Academic literature on the topic 'Projective degrees of a rational map'

AbstractFor projectivizations of rational maps, Bellon and Viallet defined the notion of algebraic entropy using the exponential growth rate of the degrees of iterates. We want to call this notion to the attention of dynamicists by computing algebraic entropy for certain rational maps of projective spaces (Theorem 6.2) and comparing it with topological entropy (Theorem 5.1). The particular rational maps we study are monomial maps (Definition 1.2), which are closely related to toral endomorphisms. Theorems 5.1 and 6.2 that imply that the algebraic entropy of a monomial map is always bounded above by its topological entropy, and that the inequality is strict if the defining matrix has more than one eigenvalue outside the unit circle. Also, Bellon and Viallet conjectured that the algebraic entropy of every rational map is the logarithm of an algebraic integer, and Theorem 6.2 establishes this for monomial maps. However, a simple example using a monomial map shows that a stronger conjecture of Bellon and Viallet is incorrect, in that the sequence of algebraic degrees of the iterates of a rational map of projective space need not satisfy a linear recurrence relation with constant coefficients.

Let be a dominant rational self-map of a smooth projective variety defined over $\overline{\mathbb{Q}}$. For each point $P\in X(\overline{\mathbb{Q}})$ whose forward $f$-orbit is well defined, Silverman introduced the arithmetic degree $\unicode[STIX]{x1D6FC}_{f}(P)$, which measures the growth rate of the heights of the points $f^{n}(P)$. Kawaguchi and Silverman conjectured that $\unicode[STIX]{x1D6FC}_{f}(P)$ is well defined and that, as $P$ varies, the set of values obtained by $\unicode[STIX]{x1D6FC}_{f}(P)$ is finite. Based on constructions by Bedford and Kim and by McMullen, we give a counterexample to this conjecture when $X=\mathbb{P}^{4}$.

Abstract We show that there is a unique hypersurface of minimal degree passing through the non-faces of a polytope which is defined by a simple hyperplane arrangement. This generalizes the construction of the adjoint curve of a polygon by Wachspress (A rational finite element basis, Academic Press, New York, 1975). The defining polynomial of our adjoint hypersurface is the adjoint polynomial introduced by Warren (Adv Comput Math 6:97–108, 1996). This is a key ingredient for the definition of Wachspress coordinates, which are barycentric coordinates on an arbitrary convex polytope. The adjoint polynomial also appears both in algebraic statistics, when studying the moments of uniform probability distributions on polytopes, and in intersection theory, when computing Segre classes of monomial schemes. We describe the Wachspress map, the rational map defined by the Wachspress coordinates, and the Wachspress variety, the image of this map. The inverse of the Wachspress map is the projection from the linear span of the image of the adjoint hypersurface. To relate adjoints of polytopes to classical adjoints of divisors in algebraic geometry, we study irreducible hypersurfaces that have the same degree and multiplicity along the non-faces of a polytope as its defining hyperplane arrangement. We list all finitely many combinatorial types of polytopes in dimensions two and three for which such irreducible hypersurfaces exist. In the case of polygons, the general such curves are elliptic. In the three-dimensional case, the general such surfaces are either K3 or elliptic.

We compare the Kontsevich moduli space [Formula: see text] of stable maps to projective space with the quasi-map space ℙ( Sym d(ℂ2) ⊗ ℂn)//SL(2). Consider the birational map [Formula: see text] which assigns to an n tuple of degree d homogeneous polynomials f1, …, fn in two variables, the map f = (f1 : ⋯ : fn) : ℙ1 → ℙn-1. In this paper, for d = 3, we prove that [Formula: see text] is the composition of three blow-ups followed by two blow-downs. Furthermore, we identify the blow-up/down centers explicitly in terms of the moduli spaces [Formula: see text] with d = 1, 2. In particular, [Formula: see text] is the SL(2)-quotient of a smooth rational projective variety. The degree two case [Formula: see text], which is the blow-up of ℙ( Sym 2ℂ2 ⊗ ℂn)//SL(2) along ℙn-1, is worked out as a preliminary example.

We define and study a family of cubic surfaces in the projectivized tangent bundle over a four-dimensional projective space associated to the root system D5. The 27 lines are rational over the base and we determine the classifying map to the moduli space of marked cubic surfaces. This map has degree two and we use it to get short proofs for some results on the Chow group of the moduli space of marked cubic surfaces.

