Related papers: Gap vectors of real projective varieties

Let $X$ be a simplicial complex with $n$ vertices. A missing face of $X$ is a simplex $\sigma\notin X$ such that $\tau\in X$ for any $\tau\subsetneq \sigma$. For a $k$-dimensional simplex $\sigma$ in $X$, its degree in $X$ is the number of…

Let us consider the group $G = < x,y \mid x^m = y^n>$ with $m$ and $n$ nonzero integers. In this paper, we study the variety of epresentations $R(G)$ and the character variety $X(G)$ in $SL(2,\C)$ of the group $G$,obtaining by elementary…

Let E and F be vector bundles over a complex projective smooth curve X, and suppose that 0 -> E -> W -> F -> 0 is a nontrivial extension. Let G be a subbundle of F, and D an effective divisor on X. We give a criterion for the subsheaf G(-D)…

Given an algebraic variety $X\subset\mathbb{P}^N$ with stabilizer $H$, the quotient $PGL_{N+1}/H$ can be interpreted a parameter space for all $PGL_{N+1}$-translates of $X$. We define $X$ to be a $\textit{homogeneous variety}$ if $H$ acts…

A basic combinatorial invariant of a convex polytope $P$ is its $f$-vector $f(P)=(f_0,f_1,\dots,f_{\dim P-1})$, where $f_i$ is the number of $i$-dimensional faces of $P$. Steinitz characterized all possible $f$-vectors of $3$-polytopes and…

Vector is a physical quantity and it does not depend on any co-ordinate system. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. The expansion in orthonormal basis is…

We announce two breakthrough results concerning important questions in the Theory of Computational Complexity. In this expository paper, a systematic and comprehensive geometric characterization of the Subset Sum Problem is presented. We…

Differential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their…

Let $L$ be a full-rank lattice in $\mathbb R^d$ and write $L^+$ for the semigroup of all vectors with nonnegative coordinates in $L$. We call a basis $X$ for $L$ positive if it is contained in $L^+$. There are infinitely many such bases,…

We introduce four invariants of algebraic varieties over imperfect fields, each of which measures either geometric non-normality or geometric non-reducedness. The first objective of this article is to establish fundamental properties of…

We find an explicit formula for the gamma vector in terms of the input polynomial in a way that extends it to arbitrary polynomials. More specifically, we find explicit linear combination in terms of coefficients of the input polynomial…

Let $G$ be a complex classical group, and let $V$ be its defining representation (possibly plus a copy of the dual). A foundational problem in classical invariant theory is to write down generators and relations for the ring of…

We construct relative Gromov--Witten theory with expanded degenerations in the normal crossings setting and establish a degeneration formula for the resulting invariants. Given a simple normal crossings pair $(X,D)$, we show that there…

In these Notes, a comprehensive description of the universal fractal geometry of conformally-invariant scaling curves or interfaces, in the plane or half-plane, is given. The present approach focuses on deriving critical exponents…

Let $X \subset \P^r$ be a linearly normal variety defined by a very ample line bundle $L$ on a projective variety $X$. Recently it is shown in \cite{HLMP} that there are many cases where $(X,L)$ satisfies property $\textsf{QR} (3)$ in the…

To each subvariety $X$ in projective $n$-space of codimension $m$ we associate an integer sequence of length $m + 1$ from $1$ to the degree of $X$ recording the maximal cardinalities of finite, reduced intersections of $X$ with linear…

We present an algorithm to find an integral vector in the polyhedral cone $\Gamma=\{X | \textbf{A}X \leq \textbf{0}\}$, without assuming the explicit knowledge of $\textbf{A}$. About the polyhedral cone, $\Gamma$, it is only given that, (i)…

The Konno invariant of a projective variety X is the minimum geometric genus of the fiber of a rational pencil on X. It was computed by Konno for surfaces in P^3, and in general can be viewed as a measure of the complexity of X. We estimate…

Let W be a projective variety of dimension n+1, L a free line bundle on W, X in $H^0(L^d)$ a hypersurface of degree d which is generic among those given by sums of monomials from $L$, and let $f : Y \to X$ be a generically finite map from a…

In this paper, two outwardly different graphs, namely, the zero divisor graph $\Gamma(C_c(X))$ and the comaximal graph $\Gamma_2^{'}(C_c(X))$ of the ring $C_c(X)$ of all real-valued continuous functions having countable range, defined on…