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Journal articles on the topic 'Projective degrees of a rational map'

Author: Grafiati

Published: 9 March 2023

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1

HASSELBLATT, BORIS, and JAMES PROPP. "Degree-growth of monomial maps." Ergodic Theory and Dynamical Systems 27, no. 5 (October 2007): 1375–97. http://dx.doi.org/10.1017/s0143385707000168.

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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.

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2

LESIEUTRE, JOHN, and MATTHEW SATRIANO. "A rational map with infinitely many points of distinct arithmetic degrees." Ergodic Theory and Dynamical Systems 40, no. 11 (April 12, 2019): 3051–55. http://dx.doi.org/10.1017/etds.2019.30.

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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 $X=\mathbb{P}^{4}$.

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3

Kohn, Kathlén, and Kristian Ranestad. "Projective Geometry of Wachspress Coordinates." Foundations of Computational Mathematics 20, no. 5 (November 11, 2019): 1135–73. http://dx.doi.org/10.1007/s10208-019-09441-z.

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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.

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4

KIEM, YOUNG-HOON, and HAN-BOM MOON. "MODULI SPACE OF STABLE MAPS TO PROJECTIVE SPACE VIA GIT." International Journal of Mathematics 21, no. 05 (May 2010): 639–64. http://dx.doi.org/10.1142/s0129167x10006264.

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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.

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COLOMBO, ELISABETTA, and BERT VAN GEEMEN. "A FAMILY OF MARKED CUBIC SURFACES AND THE ROOT SYSTEM D5." International Journal of Mathematics 18, no. 05 (May 2007): 505–25. http://dx.doi.org/10.1142/s0129167x07004163.

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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.

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6

BANAGL, MARKUS, and LAURENTIU MAXIM. "DEFORMATION OF SINGULARITIES AND THE HOMOLOGY OF INTERSECTION SPACES." Journal of Topology and Analysis 04, no. 04 (December 2012): 413–48. http://dx.doi.org/10.1142/s1793525312500185.

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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.

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7

SILVERMAN, JOSEPH H. "Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space." Ergodic Theory and Dynamical Systems 34, no. 2 (April 2012): 647–78. http://dx.doi.org/10.1017/etds.2012.144.

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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.

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8

Leshin, Jonah. "On the degree of irrationality in Noether’s problem." International Journal of Number Theory 12, no. 05 (May 10, 2016): 1209–18. http://dx.doi.org/10.1142/s1793042116500743.

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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].

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9

Mejía, Israel Moreno. "On the Image of Certain Extension Maps. I." Canadian Mathematical Bulletin 50, no. 3 (September 1, 2007): 427–33. http://dx.doi.org/10.4153/cmb-2007-041-0.

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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.

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10

Voineagu, Mircea. "Cylindrical homomorphisms and Lawson homology." Journal of K-Theory 8, no. 1 (June 8, 2010): 135–68. http://dx.doi.org/10.1017/is010004024jkt108.

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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.

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11

Chung, Kiryong, and Sanghyeon Lee. "Stable maps of genus zero in the space of stable vector bundles on a curve." International Journal of Mathematics 28, no. 11 (October 2017): 1750078. http://dx.doi.org/10.1142/s0129167x17500781.

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Let [Formula: see text] be a smooth projective curve with genus [Formula: see text]. Let [Formula: see text] be the moduli space of stable rank two vector bundles on [Formula: see text] with a fixed determinant [Formula: see text] for [Formula: see text]. In this paper, as a generalization of Kiem and Castravet’s works, we study the stable maps in [Formula: see text] with genus [Formula: see text] and degree [Formula: see text]. Let [Formula: see text] be a natural closed subvariety of [Formula: see text] which parametrizes stable vector bundles with a fixed subbundle [Formula: see text] for a line bundle [Formula: see text] on [Formula: see text]. We describe the stable map space [Formula: see text]. It turns out that the space [Formula: see text] consists of two irreducible components. One of them parameterizes smooth rational cubic curves and the other parameterizes the union of line and smooth conics.

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12

Dimca, Alexandru, and Ştefan Papadima. "Non-abelian cohomology jump loci from an analytic viewpoint." Communications in Contemporary Mathematics 16, no. 04 (July 14, 2014): 1350025. http://dx.doi.org/10.1142/s0219199713500259.

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For a space, we investigate its CJL (cohomology jump loci), sitting inside varieties of representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its CJL, sitting inside varieties of flat connections. The analytic germs at the origin 1 of representation varieties are shown to be determined by the Sullivan 1-minimal model of the space. Up to a degree q, the two types of CJL have the same analytic germs at the origins, when the space and the algebra have the same q-minimal model. We apply this general approach to formal spaces (obtaining the degeneration of the Farber–Novikov spectral sequence), quasi-projective manifolds, and finitely generated nilpotent groups. When the CDG algebra has positive weights, we elucidate some of the structure of (rank one complex) topological and algebraic CJL: all their irreducible components passing through the origin are connected affine subtori, respectively rational linear subspaces. Furthermore, the global exponential map sends all algebraic CJL into their topological counterpart.

