Related papers: Finite $0$-dimensional multiprojective schemes and…
Let $M$ be a finitely generated module over a local complete intersection $R$ of characteristic $p>0$. The property that $M$ has finite projective dimension can be characterized by the vanishing of $\ext_R^i({}^{f^n} R,M)$ for some $i>0$…
We study the projections in vector spaces over finite fields. We prove finite fields analogues of the bounds on the dimensions of the exceptional sets for Euclidean projection mapping. We provide examples which do not have exceptional…
An algebraic scheme is suggested in which discretized spacetime turns out to be a quantum observable. As an example, a toy model producing spacetimes of four points with different topologies is presented. The possibility of incorporating…
A space for gauge theories is defined, using projective limits as subsets of Cartesian products of homomorphisms from a lattice on the structure group. In this space, non-interacting and interacting measures are defined as well as functions…
In the field of numerical algebraic geometry, positive-dimensional solution sets of systems of polynomial equations are described by witness sets. In this paper, we define multiprojective witness sets which encode the multidegree…
We prove an additivity for evenly (oddly) finite dimensional objects in distinguished triangles in a triangulated monoidal category structured by an underlying model monoidal category. In particular, the result holds in the Q-localized…
We investigate the enumerative geometry of point configurations in projective space. We define "projective configuration counts": these enumerate configurations of points in projective space such that certain specified subsets are in fixed…
This paper deals with projective shape analysis, which is a study of finite configurations of points modulo projective transformations. The topic has various applications in machine vision. We introduce a convenient projective shape space,…
We study conditions on a commutative ring R which are equivalent to the following requirement; whenever X is a projective scheme over S = Spec(R) of fiber dimension \leq d for some integer d \geq 0, there is a finite morphism from X to…
We study initial degrees of symbolic powers of ideals of arbitrary finite sets of points in the projective plane over an algebraically closed field of characteristic zero. We show, how bounds on the growth of these degrees determine the…
For an ideal $I$ in a polynomial ring over a field, a monomial support of $I$ is the set of monomials that appear as terms in a set of minimal generators of $I$. Craig Huneke asked whether the size of a monomial support is a bound for the…
A ringed finite space is a ringed space whose underlying topological space is finite. The category of ringed finite spaces contains, fully faithfully, the category of finite topological spaces and the category of affine schemes. Any ringed…
Zero-schemes on smooth complex projective varieties, forcing all elements of ample and free linear systems to be reducible are studied. Relationships among the minimal length of such zero-schemes, the positivity of the line bundle…
There are two main examples where a version of the Minimal Model Program can, at least conjecturally, be performed successfully: the first is the classical MMP associated to the canonical divisor, and the other is Mori Dream Spaces. In this…
The purpose of this paper is to prove the finiteness theorems for meromorphic mappings of a complete connected K\"{a}hler manifold into projective space sharing few hyperplanes in subgeneral position without counting multiplicity, where all…
The purpose of this note is a wide generalization of the topological results of various classes of ideals of rings, semirings, and modules, endowed with Zariski topologies, to strongly irreducible ideals (endowed with Zariski topologies) of…
We introduce a family of mathematical objects called $\mathcal{P}$-schemes, where $\mathcal{P}$ is a poset of subgroups of a finite group $G$. A $\mathcal{P}$-scheme is a collection of partitions of the right coset spaces $H\backslash G$,…
The aim of this article is to give a rigorous geometric interpretation of the completion of a ring with respect to an ideal. To this end, we define the infinitesimal neighbourhood of an immersion of formal schemes as the largest possible…
The purpose of this note is to give a survey of the algebraic properties of multiplier ideals, and illustrate some of their applications to classical projective geometry.
A `polydisperse' system has an infinite number of conserved densities. We give a rational procedure for projecting its infinite-dimensional free energy surface onto a subspace comprising a finite number of linear combinations of densities…