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Journal articles on the topic 'Divisors (Algebraic Geometry)'

Author: Grafiati

Published: 4 June 2021

Last updated: 6 September 2023

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1

Ilten, Nathan Owen, and Hendrik Süß. "Algebraic geometry codes from polyhedral divisors." Journal of Symbolic Computation 45, no. 7 (July 2010): 734–56. http://dx.doi.org/10.1016/j.jsc.2010.03.008.

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2

Lanteri, Antonio, and Barbara Rondena. "Numerically positive divisors on algebraic surfaces." Geometriae Dedicata 53, no. 2 (November 1994): 145–54. http://dx.doi.org/10.1007/bf01264018.

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3

Fulger, Mihai. "Local volumes of Cartier divisors over normal algebraic varieties." Annales de l’institut Fourier 63, no. 5 (2013): 1793–847. http://dx.doi.org/10.5802/aif.2815.

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4

GHOSH, DIPANKAR, and TONY J. PUTHENPURAKAL. "Asymptotic prime divisors over complete intersection rings." Mathematical Proceedings of the Cambridge Philosophical Society 160, no. 3 (February 2, 2016): 423–36. http://dx.doi.org/10.1017/s0305004115000778.

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AbstractLet A be a local complete intersection ring. Let M, N be two finitely generated A-modules and I an ideal of A. We prove that $$\bigcup_{i\geqslant 0}\bigcup_{n \geqslant 0}{\rm Ass}_A\left({\rm Ext}_A^i(M,N/I^n N)\right)$$ is a finite set. Moreover, we prove that there exist i0, n0 ⩾ 0 such that for all i ⩾ i0 and n ⩾ n0, we have $$\begin{linenomath}\begin{subeqnarray*} {\rm Ass}_A\left({\rm Ext}_A^{2i}(M,N/I^nN)\right) &=& {\rm Ass}_A\left({\rm Ext}_A^{2 i_0}(M,N/I^{n_0}N)\right), \\ {\rm Ass}_A\left({\rm Ext}_A^{2i+1}(M,N/I^nN)\right) &=& {\rm Ass}_A\left({\rm Ext}_A^{2 i_0 + 1}(M,N/I^{n_0}N)\right). \end{subeqnarray*}\end{linenomath}$$ We also prove the analogous results for complete intersection rings which arise in algebraic geometry. Further, we prove that the complexity cxA(M, N/InN) is constant for all sufficiently large n.

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5

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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6

Kanel-Belov, Alexei, Alexei Chilikov, Ilya Ivanov-Pogodaev, Sergey Malev, Eugeny Plotkin, Jie-Tai Yu, and Wenchao Zhang. "Nonstandard Analysis, Deformation Quantization and Some Logical Aspects of (Non)Commutative Algebraic Geometry." Mathematics 8, no. 10 (October 2, 2020): 1694. http://dx.doi.org/10.3390/math8101694.

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This paper surveys results related to well-known works of B. Plotkin and V. Remeslennikov on the edge of algebra, logic and geometry. We start from a brief review of the paper and motivations. The first sections deal with model theory. In the first part of the second section we describe the geometric equivalence, the elementary equivalence, and the isotypicity of algebras. We look at these notions from the positions of universal algebraic geometry and make emphasis on the cases of the first order rigidity. In this setting Plotkin’s problem on the structure of automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of categories is pretty natural and important. The second part of the second section is dedicated to particular cases of Plotkin’s problem. The last part of the second section is devoted to Plotkin’s problem for automorphisms of the group of polynomial symplectomorphisms. This setting has applications to mathematical physics through the use of model theory (non-standard analysis) in the studying of homomorphisms between groups of symplectomorphisms and automorphisms of the Weyl algebra. The last sections deal with algorithmic problems for noncommutative and commutative algebraic geometry.The first part of it is devoted to the Gröbner basis in non-commutative situation. Despite the existence of an algorithm for checking equalities, the zero divisors and nilpotency problems are algorithmically unsolvable. The second part of the last section is connected with the problem of embedding of algebraic varieties; a sketch of the proof of its algorithmic undecidability over a field of characteristic zero is given.

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7

JARVIS, TYLER J. "GEOMETRY OF THE MODULI OF HIGHER SPIN CURVES." International Journal of Mathematics 11, no. 05 (July 2000): 637–63. http://dx.doi.org/10.1142/s0129167x00000325.

