Littérature scientifique sur le sujet « Multiprojective space »

AbstractIn this paper we study the arithmetically Cohen-Macaulay (ACM) property for sets of points in multiprojective spaces. Most of what is known is for ℙ1× ℙ1and, more recently, in (ℙ1)r. In ℙ1× ℙ1the so called inclusion property characterises the ACM property. We extend the definition in any multiprojective space and we prove that the inclusion property implies the ACM property in ℙm× ℙn. In such an ambient space it is equivalent to the so-called (⋆)-property. Moreover, we start an investigation of the ACM property in ℙ1× ℙn. We give a new construction that highlights how different the behavior of the ACM property is in this setting.

We consider the bit complexity of computing Chow forms of projective varieties defined over integers and their generalization to multiprojective spaces. We develop a deterministic algorithm using resultants and obtain a single exponential complexity upper bound. Earlier computational results for Chow forms were in the arithmetic complexity model; thus, our result represents the first bit complexity bound. We also extend our algorithm to Hurwitz forms in projective space and we explore connections between multiprojective Hurwitz forms and matroid theory. The motivation for our work comes from incidence geometry where intriguing computational algebra problems remain open.

AbstractThe main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on n-dimensional smooth projective varieties X with an n-block collection B which generates the bounded derived category To this end, we use the theory of n-blocks and Beilinson type spectral sequence to define the notion of regularity of a coherent sheaf F on X with respect to the n-block collection B. We show that the basic formal properties of the Castelnuovo-Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we compare our definition of regularity with previous ones. In particular, we show that in case of coherent sheaves on ℙn and for the n-block collection Castelnuovo-Mumford regularity and our new definition of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a multiprojective space ℙn1x…x ℙnr with respect to a suitable n1 +…+ nr-block collection and we compare it with the multigraded variant of the Castelnuovo-Mumford regularity given by Hoffman and Wang in [14].

We study the Terracini loci of an irreducible variety X embedded in a projective space: non-emptiness, dimensions and the geometry of their maximal dimension’s irreducible components. These loci were studied because they describe where the differential of an important geometric map drops rank. Our best results are if X is either a Veronese embedding of a projective space of arbitrary dimension (the set-up for the additive decomposition of homogeneous polynomials) or a Segre–Veronese embedding of a multiprojective space (the set-up for partially symmetric tensors). For an arbitrary X, we give several examples in which all Terracini loci are empty, several criteria for non-emptiness and examples with the maximal defect possible a priori of an element of a minimal Terracini locus. We raise a few open questions.

Fix a format (n1+1)×⋯×(nk+1), k>1, for real or complex tensors and the associated multiprojective space Y. Let V be the vector space of all tensors of the prescribed format. Let S(Y,x) denote the set of all subsets of Y with cardinality x. Elements of S(Y,x) are associated to rank 1 decompositions of tensors T∈V. We study the dimension δ(2S,Y) of the kernel at S of the differential of the associated algebraic map S(Y,x)→PV. The set T1(Y,x) of all S∈S(Y,x) such that δ(2S,Y)>0 is the largest and less interesting x-Terracini locus for tensors T∈V. Moreover, we consider the one (minimally Terracini) such that δ(2A,Y)=0 for all A⊈S. We define and study two different types of subsets of T1(Y,x) (primitive Terracini and solution sets). A previous work (Ballico, Bernardi, and Santarsiero) provided a complete classification for the cases x=2,3. We consider the case x=4 and several extremal cases for arbitrary x.

AbstractLet R = k[x1,…,xn] be a polynomial ring over a field k. Let J = {j1,…,jt} be a subset of {1,…, n}, and let mJ ⊂ R denote the ideal (xj1,…,xjt). Given subsets J1,…,Js of {1,…, n} and positive integers a1,…,as, we study ideals of the form These ideals arise naturally, for example, in the study of fat points, tetrahedral curves, and Alexander duality of squarefree monomial ideals. Our main focus is determining when ideals of this form are componentwise linear. Using polymatroidality, we prove that I is always componentwise linear when s ≤ 3 or when Ji ∪ Jj = [n] for all i ≠ j. When s ≥ 4, we give examples to show that I may or may not be componentwise linear. We apply these results to ideals of small sets of general fat points in multiprojective space, and we extend work of Fatabbi, Lorenzini, Valla, and the first author by computing the graded Betti numbers in the s = 2 case. Since componentwise linear ideals satisfy the Multiplicity Conjecture of Herzog, Huneke, and Srinivasan when char(k) = 0, our work also yields new cases in which this conjecture holds.