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13

Cid-Ruiz, Yairon. "Mixed multiplicities and projective degrees of rational maps." Journal of Algebra 566 (January 2021): 136–62. http://dx.doi.org/10.1016/j.jalgebra.2020.08.037.

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14

Goldberg, Bryan, and Rongwei Yang. "Self-similarity and spectral dynamics." Journal of Operator Theory 87, no. 1 (March 15, 2022): 355–88. http://dx.doi.org/10.7900/jot.2020sep27.2329.

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This paper investigates a connection between self-similar group representations and induced rational maps on the projective space which preserve the projective spectrum of the group. The focus is on the infinite dihedral group D∞. The main theorem states that the Julia set of the induced rational map F on P2 for D∞ is the union of the projective spectrum with F's extended indeterminacy set. Moreover, the limit function of the iteration sequence {F∘n} on the Fatou set is fully described. This discovery finds an application to the Grigorchuk group G of intermediate growth and its induced rational map G on P4. In the end, the paper proposes the conjecture that G's projective spectrum is contained in the Julia set of G.

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15

Brown, Morgan, and Tyler Foster. "Rational connectivity and analytic contractibility." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 747 (February 1, 2019): 45–62. http://dx.doi.org/10.1515/crelle-2016-0019.

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Abstract Let {{k}} be an algebraically closed field of characteristic 0, and let {f:X\to Y} be a morphism of smooth projective varieties over the ring {k((t))} of formal Laurent series. We prove that if a general geometric fiber of f is rationally connected, then the induced map {f^{\mathrm{an}}:X^{\mathrm{an}}\to Y^{\mathrm{an}}} between the Berkovich analytifications of X and Y is a homotopy equivalence. Two important consequences of this result are that the Berkovich analytification of any {\mathbb{P}^{n}} -bundle over a smooth projective {k((t))} -variety is homotopy equivalent to the Berkovich analytification of the base, and that the Berkovich analytification of a rationally connected smooth projective variety over {k((t))} is contractible.

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16

Dang, Nguyen‐Bac. "Degrees of iterates of rational maps on normal projective varieties." Proceedings of the London Mathematical Society 121, no. 5 (July 14, 2020): 1268–310. http://dx.doi.org/10.1112/plms.12366.

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17

Achter, Jeffrey D., Sebastian Casalaina-Martin, and Charles Vial. "On descending cohomology geometrically." Compositio Mathematica 153, no. 7 (May 10, 2017): 1446–78. http://dx.doi.org/10.1112/s0010437x17007151.

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In this paper, motivated by a problem posed by Barry Mazur, we show that for smooth projective varieties over the rationals, the odd cohomology groups of degree less than or equal to the dimension can be modeled by the cohomology of an abelian variety, provided the geometric coniveau is maximal. This provides an affirmative answer to Mazur’s question for all uni-ruled threefolds, for instance. Concerning cohomology in degree three, we show that the image of the Abel–Jacobi map admits a distinguished model over the rationals.

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18

Hoa, Tran Quang, and Ho Vu Ngoc Phuong. "Fibers of rational maps and Rees algebras of their base ideals." Hue University Journal of Science: Natural Science 129, no. 1B (June 22, 2020): 5–14. http://dx.doi.org/10.26459/hueuni-jns.v129i1b.5349.

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We consider a ratinonal map $\phi$ from m-dimensional projective space to n-dimensional projective space that is a parameterization of m-dimensional variety. Our main goal is to study the (m-1)-dimensional fibers of $\phi$ in relation with the m-th local cohomology modules of Rees algebra of its base ideal.

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19

Krylov, Igor, and Takuzo Okada. "Stable Rationality of del Pezzo Fibrations of Low Degree Over Projective Spaces." International Mathematics Research Notices 2020, no. 23 (October 25, 2018): 9075–119. http://dx.doi.org/10.1093/imrn/rny252.

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Abstract The main aim of this article is to show that a very general three-dimensional del Pezzo fibration of degrees 1, 2, and 3 is not stably rational except for a del Pezzo fibration of degree 3 belonging to explicitly described two families. Higher-dimensional generalizations are also discussed and we prove that a very general del Pezzo fibration of degrees 1, 2, and 3 defined over the projective space is not stably rational provided that the anti-canonical divisor is not ample.

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20

Fan, Shilei, and Lingmin Liao. "Rational map ax + 1/x on the projective line over ℚ2." Science China Mathematics 61, no. 12 (November 19, 2018): 2221–36. http://dx.doi.org/10.1007/s11425-017-9229-3.

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21

HINDES, WADE. "Dynamical and arithmetic degrees for random iterations of maps on projective space." Mathematical Proceedings of the Cambridge Philosophical Society 171, no. 2 (February 26, 2021): 369–85. http://dx.doi.org/10.1017/s0305004120000250.

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AbstractWe show that the dynamical degree of an (i.i.d) random sequence of dominant, rational self-maps on projective space is almost surely constant. We then apply this result to height growth and height counting problems in random orbits.