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This article treats various aspects of the geometry of the moduli [Formula: see text] of r-spin curves and its compactification [Formula: see text]. Generalized spin curves, or r-spin curves, are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), and have been of interest lately because of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. In particular, these spaces are the subject of a remarkable conjecture of E. Witten relating their intersection theory to the Gelfand–Dikii (KdVr) heirarchy. There is also a W-algebra conjecture for these spaces [16] generalizing the Virasoro conjecture of quantum cohomology. For any line bundle [Formula: see text] on the universal curve over the stack of stable curves, there is a smooth stack [Formula: see text] of triples (X, ℒ, b) of a smooth curve X, a line bundle ℒ on X, and an isomorphism [Formula: see text]. In the special case that [Formula: see text] is the relative dualizing sheaf, then [Formula: see text] is the stack [Formula: see text] of r-spin curves. We construct a smooth compactification [Formula: see text] of the stack [Formula: see text], describe the geometric meaning of its points, and prove that its coarse moduli is projective. We also prove that when r is odd and g>1, the compactified stack of spin curves [Formula: see text] and its coarse moduli space [Formula: see text] are irreducible, and when r is even and [Formula: see text] is the disjoint union of two irreducible components. We give similar results for n-pointed spin curves, as required for Witten's conjecture, and also generalize to the n-pointed case the classical fact that when [Formula: see text] is the disjoint union of d(r) components, where d(r) is the number of positive divisors of r. These irreducibility properties are important in the study of the Picard group of [Formula: see text] [15], and also in the study of the cohomological field theory related to Witten's conjecture [16, 34].

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8

Malygina, E. S. "Investigation of automorphism group for code associated with optimal curve of genus three." Prikladnaya Diskretnaya Matematika, no. 56 (2022): 5–16. http://dx.doi.org/10.17223/20710410/56/1.

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The main result of this paper is contained in two theorems. In the first theorem, it is proved that the mapping λ : ℒ(mP∞) → ℒ(mP∞) has the multiplicative property on the corresponding Riemann - Roch space associated with the divisor mP∞ which defines some algebraic-geometric code if the number of points of degree one in the function field of genus three optimal curve over finite field with a discriminant {-19, -43, -67, -163} has the lower bound 12m/(m-3). Using an explicit calculation with the valuations of the pole divisors of the images of the basis functions x, y, z in the function field of the curve via the mapping λ, we have proved that the automorphism group of the function field of our curve is a subgroup in the automorphism group of the corresponding algebraic-geometric code. In the second theorem, it is proved that if m≥ 4 and n > 12m/(m - 3), then the automorphism group of the function field of our curve is isomorphic to the automorphism group of the algebraic-geometric code n associated with divisors ∑ Pi и mP∞, where Pi are points of the degree one.

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9

Lindner, Sebastian, Laurent Imbert, and Michael J. Jacobson. "Improved divisor arithmetic on generic hyperelliptic curves." ACM Communications in Computer Algebra 54, no. 3 (September 2020): 95–99. http://dx.doi.org/10.1145/3457341.3457345.

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The divisor class group of a hyperelliptic curve defined over a finite field is a finite abelian group at the center of a number of important open questions in algebraic geometry, number theory and cryptography. Many of these problems lend themselves to numerical investigation, and as emphasized by Sutherland [14, 13], fast arithmetic in the divisor class group is crucial for their efficiency. Besides, implementations of these fundamental operations are at the core of the algebraic geometry packages of widely-used computer algebra systems such as Magma and Sage.

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10

Ballaÿ, François. "Successive minima and asymptotic slopes in Arakelov geometry." Compositio Mathematica 157, no. 6 (June 2021): 1302–39. http://dx.doi.org/10.1112/s0010437x21007156.

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Let $X$ be a normal and geometrically integral projective variety over a global field $K$ and let $\bar {D}$ be an adelic ${\mathbb {R}}$-Cartier divisor on $X$. We prove a conjecture of Chen, showing that the essential minimum $\zeta _{\mathrm {ess}}(\bar {D})$ of $\bar {D}$ equals its asymptotic maximal slope under mild positivity assumptions. As an application, we see that $\zeta _{\mathrm {ess}}(\bar {D})$ can be read on the Okounkov body of the underlying divisor $D$ via the Boucksom–Chen concave transform. This gives a new interpretation of Zhang's inequalities on successive minima and a criterion for equality generalizing to arbitrary projective varieties a result of Burgos Gil, Philippon and Sombra concerning toric metrized divisors on toric varieties. When applied to a projective space $X = {\mathbb {P}}_K^{d}$, our main result has several applications to the study of successive minima of hermitian vector spaces. We obtain an absolute transference theorem with a linear upper bound, answering a question raised by Gaudron. We also give new comparisons between successive slopes and absolute minima, extending results of Gaudron and Rémond.