While intersection cohomology is stable under small resolutions, both ordinary and intersection cohomology are unstable under smooth deformation of singularities. For complex projective algebraic hypersurfaces with an isolated singularity, we show that the first author's cohomology of intersection spaces is stable under smooth deformations in all degrees except possibly the middle, and in the middle degree precisely when the monodromy action on the cohomology of the Milnor fiber is trivial. In many situations, the isomorphism is shown to be a ring homomorphism induced by a continuous map. This is used to show that the rational cohomology of intersection spaces can be endowed with a mixed Hodge structure compatible with Deligne's mixed Hodge structure on the ordinary cohomology of the singular hypersurface. Regardless of monodromy, the middle degree homology of intersection spaces is always a subspace of the homology of the deformation, yet itself contains the middle intersection homology group, the ordinary homology of the singular space, and the ordinary homology of the regular part.

AbstractLet φ:ℙN⤏ℙN be a dominant rational map. The dynamical degree of φ is the quantity δφ=lim (deg φn)1/n. When φ is defined over ${\bar {{\mathbb {Q}}}}$, we define the arithmetic degree of a point $P\in {\mathbb {P}}^N({\bar {{\mathbb {Q}}}})$ to be αφ(P)=lim sup h(φn(P))1/n and the canonical height of P to be $\hat {h}_\varphi (P)=\limsup \delta _\varphi ^{-n}n^{-\ell _\varphi }h(\varphi ^n(P))$ for an appropriately chosen ℓφ. We begin by proving some elementary relations and making some deep conjectures relating δφ, αφ(P) , ${\hat h}_\varphi (P)$, and the Zariski density of the orbit 𝒪φ(P) of P. We then prove our conjectures for monomial maps.

Noether’s problem asks whether, for a given field [Formula: see text] and finite group [Formula: see text], the fixed field [Formula: see text] is a purely transcendental extension of [Formula: see text], where [Formula: see text] acts on the [Formula: see text] by [Formula: see text]. The field [Formula: see text] is naturally the function field for a quotient variety [Formula: see text]. We study the degree of irrationality [Formula: see text] of [Formula: see text] for an abelian group [Formula: see text], which is defined to be the minimal degree of a dominant rational map from [Formula: see text] to projective space. In particular, we give bounds for [Formula: see text] in terms of the arithmetic of cyclotomic extensions [Formula: see text].

AbstractLet X be a smooth complex projective curve of genus g ≥ 1. Let ξ ∈ J1(X) be a line bundle on X of degree 1. LetW = Ext1(ξn, ξ–1) be the space of extensions of ξn by ξ–1. There is a rational map Dξ : G(n,W) → SUX(n + 1), where G(n,W) is the Grassmannian variety of n-linear subspaces of W and SUX(n + 1) is the moduli space of rank n + 1 semi-stable vector bundles on X with trivial determinant. We prove that if n = 2, then Dξ is everywhere defined and is injective.

AbstractWe use the cylindrical homomorphism and a geometric construction introduced by J. Lewis to study the Lawson homology groups of certain hypersurfaces X ⊂ ℙn + 1 of degree d ℙ n + 1. As an application, we compute the rational semi-topological K-theory of generic cubics of dimensions 5, 6 and 8 and, using the Bloch-Kato conjecture, we prove Suslin's conjecture for these varieties. Using generic cubic sevenfolds, we show that there are smooth projective varieties such that the lowest nontrivial step in their s-filtration is infinitely generated and undetected by the Abel-Jacobi map.

Cette thèse est consacrée à l’étude des courbes sextiques nodales, et en particulier des sextiques rationnelles, dans le plan projectif réel. Deux sextiques nodales réelles ayant k points doubles sont dites rigidement isotopes si elles peuvent être reliées par un chemin dans l’espace des sextiques nodales réelles ayant k points doubles. Le résultat principal de la première partie de la thèse donne une classification à isotopie rigide près des sextiques nodales irréductibles sans points doubles réels, généralisant la classification des sextiques non-singulières obtenue par Nikulin. La seconde partie porte sur les sextiques ayant des points doubles réels : une classification est obtenue pour les sextiques nodales séparantes, c’est-à-dire celles dont la partie réelle sépare leur complexification (l’ensemble des points complexes) en deux composantes connexes. Ce résultat est appliqué au cas des sextiques rationnelles réelles pouvant être perturbées en des sextiques maximales ou presque maximales (dans le sens de l’inégalité de Harnack). L’approche retenue repose sur l’étude des périodes des surfaces K3, se basant notamment sur le Théorème de Torelli Global de Piatetski-Shapiro et Shafarevich et le Théorème de Surjectivité de Kulikov, ainsi que sur les résultats de Nikulin portant sur les formes bilinéaires symétriques intégrales
This thesis studies nodal sextics (algebraic curves of degree six), and in particular rational sextics, in the real projective plane. Two such sextics with k nodes are called rigidly isotopic if they can be joined by a path in the space of real nodal sextics with k nodes. The main result of the first part of the thesis is a rigid isotopy classification of real nodal sextics without real nodes, generalizing Nikulin’s classification of non-singular sextics. In the second part we study sextics with real nodes and we describe the rigid isotopy classes of such sextics in the case where the sextics are dividing, i.e., their real part separates the complexification (the set of complex points) into two halves. As a main application, we give a rigid isotopy classification for those nodal real rational sextics which can be perturbed to maximal or next-to-maximal sextics in the sense of Harnack’s inequality. Our approach is based on the study of periods of K3 surfaces, drawing on the Global Torelli Theorem by Piatetski-Shapiro and Shafarevich and Kulikov’s surjectivity theorem, as well as Nikulin’s results on symmetric integral bilinear forms