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22

Okonek, Christian, and Andrei Teleman. "A wall-crossing formula for degrees of Real central projections." International Journal of Mathematics 25, no. 04 (April 2014): 1450038. http://dx.doi.org/10.1142/s0129167x14500384.

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The main result is a wall-crossing formula for central projections defined on submanifolds of a Real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to Real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a ℤ-valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the Real subspace problem.

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23

Fisette, Robert, and Alexander Polishchuk. "-algebras associated with curves and rational functions on . I." Compositio Mathematica 150, no. 4 (March 10, 2014): 621–67. http://dx.doi.org/10.1112/s0010437x13007574.

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AbstractWe consider the natural$A_{\infty }$-structure on the$\mathrm{Ext}$-algebra$\mathrm{Ext}^*(G,G)$associated with the coherent sheaf$G=\mathcal{O}_C\oplus \mathcal{O}_{p_1}\oplus \cdots \oplus \mathcal{O}_{p_n}$on a smooth projective curve$C$, where$p_1,\ldots,p_n\in C$are distinct points. We study the homotopy class of the product$m_3$. Assuming that$h^0(p_1+\cdots +p_n)=1$, we prove that$m_3$is homotopic to zero if and only if$C$is hyperelliptic and the points$p_i$are Weierstrass points. In the latter case we show that$m_4$is not homotopic to zero, provided the genus of$C$is greater than$1$. In the case$n=g$we prove that the$A_{\infty }$-structure is determined uniquely (up to homotopy) by the products$m_i$with$i\le 6$. Also, in this case we study the rational map$\mathcal{M}_{g,g}\to \mathbb{A}^{g^2-2g}$associated with the homotopy class of$m_3$. We prove that for$g\ge 6$it is birational onto its image, while for$g\le 5$it is dominant. We also give an interpretation of this map in terms of tangents to$C$in the canonical embedding and in the projective embedding given by the linear series$|2(p_1+\cdots +p_g)|$.

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24

Golla, Marco, and Laura Starkston. "The symplectic isotopy problem for rational cuspidal curves." Compositio Mathematica 158, no. 7 (July 2022): 1595–682. http://dx.doi.org/10.1112/s0010437x2200762x.

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We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology.

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25

Gasbarri, Carlo. "Transcendental Liouville Inequalities on Projective Varieties." International Mathematics Research Notices 2020, no. 24 (November 15, 2019): 9844–86. http://dx.doi.org/10.1093/imrn/rnz252.

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Abstract Let $p$ be an algebraic point of a projective variety $X$ defined over a number field. Liouville inequality tells us that the norm at $p$ of a non-vanishing integral global section of a hermitian line bundle over $X$ is zero or it cannot be too small with respect to the $\sup $ norm of the section itself. We study inequalities similar to Liouville’s for subvarietes and for transcendental points of a projective variety defined over a number field. We prove that almost all transcendental points verify a good inequality of Liouville type. We also relate our methods to a (former) conjecture by Chudnovsky and give two applications to the growth of the number of rational points of bounded height on the image of an analytic map from a disk to a projective variety.

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26

Bogomolov, F. A., and Yu Tschinkel. "On the density of rational points on elliptic fibrations." Journal für die reine und angewandte Mathematik (Crelles Journal) 1999, no. 511 (June 25, 1999): 87–93. http://dx.doi.org/10.1515/crll.1999.511.87.

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1. Introduction Let X be an algebraic variety defined over a number field F. We will say that rational points are potentially dense if there exists a finite extension K/F such that the set of K-rational points X(K) is Zariski dense in X. The main problem is to relate this property to geometric invariants of X. Hypothetically, on varieties of general type rational points are not potentially dense. In this paper we are interested in smooth projective varieties such that neither they nor their unramified coverings admit a dominant map onto varieties of general type. For these varieties it seems plausible to expect that rational points are potentially dense (see [2]).

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27

ONO, ASUKA, and TOHRU KOHDA. "SOLVABLE THREE-DIMENSIONAL RATIONAL CHAOTIC MAP DEFINED BY JACOBIAN ELLIPTIC FUNCTIONS." International Journal of Bifurcation and Chaos 17, no. 10 (October 2007): 3645–50. http://dx.doi.org/10.1142/s0218127407019500.

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Cryptanalysis needs a lot of pseudo-random numbers. In particular, a sequence of independent and identically distributed (i.i.d.) binary random variables plays an important role in modern digital communication systems. Sufficient conditions have been recently provided for a class of ergodic maps of an interval onto itself: R1 → R1 and its associated binary function to generate a sequence of i.i.d. random variables. In order to get more i.i.d. binary random vectors, Jacobian elliptic Chebyshev rational map, its derivative and second derivative which define a Jacobian elliptic space curve have been introduced. Using duplication formula gives three-dimensional real-valued sequences on the space curve onto itself: R3 → R3. This also defines three projective onto mappings, represented in the form of rational functions of xn, yn, zn. These maps generate a three-dimensional sequence of i.i.d. random vectors.

