that the complex projective plane blown up at 6, 7, or 8 points has infinitely many distinct smooth structures. None of these smooth structures admit smoothly embedded spheres with…

A recent line of work on VC set systems in minor-free (undirected) graphs, starting from Li and Parter, who constructed a new VC set system for planar graphs, has given surprising algorithmic results. In this work, we initialize a more…