Cette thèse explore deux sujets distincts à l'intersection de l'algèbre commutative, de la géométrie algébrique, de l'algèbre multilinéaire et de la modélisation géométrique :1. L'étude et la construction effective des transformations birationnelles multilinéaires 2. L'extraction d'informations à partir de données discrètes à l'aide de modèles polynomiaux. La partie principale de ce travail est consacrée aux transformations birationnelles multilinéaires.Une transformation birationnelle multilinéaire est une transformation rationnelle phi : (mathbb{P}^1)^n dashrightarrow{} mathbb{P}^n, définie par des polynômes multilinéaires, qui admet une transformation rationnelle inverse.Les transformations birationnelles entre espaces projectifs constituent un sujet d'étude important de la géométrie algébrique, initié par les travaux fondateurs de Cremona, qui a connu des avancées significatives au cours des dernières décennies.Plus récemment, les transformation birationnelles multiprojectives, c'est-à-dire définies par des polynômes multi-homogènes, ont récemment suscité un regain d'intérêt, motivé notamment par l'étude des structures multigraduées en algèbre commutative et leurs applications pratique en modélisation géométrique.Dans la première partie, nous étudions les aspects algébriques et géométriques des transformations birationales multilinéaires.Nous nous concentrons principalement sur les transformations birationnelles trilinéaires phi : (mathbb{P}^1)^3 dashrightarrow{} mathbb{P}^3 dont nous établissons une classification en fonction de la structure algébrique de leur espace, du lieu base, et des résolutions libres graduées minimales de l'idéal engendré par les polynômes de définition.En outre, nous développons les premières méthodes qui permettent de construire et de manipuler des transformations birationnelles non linéaires en dimension 3 avec une flexibilité suffisante pour les applications visées en modélisation géométrique.De plus, nous établissons une caractérisation de la birationalité basée sur le rang de tenseurs, qui permet de construire efficacement et ouvre la voie à l'application des outils de l'algèbre tensorielle à la birationnalité.Nous étendons également nos résultats aux transformations birationnelles multilinéaires en dimension arbitraire, dans le cas où il existe un inverse multilinéaire.Dans la deuxième partie, nous nous concentrons sur l'application des polynômes à l'analyse des données discrètes.Nous nous attaquons à deux problèmes distincts.Tout d'abord, nous dérivons des bornes pour la taille des (1-epsilon)-nets pour les ensembles de non-négativité de polynômes réels.Nos résultats nous permettent d'étendre le théorème classique du point central aux inégalités polynomiales de degré supérieur.Ensuite, nous abordons la classification des cylindres réels qui passent par cinq points qui sont tels que quatre d'entre eux sont cocycliques, c'est-à-dire qu'ils se trouvent sur un cercle.Il s'agit d'un cas particulier de problèmes plus généraux que sont la classification des racines réelles des systèmes de polynômes réels et l'extraction de primitives algébriques à partir de données brutes
This thesis explores two distinct subjects at the intersection of commutative algebra, algebraic geometry, multilinear algebra, and computer-aided geometric design:1. The study and effective construction of multilinear birational maps2. The extraction of information from measures and data using polynomialsThe primary and most extensive part of this work is devoted to multilinear birational maps.A multilinear birational map is a rational map phi: (mathbb{P}^1)^n dashrightarrow{} mathbb{P}^n, defined by multilinear polynomials, which admits an inverse rational map. Birational transformations between projective spaces have been a central theme in algebraic geometry, tracing back to the seminal works of Cremona, which has witnessed significant advancement in the last decades. Additionally, there has been a recent surge of interest in tensor-product birational maps, driven by the study of multiprojective spaces in commutative algebra and their practical application in computer-aided geometric design.In the first part, we address algebraic and geometric aspects of multilinear birational maps.We primarily focus on trilinear birational maps phi: (mathbb{P}^1)^3 dashrightarrow{} mathbb{P}^3, that we classify according to the algebraic structure of their space, base loci, and the minimal graded free resolutions of the ideal generated by the defining polynomials. Furthermore, we develop the first methods for constructing and manipulating nonlinear birational maps in 3D with sufficient flexibility for geometric modeling and design.Interestingly, we discover a characterization of birationality based on tensor rank, which yields effective constructions and opens the door to the application of tools from tensors to birationality. We also extend our results to multilinear birational maps in arbitrary dimension, in the case that there is a multilinear inverse.In the second part, our focus shifts to the application of polynomials in analyzing data and measures.We tackle two distinct problems. Firstly, we derive bounds for the size of (1-epsilon)-nets for superlevel sets of real polynomials. Our results allow us to extend the classical centerpoint theorem to polynomial inequalities of higher degree. Secondly, we address the classification of real cylinders through five-point configurations where four points are cocyclic, i.e. they lie on a circumference. This is an instance of the more general problems of real root classification of systems of real polynomials and the extraction of algebraic primitives from raw data