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28

Korshunov, R. A. "Using the fictive points to transform entities in the process of map updating." Geodesy and Cartography 924, no. 6 (July 20, 2017): 17–24. http://dx.doi.org/10.22389/0016-7126-2017-924-6-17-24.

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The method of transformation of a vectorized situation from remote-sensed data to digital map, using fictive points, by means of rational polynoms of different degrees is reported. Those points are not really existing corresponding points of entities and their images in the picture. Actually they are the intersections of appropriate corresponding lines both on the map and on the picture (photo). These lines have been drawn through the identical edges of contours and also new vectors, formed by fictive points. Those ones were used to calculate transform parameters of rational polynoms. We may take the degrees of polynoms (the numerator and denominator) separately from the first to the third degree. The degree of rational polynoms be applied depends on situation in the map and on the geometry of material in use. If any coefficient in particular polynoms of 1t, 2nd or 3d degree, making near zero contribution, useful to eliminate it. This enlarges degree of freedom in resolution. These parameters are used for transfer entity of actualization direct from picture on the digital map. Method was tested on the special maid application ad models and on real images and maps.

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29

Lodha, Yash. "An upper bound for the Tarski numbers of nonamenable groups of piecewise projective homeomorphisms." International Journal of Algebra and Computation 27, no. 03 (April 11, 2017): 315–21. http://dx.doi.org/10.1142/s0218196717500151.

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The Tarski number of a nonamenable group is the smallest number of pieces needed for a paradoxical decomposition of the group. Nonamenable groups of piecewise projective homeomorphisms were introduced in [N. Monod, Groups of piecewise projective homeomorphisms, Proc. Natl. Acad. Sci. 110(12) (2013) 4524–4527], and nonamenable finitely presented groups of piecewise projective homeomorphisms were introduced in [Y. Lodha and J. T. Moore, A finitely presented non amenable group of piecewise projective homeomorphisms, Groups, Geom. Dyn. 10(1) (2016) 177–200]. These groups do not contain non-abelian free subgroups. In this paper, we prove that the Tarski number of all groups in both families is at most 25. In particular, we demonstrate the existence of a paradoxical decomposition with 25 pieces. Our argument also applies to any group of piecewise projective homeomorphisms that contains as a subgroup the group of piecewise [Formula: see text] homeomorphisms of [Formula: see text] with rational breakpoints and an affine map that is a not an integer translation.

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30

nsiper, Hurit. "On the structure of generalized Albanese varieties." Mathematical Proceedings of the Cambridge Philosophical Society 111, no. 2 (March 1992): 267–72. http://dx.doi.org/10.1017/s0305004100075356.

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Given a smooth projective surface X over an algebraically closed field k and a modulus (an effective divisor) m on X, one defines the idle class group Cm(X) of X with modulus m (see 1, chapter III, section 4). The corresponding generalized Albanese variety Gum and the generalized Albanese map um:X|m|Gum have the following universal mapping property (2): if :XG is a rational map into a commutative algebraic group which induces a homomorphism Cm(X)G(k) (1, chapter III, proposition 1), then factors uniquely through um.

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31

KOCH, SARAH, and ROLAND K. W. ROEDER. "Computing dynamical degrees of rational maps on moduli space." Ergodic Theory and Dynamical Systems 36, no. 8 (July 21, 2015): 2538–79. http://dx.doi.org/10.1017/etds.2015.29.

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The dynamical degrees of a rational map$f:X{\dashrightarrow}X$are fundamental invariants describing the rate of growth of the action of iterates of$f$on the cohomology of$X$. When$f$has non-empty indeterminacy set, these quantities can be very difficult to determine. We study rational maps$f:X^{N}{\dashrightarrow}X^{N}$, where$X^{N}$is isomorphic to the Deligne–Mumford compactification$\overline{{\mathcal{M}}}_{0,N+3}$. We exploit the stratified structure of$X^{N}$to provide new examples of rational maps, in arbitrary dimension, for which the action on cohomology behaves functorially under iteration. From this, all dynamical degrees can be readily computed (given enough book-keeping and computing time). In this paper, we explicitly compute all of the dynamical degrees for all such maps$f:X^{N}{\dashrightarrow}X^{N}$, where$\text{dim}(X^{N})\leq 3$and the first dynamical degrees for the mappings where$\text{dim}(X^{N})\leq 5$. These examples naturally arise in the setting of Thurston’s topological characterization of rational maps.

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32

Liu, Yongqiang, Laurenţiu Maxim, and Botong Wang. "Aspherical manifolds, Mellin transformation and a question of Bobadilla–Kollár." Journal für die reine und angewandte Mathematik (Crelles Journal) 2021, no. 781 (October 16, 2021): 1–18. http://dx.doi.org/10.1515/crelle-2021-0055.

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Abstract In their paper from 2012, Bobadilla and Kollár studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this paper, we answer positively the integral homology version of their question in the case of abelian varieties, and the rational homology version in the case of compact ball quotients. We also propose several conjectures in relation to the Singer–Hopf conjecture in the complex projective setting.

