Related papers: Grothendieck-Lefschetz Theory, Set-Theoretic Compl…
We consider smooth codimension two subcanonical subvarieties in $\mathbb{P}^n$ with $n \geq 5$, lying on a hypersurface of degree $s$ having a linear subspace of multiplicity $(s-2)$. We prove that such varieties are complete intersections.…
We prove Grothendieck's existence theorem for relatively perfect complexes on an algebraic stack that is proper and flat over an $I$-adically complete Noetherian ring $A$. This generalizes an earlier result of Lieblich in the setting of…
A subset $Y$ of the general linear group $\operatorname{GL}(n,q)$ is called $t$-intersecting if $\operatorname{rk}(x-y)\le n-t$ for all $x,y\in Y$, or equivalently $x$ and $y$ agree pointwise on a $t$-dimensional subspace of…
Samuel conjectured in 1961 that a (Noetherian) local complete intersection ring that is a UFD in codimension at most three is itself a UFD. It is said that Grothendieck invented local cohomology to prove this fact. Following the philosophy…
Finite rational $\cw$ algebras are very natural structures appearing in coset constructions when a Kac-Moody subalgebra is factored out. In this letter we address the problem of relating these algebras to integrable hierarchies of…
Let $S_n$ be the symmetric group on $n$ points. Deza and Frankl [M. Deza and P. Frankl, On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A 22 (1977) 352--360] proved that if ${\cal F}$ is…
This is a short survey about the theory of stable polynomials and its applications. It gives self-contained proofs of two theorems of Schrijver. One of them asserts that for a $d$--regular bipartite graph $G$ on $2n$ vertices, the number of…
A variety is rationally connected if two general points can be joined by a rational curve. A higher version of this notion is rational simple connectedness, which requires suitable spaces of rational curves through two points to be…
Let $Z$ be a closed subscheme of a smooth complex projective complete intersection variety $Y\subseteq \Ps^N$, with $dim Y=2r+1\geq 3$. We describe the N\'eron-Severi group $NS_r(X)$ of a general smooth hypersurface $X\subset Y$ of…
We study the parameterized complexity of dominating sets in geometric intersection graphs. In one dimension, we investigate intersection graphs induced by translates of a fixed pattern Q that consists of a finite number of intervals and a…
Continuous spline functions are defined as piecewise polynomials on the faces of a polyhedral complex that agree on the intersections of two faces. Splines are used in approximation theory and numerical analysis, with applications in data…
We find a relation between the spectrum of solitons of massive $N=2$ quantum field theories in $d=2$ and the scaling dimensions of chiral fields at the conformal point. The condition that the scaling dimensions be real imposes restrictions…
In this article we introduce and study the intersection graph of graded ideals of graded rings. The intersection graph of $G-$graded ideals of a graded ring $(R,G)$ is a simple graph, denoted by $Gr_G(R)$, whose vertices are the nontrivial…
In this note, we introduce a new concept of a {\it generalized algebraic rational identity} to investigate the structure of division rings. The main theorem asserts that if a non-central subnormal subgroup $N$ of the multiplicative group…
The mincut graph bisection problem involves partitioning the n vertices of a graph into disjoint subsets, each containing exactly n/2 vertices, while minimizing the number of "cut" edges with an endpoint in each subset. When considered over…
We introduce a natural partial order on the set Cones(d) of rational cones in R^d. The poset NPol(d-1) of normal polytopes in R^{d-1} embeds into Cones(d) via the homogenization map. The order in Cones(d) is conjecturally the inclusion…
We define a Grothendieck ring for basic real semialgebraic formulas, that is for systems of real algebraic equations and inequalities. In this ring the class of a formula takes into consideration the algebraic nature of the set of points…
We introduce a Grothendieck ring of higher Artin stacks generalizing the Grothendieck ring of algebraic varieties. We show that this ring is not trivial by noticing that it factors the invariant "number of rational points over a finite…
The sl_2-triples play a fundamental role for the structure theory of Lie algebras, and representation theory in general. Here we investigate sl_2-triples of global vector fields on schemes X in positive characteristics p>0, and develop a…
Given a finite abelian group $G$ and $t\in \mathbb{N}$, there are two natural types of subsets of the Cartesian power $G^t$; namely, Cartesian powers $S^t$ where $S$ is a subset of $G$, and (cosets of) subgroups $H$ of $G^t$. A basic…