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11

Timashev, D. A. "Cartier divisors and geometry of normalG-varieties." Transformation Groups 5, no. 2 (June 2000): 181–204. http://dx.doi.org/10.1007/bf01236468.

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12

Li, Zhu. "Algebro-Geometric Solutions of the Harry Dym Hierarchy." International Journal of Nonlinear Sciences and Numerical Simulation 18, no. 2 (April 1, 2017): 129–36. http://dx.doi.org/10.1515/ijnsns-2016-0057.

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AbstractThe Harry Dym hierarchy is derived with the help of Lenard recursion equations and zero curvature equation. Based on the Lax matrix, an algebraic curve $\mathcal{K}_{n}$ of arithmetic genus $n$ is introduced, from which the corresponding meromorphic function $\phi$ and Dubrovin-type equations are given. Further, the divisor and asymptotic properties of $\phi$ are studied. Finally, algebro-geometric solutions for the entire hierarchy are obtained according to above results and the theory of algebraic curve.

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13

Gesztesy, F., and R. Ratnaseelan. "An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy." Reviews in Mathematical Physics 10, no. 03 (April 1998): 345–91. http://dx.doi.org/10.1142/s0129055x98000112.

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We develop an alternative systematic approach to the AKNS hierarchy based on elementary algebraic methods. In particular, we recursively construct Lax pairs for the entire AKNS hierarchy by introducing a fundamental polynomial formalism and establish the basic algebro-geometric setting including associated Burchnall–Chaundy curves, Baker–Akhiezer functions, trace formulas, Dubrovin-type equations for analogs of Dirichlet and Neumann divisors, and theta function representations for algebro-geometric solutions.

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14

Katz, Eric, and Stefano Urbinati. "Newton–Okounkov Bodies over Discrete Valuation Rings and Linear Systems on Graphs." International Mathematics Research Notices 2019, no. 14 (January 3, 2018): 4516–48. http://dx.doi.org/10.1093/imrn/rnx248.

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Abstract The theory of Newton–Okounkov bodies attaches a convex body to a line bundle on a variety equipped with a flag of subvarieties. This convex body encodes the asymptotic properties of sections of powers of the line bundle. In this article, we study Newton–Okounkov bodies for schemes defined over discrete valuation rings. We give the basic properties and then focus on the case of toric schemes and semistable curves. We provide a description of the Newton–Okounkov bodies for semistable curves in terms of the Baker–Norine theory of linear systems on graphs, finding a connection with tropical geometry. We do this by introducing an intermediate object, the Newton–Okounkov linear system of a divisor on a curve. We prove that it is equal to the set of effective elements of the real Baker–Norine linear system of the specialization of that divisor on the dual graph of the curve. As a bonus, we obtain an asymptotic algebraic geometric description of the Baker–Norine linear system.

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15

Chen, Xuemiao, and Song Sun. "Algebraic Tangent Cones of Reflexive Sheaves." International Mathematics Research Notices 2020, no. 24 (December 6, 2018): 10042–63. http://dx.doi.org/10.1093/imrn/rny276.

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Abstract We study the notion of algebraic tangent cones at singularities of reflexive sheaves. These correspond to extensions of reflexive sheaves across a negative divisor. We show the existence of optimal extensions in a constructive manner, and we prove the uniqueness in a suitable sense. The results here are an algebro-geometric counterpart of our previous study on singularities of Hermitian–Yang–Mills connections.

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16

Rito, Carlos. "A surface with canonical map of degree 24." International Journal of Mathematics 28, no. 06 (April 19, 2017): 1750041. http://dx.doi.org/10.1142/s0129167x17500410.

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We construct a complex algebraic surface with geometric genus [Formula: see text], irregularity [Formula: see text], self-intersection of the canonical divisor [Formula: see text] and canonical map of degree [Formula: see text] onto [Formula: see text].

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17

Koam, Ali N. A., Ali Ahmad, and Azeem Haider. "On Eccentric Topological Indices Based on Edges of Zero Divisor Graphs." Symmetry 11, no. 7 (July 12, 2019): 907. http://dx.doi.org/10.3390/sym11070907.