This chapter uses algebraic cobordism to establish some degree formulas. It presents δ‎ as a function from a class of smooth projective varieties over a field 𝑘 to some abelian group. Here, a degree formula for δ‎ is a formula relating δ‎(𝑋), δ‎(𝑌), and deg(𝑓) for any generically finite map 𝑓 : 𝑌 → 𝑋 in this class. The formula is usually δ‎(𝑌)=deg(𝑓)δ‎(𝑋). These degree formulas are used to prove that any norm variety over 𝑘 is a ν‎ _n−1-variety. Using a standard result for the complex bordism ring 𝑀𝑈_*, which uses a gluing argument of equivariant bordism theory, this chapter establishes Rost's DN (Degree and Norm Principle) Theorem for degrees, and defines the invariant η‎(𝑋/𝑆) of a pseudo-Galois cover.

Digital diagrams are constituted as strategic-communicative-productive intermediate space-matrices among architecture, the architect and the digital machine, and between architecture and other disciplinary fields. According to Stan Allen, diagrams are a map of the possible worlds, a description of potential relationships, where a plethora of functions, actions and configurations are implicit in time, subject to continuous modifications.Alluding to the notions of diagrams, machinic and figural of Deleuze and Guattari, Eisenman’s conceptual-generative diagram functions as a hypertext and a creative and affective interface between intention, randomness and imagination, and architectural space and form. Proposing a new type of reality in a permanent state of evolution and of form-thinking and form-making, they have become a technique or poetic operation that, in addition to representing, also present and evoke, in between order and chaos, intention and the unexpected, mechanical and organic, real and virtual, presence and absence. For Eisenman, diagrams function as a heuristic instrument of criticism in the design process, in search of other spatial and conceptual qualities for a reflection within the discipline of architecture itself. Diagrams are a space-matrix, or as Eisenman puts it, a “meta-writing” in terms of the field of orientations and possibilities to be apprehended and inscribed, first in the project and later in the construction of the architectural place. These possibilities and orientations, guidelines or meanings are not totally contained within the diagram itself, rather they also reside in the intermediate space between the diagram and the observer, creator or architect. Diagrams are an evocative and inspiring space-formal matrix, the contents or evocations of which are not found “embedded” or “enclosed” in their shape or material, rather they are indicated or outlined, in varying degrees of explicitness, as signals or traces, evoking multiple interpretations and reflections. Eisenman uses digital diagrams as a mediating agent or instrument to investigate, explore, create and draw the architectural space within the thematic basis of the interstitial–“the “in-between”. He does so through a process that is intentional, random, interpretive, esthetic and poetic, all at the same time. In contrast to the traditional quest for form that is synthesized in the idea of a box or container, Eisenman proposes an alternative means, through which form or space can be found through a long process in which rational approaches and computerized drawing intermingle, introducing formal randomness, in which the diagram is the mediator. Eisenman refers to that procedure as “spacing”, ”espacement” or ”espaciamiento”, in opposition to “forming”.Eisenman’s conceptual-generative diagram constructs and develops a matrix field of forces and geometries that, acting in the project as a spatial-formal guide, opens up from the first record or first intention, to many possibilities of configuration/definition of the object or architectural place. Consequently, it makes possible the exploration and discovery in architecture of other ways of thinking, imagining and manifesting forms and spaces, that investigate new ways of occupancy and promote other possible ways of life. It is an architecture in which diagrams are constituted as an expression of the figural/imprecise/blurred condition, the traces of which persist in the space-form of the building; a diagram that is both a creative interface between the intrinsic exploration of its defined concepts and the final configured complexity of its spatialities and functional superpositions.