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33

Livorni, Elvira Laura. "Classification of Algebraic Surfaces with Sectional Genus less than or Equal to Six. II: Ruled Surfaces with dim ϕKx⊗L(x) = l." Canadian Journal of Mathematics 38, no. 5 (October 1, 1986): 1110–21. http://dx.doi.org/10.4153/cjm-1986-055-5.

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Let L be a very ample line bundle on a smooth, connected, projective, ruled not rational surface X. We have considered the problem of classifying biholomorphically smooth, connected, projected, ruled, non rational surfaces X with smooth hyperplane section C such that the genus g = g(C) is less than or equal to six and dim where is the map associated to . L. Roth in [10] had given a birational classification of such surfaces. If g = 0 or 1 then X has been classified, see [8].If g = 2 ≠ hl,0(X) by [12, Lemma (2.2.2) ] it follows that X is a rational surface. Thus we can assume g ≦ 3.Since X is ruled, h2,0(X) = 0 andsee [4] and [12, p. 390].

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34

Jacobi, Martin N. "Hierarchical Organization in Smooth Dynamical Systems." Artificial Life 11, no. 4 (September 2005): 493–512. http://dx.doi.org/10.1162/106454605774270598.

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This article is concerned with defining and characterizing hierarchical structures in smooth dynamical systems. We define transitions between levels in a dynamical hierarchy by smooth projective maps from a phase space on a lower level, with high dimensionality, to a phase space on a higher level, with lower dimensionality. It is required that each level describe a self-contained deterministic dynamical system. We show that a necessary and sufficient condition for a projective map to be a transition between levels in the hierarchy is that the kernel of the differential of the map is tangent to an invariant manifold with respect to the flow. The implications of this condition are discussed in detail. We demonstrate two different causal dependences between degrees of freedom, and how these relations are revealed when the dynamical system is transformed into global Jordan form. Finally these results are used to define functional components on different levels, interaction networks, and dynamical hierarchies.

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35

Truong, Tuyen Trung. "Relative dynamical degrees of correspondences over a field of arbitrary characteristic." Journal für die reine und angewandte Mathematik (Crelles Journal) 2020, no. 758 (January 1, 2020): 139–82. http://dx.doi.org/10.1515/crelle-2017-0052.

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AbstractLet {\mathbb{K}} be an algebraically closed field of arbitrary characteristic, X and Y irreducible possibly singular algebraic varieties over {\mathbb{K}}. Let {f:X\vdash X} and {g:Y\vdash Y} be dominant correspondences, and {\pi:X\dashrightarrow Y} a dominant rational map which semi-conjugate f and g, i.e. so that {\pi\circ f=g\circ\pi}. We define relative dynamical degrees {\lambda_{p}(f|\pi)\geq 1} for any {p=0,\dots,\dim(X)-\dim(Y)}. These degrees measure the relative growth of positive algebraic cycles, satisfy a product formula when Y is smooth and g is a multiple of a rational map, and are birational invariants. More generally, a weaker product formula is proven for more general semi-conjugacies, and for any generically finite semi-conjugacy {(\varphi,\psi)} from {\pi_{2}:(X_{2},f_{2})\rightarrow(Y_{2},g_{2})} to {\pi_{1}:(X_{1},f_{1})\rightarrow(Y_{1},g_{1})} we have {\lambda_{p}(f_{1}|\pi_{1})\geq\lambda_{p}(f_{2}|\pi_{2})} for all p. Many of our results are new even when {\mathbb{K}=\mathbb{C}}. Self-correspondences are abundant, even on varieties having not many self rational maps, hence these results can be applied in many situations. In the last section of the paper, we will discuss recent new applications of this to algebraic dynamics, in particular to pullback on l-adic cohomology groups in positive characteristics.

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36

ARCARA, D. "EXTENSIONS AND RANK-2 VECTOR BUNDLES ON IRREDUCIBLE NODAL CURVES." International Journal of Mathematics 16, no. 10 (November 2005): 1081–118. http://dx.doi.org/10.1142/s0129167x05003284.

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We generalize Bertram's work on rank two vector bundles to an irreducible projective nodal curve C. We use the natural rational map [Formula: see text] defined by [Formula: see text] to study a compactification [Formula: see text] of the moduli space [Formula: see text] of semi-stable vector bundles of rank 2 and determinant L on C. In particular, we resolve the indeterminancy of ϕL in the case deg L = 3,4 via a sequence of three blow-ups with smooth centers.

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37

Choudhry, Ajai. "A new method of solving certain quartic and higher degree diophantine equations." International Journal of Number Theory 14, no. 08 (August 22, 2018): 2129–54. http://dx.doi.org/10.1142/s1793042118501282.

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In this paper, we present a new method of solving certain quartic and higher degree homogeneous polynomial diophantine equations in four variables. The method can also be applied to some diophantine systems in five or more variables. Under certain conditions, the method yields an arbitrarily large number of integer solutions of such diophantine equations and diophantine systems, two examples being a sextic equation in four variables and two simultaneous equations of degrees four and six in six variables. We also simultaneously obtain arbitrarily many rational solutions of certain related nonhomogeneous equations of high degree. We obtain these solutions without finding a curve of genus 0 or 1 on the variety defined by the equations concerned. It appears that there exist projective varieties on which there are an arbitrarily large number of rational points and which do not contain a curve of genus 0 or 1 with infinitely many rational points.

