Related papers: Uniform Symbolic Topologies in Normal Toric Rings

The $F$-signature is a numerical invariant defined by the number of free direct summands in the Frobenius push-forward, and it measures singularities in positive characteristic. It can be generalized by focussing on the number of non-free…

Let $D$ be a subset of a finite commutative ring $R$ with identity. Let $f(x)\in R[x]$ be a polynomial of positive degree $d$. For integer $0\leq k \leq |D|$, we study the number $N_f(D,k,b)$ of $k$-subsets $S\subseteq D$ such that…

Let S be a subsemigroup of an abelian torsion-free group G. If S is a positive cone of G, then all C*-algebras generated by faithful isometrical non-unitary representations of S are canonically isomorphic. Proved by Murphy, this statement…

Let $R=\mathbb{K}[x_1,\dots,x_n]$ and let $\mathfrak{a}_1,\dots,\mathfrak{a}_m$ be homogeneous ideals satisfying certain properties, which include a description of the Noetherian symbolic Rees algebra. We give a solution to a question of…

Let $F$ be a number field and $D$ a quaternion algebra over $F$. Take a cuspidal automorphic representation $\pi$ of $D_{\mathbb{A}}^\times$ with trivial central character and a cusp form $\phi$ in $\pi$. Using the prehomogeneous zeta…

Let D = {D_{1},...,D_{l}} be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space P^n and let \Omega^{1}_{P^n}(log D) be the logarithmic bundle attached to it. Following [1], we show that…

Let $\Delta$ be a simplicial complex of a matroid $M$. In this paper, we explicitly compute the regularity of all the symbolic powers of a Stanley-Reisner ideal $I_\Delta$ in terms of combinatorial data of the matroid $M$. In order to do…

By the classical result of Milnor and Novikov, the unitary cobordism ring is isomorphic to a graded polynomial ring with countably many generators: $\Omega^U_*\simeq \mathbb Z[a_1,a_2,\dots]$, ${\rm deg}(a_i)=2i$. In this paper we solve a…

A set $\mathcal{S}$ of points in $\mathbb{R}^n$ is called a rationally parameterisable hypersurface if $\mathcal{S}=\{\boldsymbol{\sigma}(\mathbf{t}): \mathbf{t} \in D\}$, where $\boldsymbol{\sigma}: \mathbb{R}^{n-1} \rightarrow…

The theory of standard bases in polynomial rings with coefficients in a ring R with respect to local orderings is developed. R is a commutative Noetherian ring with 1 and we assume that linear equations are solvable in R.

Fix a pair of positive integers d and n. We create a ring R and a complex G of R-modules with the following universal property. Let P be a polynomial ring in d variables over a field and let I be a grade d Gorenstein ideal in P which is…

Symbolic ultrametrics define edge-colored complete graphs K_n and yield a simple tree representation of K_n. We discuss, under which conditions this idea can be generalized to find a symbolic ultrametric that, in addition, distinguishes…

We continue the study of intersection algebras $\mathcal B = \mathcal B_R(I, J)$ of two ideals $I, J$ in a commutative Noetherian ring $R$. In particular, we exploit the semigroup ring and toric structures in order to calculate various…

Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we show that all the symbolic and ordinary powers of $I(\mathcal{D})$ coincide when $\mathcal{D}$ is a weighted oriented certain class of…

Let p be a prime, k be a field of characteristic different from p containing a primitive p-th root of unity and N be the normalizer of the maximal torus in the projective linear group PGLn. We compute the exact value of the essential…

Let $S$ be a positively graded polynomial ring over a field of characteristic 0, and $I\subset S$ a proper graded ideal. In this note it is shown that $S/I$ is Golod if $\partial(I)^2\subset I$. Here $\partial(I)$ denotes the ideal…

Let A be an ample line bundle on a projective toric variety X of dimension n. We show that if l>=n-1+p, then A^l satisfies the property N_p. Applying similar methods, we obtain a combinatorial theorem: For a given lattice polytope P we give…

Let X be a complete toric variety with homogeneous coordinate ring S. In this article, we compute upper and lower bounds for the codimension in the critical degree of ideals of S generated by dim(X)+1 homogeneous polynomials that don't…

When the standard representation of a crystallographic Coxeter group G (with string diagram) is reduced modulo the integer d>1, one obtains a finite group G^d which is often the automorphism group of an abstract regular polytope. Building…

Let K be a number field with euclidean ring of integers O. Let G be a finite-index torsion-free subgroup of Sp(2n, O). We exhibit a finite, geometrically defined spanning set of the top dimensional integral cohomology of G by generalizing…