Related papers: Shellability of the higher pinched Veronese posets
We characterize the double Veronese embedding of P^n as the only variety that, under certain general conditions, can be isomorphically projected from the Grassmannian of lines in P^{2n+1} to the Grassmannian of lines in P^{n+1}.
We determine the cone of nef divisors on the Voronoi compactification A_g^* of the moduli space A_g of principally polarized abelian varieties of dimension g for genus g=2,3. As a corollary we obtain that the spaces A_g^*(n) with level-n…
The construction of the Bier sphere Bier(K) for a simplicial complex K is due to Bier. Bj\"orner, Paffenholz, Sj\"ostrand and Ziegler generalize this construction to obtain a Bier poset Bier(P,I) from any bounded poset P and any proper…
In 1977 Stanley conjectured that the $h$-vector of a matroid independence complex is a pure $O$-sequence. In this paper we use lexicographic shellability for matroids to motivate a combinatorial strengthening of Stanley's conjecture. This…
A point $p\in\mathbb{P}^N$ of a projective space is $h$-identifiable, with respect to a variety $X\subset\mathbb{P}^N$, if it can be written as linear combination of $h$ elements of $X$ in a unique way. Identifiability is implied by…
Given a polarizable $\mathbb{Z}$-variation of Hodge structures $\mathbb{V}$ over a complex smooth quasi-projective base $S$, a classical result of Cattani, Deligne and Kaplan says that its Hodge locus (i.e. the locus where exceptional Hodge…
The partition problem is a well-known basic NP-complete problem. We mainly consider the optimization version of it in this paper. The problem has been investigated from various perspectives for a long time and can be solved efficiently in…
We introduce the notion of a \emph{Whitney dual} of a graded poset. Two posets are Whitney duals to each other if (the absolute value of) their Whitney numbers of the first and second kind are interchanged between the two posets. We define…
The Dushnik-Miller dimension of a partially-ordered set $P$ is the smallest $d$ such that one can embed $P$ into a product of $d$ linear orders. We prove that the dimension of the divisibility order on the interval $\{1, \dotsc, n\}$, is…
We prove that the subposet induced by the fixed elements of any automorphism of a pircon is also a pircon. By a result of Abdallah, Hansson, and Hultman, the order complex of any open interval in a pircon is a PL ball or a PL sphere. We…
It was shown by Gruslys, Leader and Tan that any finite subset of $\mathbb{Z}^n$ tiles $\mathbb{Z}^d$ for some $d$. The first non-trivial case is the punctured interval, which consists of the interval $\{-k,\ldots,k\} \subset \mathbb{Z}$…
In 1989, Ne\v{s}et\v{r}il and Pudl\'ak posed the following challenging question: Do planar posets have bounded Boolean dimension? We show that every poset with a planar cover graph and a unique minimal element has Boolean dimension at most…
It is known that the variety parametrizing pairs of commuting nilpotent matrices is irreducible and that this provides a proof of the irreducibility of the punctual Hilbert scheme in the plane. We extend this link to the nilpotent commuting…
In this paper we study some cohomological properties of non-standard multigraded modules and Veronese transforms of them. Among others numerical characters, we study the generalized depth of a module and we see that it is invariant by…
In this paper, we investigate the solubility of homogeneous polynomial equations. The work of Browning, Le boudec, Sawin [3] shows that almost all homogeneous equations of degree $d\geq 4$ in $d+1$ or more variables satisfy the Hasse…
For a hypergraph $\mathbb{H}$ on $[n]$, the hypergraphic poset $P_\mathbb{H}$ is the transitive closure of the oriented skeleton of the hypergraphic polytope $\triangle_\mathbb{H}$ (the Minkowski sum of the standard simplices $\triangle_H$…
Let $P(n)$ be the set of all posets with $n$ elements. Let $P^{(j)}(n)$, $1\leq j\leq 2^n,$ be the number of all posets with $n$ elements possessing exactly $j$ antichains. We have determined the numbers $P^{(j)}(7),$ $1\leq j\leq 128$, and…
Let $h\geq 2$ be a positive integer. For any subset $\mathcal{A}\subset \mathbb{Z}_n$, let $h^{\wedge}\mathcal{A}$ be the set of the elements of $\mathbb{Z}_n$ which are sums of $h$ distinct elements of $\mathcal{A}$. In this paper, we…
The off-shell vector-tensor multiplet is considered in an arbitrary background of N=2 vector supermultiplets. We establish the existence of two inequivalent versions, characterized by different Chern-Simons couplings. In one version the…
We study a natural stratification of certain affine slices of univariate hyperbolic polynomials. We look into which posets of strata can be realized and show that the dual of the poset of strata is a shellable simplicial complex and in…