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38

Zhu, Huanping. "The canonical volume of minimal 3-folds of general type." International Journal of Mathematics 29, no. 03 (March 2018): 1850023. http://dx.doi.org/10.1142/s0129167x18500234.

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Let [Formula: see text] be a nonsingular projective [Formula: see text]-fold of general type. Denote by [Formula: see text] the [Formula: see text]-canonical map of [Formula: see text] which is the rational map naturally associated to the complete linear system [Formula: see text]. Suppose that [Formula: see text] be a minimal [Formula: see text]-fold of [Formula: see text] and [Formula: see text] the pluricanonical section index. In this paper, we obtain the lower bounds of the canonical volume [Formula: see text] in term of [Formula: see text] for [Formula: see text]. In addition, we also classify the weighted baskets [Formula: see text] of [Formula: see text] satisfying [Formula: see text].

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39

Blekherman, Grigoriy, Jonathan Hauenstein, John Christian Ottem, Kristian Ranestad, and Bernd Sturmfels. "Algebraic boundaries of Hilbert’s SOS cones." Compositio Mathematica 148, no. 6 (October 15, 2012): 1717–35. http://dx.doi.org/10.1112/s0010437x12000437.

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AbstractWe study the geometry underlying the difference between non-negative polynomials and sums of squares (SOS). The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether–Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized, respectively, by the Severi variety of plane rational sextics and by the variety of quartic symmetroids.

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40

NEVINS, THOMAS A. "MODULI SPACES OF FRAMED SHEAVES ON CERTAIN RULED SURFACES OVER ELLIPTIC CURVES." International Journal of Mathematics 13, no. 10 (December 2002): 1117–51. http://dx.doi.org/10.1142/s0129167x02001599.

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Fix a ruled surface S obtained as the projective completion of a line bundle L on a complex elliptic curve C; we study the moduli problem of parametrizing certain pairs consisting of a sheaf ℰ on S and a map of ℰ to a fixed reference sheaf on S. We prove that the full moduli stack for this problem is representable by a scheme in some cases. Moreover, the moduli stack admits an action by the group C*, and we determine its fixed-point set, which leads to explicit formulas for the rational homology of the moduli space.

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41

ACUS, A., E. NORVAIŠAS, and D. O. RISKA. "NUCLEI AS SUPERPOSITION OF TOPOLOGICAL SOLITONS." International Journal of Modern Physics E 17, no. 01 (January 2008): 212–16. http://dx.doi.org/10.1142/s0218301308009719.

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The rational map approximation provides an opportunity to describe light nuclei as classical solitons with baryon number B > 1 in the framework of the Skyrme model. The rational map ansatz yields a possibility of factorization of S3 baryon charge into S1 and S2 parts, the phenomenology of the model being strongly affected by the chosen factorization. Moreover, in the fundamental representation superposition of two different soliton factorizations can be used as solution ansatz. The canonical quantization procedure applied to collective degrees of freedom of the classical soliton leads to anomalous breaking of the chiral symmetry and exponential falloff of the energy density of the soliton at large distance, without explicit symmetry breaking terms included. The evolution of the shape of electric form factor as a function of two different factorization soliton mix ratio is investigated. Numerical results are presented.

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42

HOFFMAN, J. WILLIAM, and HAOHAO WANG. "DEFINING EQUATIONS OF THE REES ALGEBRA OF CERTAIN PARAMETRIC SURFACES." Journal of Algebra and Its Applications 09, no. 06 (December 2010): 1033–49. http://dx.doi.org/10.1142/s0219498810004385.

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Let f0, f1, f2, f3 be linearly independent nonzero homogeneous polynomials in the standard ℤ-graded ring R ≔ 𝕂[s, t, u] of the same degree d, and gcd (f0, f1, f2, f3) = 1. This defines a rational map ℙ2 → ℙ3. The Rees algebra Rees (I) = R ⊕ I ⊕ I2 ⊕ ⋯ of the ideal I = 〈f0, f1, f2, f3〉 is the graded R-algebra which can be described as the image of the R-algebra homomorphism h: R[x, y, z, w ] → Rees (I). This paper discusses one result concerning the structure of the kernel of the map h when I is a saturated local complete intersection ideal with V(I) ≠ ∅ and μ-basis of degrees (1,1,d - 2).

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43

de Jeu, Rob, and James D. Lewis. "Beilinson's Hodge Conjecture for Smooth Varieties." Journal of K-Theory 11, no. 2 (March 6, 2013): 243–82. http://dx.doi.org/10.1017/is013001030jkt212.