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This article is devoted to the determination of edge-based eccentric topological indices of a zero divisor graph of some algebraic structures. In particular, we computed the first Zagreb eccentricity index, third Zagreb eccentricity index, geometric-arithmetic eccentricity index, atom-bond connectivity eccentricity index and a fourth type of eccentric harmonic index for zero divisor graphs associated with a class of finite commutative rings.

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18

Rana, Julie. "A boundary divisor in the moduli spaces of stable quintic surfaces." International Journal of Mathematics 28, no. 04 (April 2017): 1750021. http://dx.doi.org/10.1142/s0129167x17500215.

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We give a bound on which singularities may appear on Kollár–Shepherd-Barron–Alexeev stable surfaces for a wide range of topological invariants and use this result to describe all stable numerical quintic surfaces (KSBA-stable surfaces with [Formula: see text]) whose unique non-Du Val singularity is a Wahl singularity. We then extend the deformation theory of Horikawa to the log setting in order to describe the boundary divisor of the moduli space [Formula: see text] corresponding to these surfaces. Quintic surfaces are the simplest examples of surfaces of general type and the question of describing their moduli is a long-standing question in algebraic geometry.

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19

Nikulin, Viacheslav V. "A remark on algebraic surfaces with polyhedral Mori cone." Nagoya Mathematical Journal 157 (2000): 73–92. http://dx.doi.org/10.1017/s0027763000007194.

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We denote by FPMC the class of all non-singular projective algebraic surfaces X over ℂ with finite polyhedral Mori cone NE(X) ⊂ NS(X) ⊗ ℝ. If ρ(X) = rk NS(X) ≥ 3, then the set Exc(X) of all exceptional curves on X ∈ FPMC is finite and generates NE(X). Let δE(X) be the maximum of (-C2) and pE(X) the maximum of pa(C) respectively for all C ∈ Exc(X). For fixed ρ ≥ 3, δE and pE we denote by FPMCρ,δE,pE the class of all algebraic surfaces X ∈ FPMC such that ρ(X) = ρ, δE(X) = δE and pE(X) = pE. We prove that the class FPMCρ,δE,pE is bounded in the following sense: for any X ∈ FPMCρ,δE,pE there exist an ample effective divisor h and a very ample divisor h′ such that h2 ≤ N(ρ, δE) and h′2 ≤ N′(ρ, δE, pE) where the constants N(ρ, δE) and N′(ρ, δE, pE) depend only on ρ, δE and ρ, δE, pE respectively.One can consider Theory of surfaces X ∈ FPMC as Algebraic Geometry analog of the Theory of arithmetic reflection groups in hyperbolic spaces.

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20

Dusan Vallo, Lucia Rumanová, and Veronika Bočková. "Elements of Algorithmic Thinking in the Teaching of School Geometry through the Application of Geometric Problems." International Journal of Emerging Technologies in Learning (iJET) 18, no. 14 (July 31, 2023): 229–43. http://dx.doi.org/10.3991/ijet.v18i14.40341.

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Algorithmic thinking and the creation of algorithms have traditionally been associated with mathematics. It is based on the general perception of an algorithm as a logically unambiguous and precise prescription for performing a certain set of operations, through which we reach a result in real time in a finite number of steps. There are well-known examples from history, such as the division algorithm used by ancient Babylonian mathematicians, Eratosthenes algorithm for finding prime numbers, Euclid’s algorithm for finding the greatest common divisor of two numbers, and cryptographic algorithm for coding and breaking, invented by Arabic mathematicians in the 9th century. Although the usage of algorithms and the development of algorithmic thinking currently fall within the domain of computer science, algorithms still play a role in mathematics and its teaching today. Contemporary mathematics, and especially its teaching in schools of all grades, prefers specific algorithms in arithmetic, algebra, and calculus. For example, operations with numbers, modifications of algebraic expressions, and derivation of functions. Teaching geometry in schools involves solving a variety of problems, many of which are presented as word problems. Algorithmization of school geometric tasks is therefore hardly visible and possible at first glance. However, there are ways to solve examples of a certain kind and to establish a characteristic and common algorithmic procedure for them. Algorithmic thinking in geometry and the application of algorithms in the teaching of thematic parts of school geometry are specific issue that we deal with in this study. We will focus on a detailed analysis of the possibilities of developing algorithmic thinking in school geometry and the algorithmization of geometric tasks.