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AbstractLet U/ℂ be a smooth quasi-projective variety of dimension d, CHr (U,m) Bloch's higher Chow group, andclr,m: CHr (U,m) ⊗ ℚ → homMHS (ℚ(0), H2r−m (U, ℚ(r)))the cycle class map. Beilinson once conjectured clr,m to be surjective [Be]; however, Jannsen was the first to find a counterexample in the case m = 1 [Ja1]. In this paper we study the image of clr,m in more detail (as well as at the “generic point” of U) in terms of kernels of Abel-Jacobi mappings. When r = m, we deduce from the Bloch-Kato conjecture (now a theorem) various results, in particular that the cokernel of clm,m at the generic point is the same for integral or rational coefficients.

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44

Levin, Aaron, and Julie Tzu-Yueh Wang. "On Non-Archimedean Curves Omitting Few Components and their Arithmetic Analogues." Canadian Journal of Mathematics 69, no. 1 (February 1, 2017): 130–42. http://dx.doi.org/10.4153/cjm-2015-030-1.

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AbstractLet k be an algebraically closed field completewith respect to a non-Archimedean absolute value of arbitrary characteristic. Let D1 , … , Dn be effective nef divisors intersecting transversally in an n-dimensional nonsingular projective variety X. We study the degeneracy of non-Archimedean analytic maps from k into under various geometric conditions. When X is a rational ruled surface and D1 and D2 are ample, we obtain a necessary and sufficient condition such that there is no non-Archimedean analytic map from k into . Using the dictionary between non-Archimedean Nevanlinna theory and Diophantine approximation that originated in earlier work with T. T. H. An, we also study arithmetic analogues of these problems, establishing results on integral points on these varieties over ℤ or the ring of integers of an imaginary quadratic field.

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45

DE MARCO, LAURA, and DRAGOS GHIOCA. "Rationality of dynamical canonical height." Ergodic Theory and Dynamical Systems 39, no. 9 (January 18, 2018): 2507–40. http://dx.doi.org/10.1017/etds.2017.131.

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We present a dynamical proof of the well-known fact that the Néron–Tate canonical height (and its local counterpart) takes rational values at points of an elliptic curve over a function field $k=\mathbb{C}(X)$, where $X$ is a curve. More generally, we investigate the mechanism by which the local canonical height for a map $f:\mathbb{P}^{1}\rightarrow \mathbb{P}^{1}$ defined over a function field $k$ can take irrational values (at points in a local completion of $k$), providing examples in all degrees $\deg f\geq 2$. Building on Kiwi’s classification of non-archimedean Julia sets for quadratic maps [Puiseux series dynamics of quadratic rational maps. Israel J. Math.201 (2014), 631–700], we give a complete answer in degree 2 characterizing the existence of points with irrational local canonical heights. As an application we prove that if the heights $\widehat{h}_{f}(a),\widehat{h}_{g}(b)$ are rational and positive, for maps $f$ and $g$ of multiplicatively independent degrees and points $a,b\in \mathbb{P}^{1}(\bar{k})$, then the orbits $\{f^{n}(a)\}_{n\geq 0}$ and $\{g^{m}(b)\}_{m\geq 0}$ intersect in at most finitely many points, complementing the results of Ghioca et al [Intersections of polynomials orbits, and a dynamical Mordell–Lang conjecture. Invent. Math.171 (2) (2008), 463–483].

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46

CHEN, MENG. "Canonical stability in terms of singularity index for algebraic threefolds." Mathematical Proceedings of the Cambridge Philosophical Society 131, no. 2 (September 2001): 241–64. http://dx.doi.org/10.1017/s030500410100531x.

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Throughout the ground field is always supposed to be algebraically closed of characteristic zero. Let X be a smooth projective threefold of general type, denote by ϕm the m-canonical map of X which is nothing but the rational map naturally associated with the complete linear system [mid ]mKX[mid ]. Since, once given such a 3-fold X, ϕm is birational whenever m [Gt ] 0, quite an interesting thing to find is the optimal bound for such an m. This bound is important because it is not only crucial to the classification theory, but also strongly related to other problems. For example, it can be applied to determine the order of the birational automorphism group of X [21, remark in section 1]. To fix the terminology we say that ϕm is stably birational if ϕt is birational onto its image for all t [ges ] m. It is well known that the parallel problem in the surface case was solved by Bombieri [1] and others. In the 3-dimensional case, many authors have studied the problem, in quite different ways. Because, in this paper, we are interested in the results obtained by Hanamura [7], we do not plan to mention more references here. According to 3-dimensional MMP, X has a minimal model which is a normal projective 3-fold with only ℚ-factorial terminal singularities. Though X may have many minimal models, the singularity index (namely the canonical index) of any of its minimal models is uniquely determined by X. Denote by r the canonical index of minimal models of X. When r = 1 we know that ϕ6 is stably birational by virtue of [3, 6, 13 and 14]. When r [ges ] 2, Hanamura proved the following theorem.

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47

Samokhvalova, E. V., S. N. Zudilin, and O. A. Lavrennikova. "Spatial analysis of Samara region land degradation and differentiation of antierosion territory organization." BIO Web of Conferences 27 (2020): 00142. http://dx.doi.org/10.1051/bioconf/20202700142.

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In the research, a spatial analysis of the degradation of Samara region agricultural land with the assessment of economic losses due to water erosion is carried out. A map chart of the distribution of districts with different degrees of erosion has also been developed. The values of the degradation factor coefficient and economic losses due to the influence of erosion processes are calculated. The key points of antierosion territory organization and land regulation depending on landscape nature and kind of damage are represented. The plan of action for the antierosion territory organization of a farm in Kinelsky district is proposed and its effectiveness to stop and prevent erosion processes, as well as for rational use of land and increase soil fertility is shown.

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48

SUGIYAMA, KATSUYUKI. "THREE-POINT FUNCTIONS ON THE SPHERE OF CALABI-YAU d-FOLDS." International Journal of Modern Physics A 11, no. 02 (January 20, 1996): 229–52. http://dx.doi.org/10.1142/s0217751x96000110.

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Using mirror symmetry in Calabi-Yau manifolds M, we study three-point functions of A(M) model operators on the genus 0 Riemann surface in cases of one-parameter families of d-folds realized as Fermat type hypersurfaces embedded in weighted projective spaces and a two-parameter family of d-folds embedded in a weighted projective space Pd+1 [2,2,2,...,2,2,1,1] (2 (d + 1)). These three-point functions [Formula: see text] are expanded by indeterminates [Formula: see text] associated with a set of Kähler coordinates {tl}, and their expansion coefficients count the number of maps with a definite degree which map each of the three-points 0, 1 and ∞ on the world sheet on some homology cycle of M associated with a cohomology element. From these analyses, we can read the fusion structure of Calabi-Yau A(M) model operators. In our cases they constitute a subring of a total quantum cohomology ring of the A(M) model operators. In fact we switch off all perturbation operators on the topological theories except for marginal ones associated with Kähler forms of M. For that reason, the charge conservation of operators turns out to be a classical one. Furthermore, because their first Chern classes c1 vanish, their topological selection rules do not depend on the degree of maps (in particular, a nilpotent property of operators [Formula: see text] is satisfied). Then these fusion couplings {κl} are represented as some series adding up all degrees of maps.

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49

PANTI, GIOVANNI. "Decreasing height along continued fractions." Ergodic Theory and Dynamical Systems 40, no. 3 (August 10, 2018): 763–88. http://dx.doi.org/10.1017/etds.2018.55.

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The fact that the euclidean algorithm eventually terminates is pervasive in mathematics. In the language of continued fractions, it can be stated by saying that the orbits of rational points under the Gauss map$x\mapsto x^{-1}-\lfloor x^{-1}\rfloor$eventually reach zero. Analogues of this fact for Gauss maps defined over quadratic number fields have relevance in the theory of flows on translation surfaces, and have been established via powerful machinery, ultimately relying on the Veech dichotomy. In this paper, for each commensurability class of non-cocompact triangle groups of quadratic invariant trace field, we construct a Gauss map whose defining matrices generate a group in the class; we then provide a direct and self-contained proof of termination. As a byproduct, we provide a new proof of the fact that non-cocompact triangle groups of quadratic invariant trace field have the projective line over that field as the set of cross-ratios of cusps. Our proof is based on an analysis of the action of non-negative matrices with quadratic integer entries on the Weil height of points. As a consequence of the analysis, we show that long symbolic sequences in the alphabet of our maps can be effectively split into blocks of predetermined shape having the property that the height of points which obey the sequence and belong to the base field decreases strictly at each block end. Since the height cannot decrease infinitely, the termination property follows.

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50

BAJAJ, CHANDRAJIT L., and ANDREW V. ROYAPPA. "FINITE REPRESENTATIONS OF REAL PARAMETRIC CURVES AND SURFACES." International Journal of Computational Geometry & Applications 05, no. 03 (September 1995): 313–26. http://dx.doi.org/10.1142/s0218195995000180.

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Global parameterizations of parametric algebraic curves or surfaces are defined over infinite parameter domains. Considering parameterizations in terms of rational functions that have real coefficients and vary over real parameter values, we show how to replace one global parameterization with a finite number of alternate bounded parameterizations, each defined over a fixed, bounded part of the real parameter domain space. The new bounded parameterizations together generate all real points of the old one and in particular the points corresponding to infinite parameter values in the old domain. We term such an alternate finite set of bounded parameterizations a finite representation of a real parametric curve or surface. Two solutions are presented for real parametric varieties of arbitrary dimension n. In the first method, a real parametric variety of dimension n is finitely represented in a piecewise fashion by 2n bounded parameterizations with individual pieces meeting with C∞ continuity; each bounded parameterization is a map from a unit simplex of the real parameter domain space. In the second method, only a single bounded parameterization is used; it is a map from the unit hypersphere centered at the origin of the real parameter domain space. Both methods start with an arbitrary real parameterization of a real parametric variety and apply projective domain transformations of different types to yield the new bounded parameterizations. Both these methods are implementable in a straightforward fashion. Applications of these results include displaying entire real parametric curves and surfaces (except those real points generated by complex parameter values), computing normal parameterizations of curves and surfaces (settling an open problem for quadric surfaces).

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