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21

Lesch, Matthias, Henri Moscovici, and Markus J. Pflaum. "Relative pairing in cyclic cohomology and divisor flows." Journal of K-Theory 3, no. 2 (February 11, 2008): 359–407. http://dx.doi.org/10.1017/is008001021jkt051.

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AbstractWe construct invariants of relative K-theory classes of multiparameter dependent pseudodifferential operators, which recover and generalize Melrose's divisor flow and its higher odd-dimensional versions of Lesch and Pflaum. These higher divisor flows are obtained by means of pairing the relative K-theory modulo the symbols with the cyclic cohomological characters of relative cycles constructed out of the regularized operator trace together with its symbolic boundary. Besides giving a clear and conceptual explanation to the essential features of the divisor flows, namely homotopy invariance, additivity and integrality, this construction allows to uncover the previously unknown even-dimensional counterparts. Furthermore, it confers to the totality of these invariants a purely topological interpretation, that of implementing the classical Bott periodicity isomorphisms in a manner compatible with the suspension isomorphisms in both K-theory and in cyclic cohomology. We also give a precise formulation, in terms of a natural Clifford algebraic suspension, for the relationship between the higher divisor flows and the spectral flow.

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22

Pereira, Jorge Vitório. "Fibrations, divisors and transcendental leaves." Journal of Algebraic Geometry 15, no. 1 (January 1, 2006): 87–110. http://dx.doi.org/10.1090/s1056-3911-05-00417-0.

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23

Huerta, César Lozano, and Tim Ryan. "On the birational geometry of Hilbert schemes of points and Severi divisors." Communications in Algebra 48, no. 11 (June 16, 2020): 4596–614. http://dx.doi.org/10.1080/00927872.2020.1767119.

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24

Chang, M. C., and Z. Ran. "Divisors on some generic hypersurfaces." Journal of Differential Geometry 38, no. 3 (1993): 671–78. http://dx.doi.org/10.4310/jdg/1214454486.

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25

ANDERSON, DAVE. "EFFECTIVE DIVISORS ON BOTT–SAMELSON VARIETIES." Transformation Groups 24, no. 3 (September 8, 2018): 691–711. http://dx.doi.org/10.1007/s00031-018-9493-6.

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26

Khesin, Boris, and Alexei Rosly. "Polar linkings, intersections and Weil pairing." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 461, no. 2063 (September 9, 2005): 3505–24. http://dx.doi.org/10.1098/rspa.2005.1498.

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Polar homology and linkings arise as natural holomorphic analogues in algebraic geometry of the homology groups and links in topology. For complex projective manifolds, the polar k -chains are subvarieties of complex dimension k with meromorphic forms on them, while the boundary operator is defined by taking the polar divisor and the Poincaré residue on it. We also define the corresponding analogues for the intersection and linking numbers of complex submanifolds, and show that they have properties similar to those of the corresponding topological objects. Finally, we establish the relation between the holomorphic linking and the Weil pairing of functions on a complex curve and its higher-dimensional counterparts.

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27

Buchweitz, Ragnar-Olaf, Wolfgang Ebeling, and Hans-Christian Graf von Bothmer. "Low-dimensional singularities with free divisors as discriminants." Journal of Algebraic Geometry 18, no. 2 (May 1, 2009): 371–406. http://dx.doi.org/10.1090/s1056-3911-08-00508-0.

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28

Yoshikawa, Ken-ichi. "Discriminant of theta divisors and Quillen metrics." Journal of Differential Geometry 52, no. 1 (1999): 73–115. http://dx.doi.org/10.4310/jdg/1214425217.

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29

NGUYEN, QUANG MINH. "VECTOR BUNDLES, DUALITIES AND CLASSICAL GEOMETRY ON A CURVE OF GENUS TWO." International Journal of Mathematics 18, no. 05 (May 2007): 535–58. http://dx.doi.org/10.1142/s0129167x07004230.

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Let C be a curve of genus two. We denote by [Formula: see text] the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over C, and by Jd the variety of line bundles of degree d on C. In particular, J1 has a canonical theta divisor Θ. The space [Formula: see text] is a double cover of ℙ8 = |3Θ| branched along a sextic hypersurface, the Coble sextic. In the dual [Formula: see text], where J1 is embedded, there is a unique cubic hypersurface singular along J1, the Coble cubic. We prove that these two hypersurfaces are dual, inducing a non-abelian Torelli result. Moreover, by looking at some special linear sections of these hypersurfaces, we can observe and reinterpret some classical results of algebraic geometry in a context of vector bundles: the duality of the Segre–Igusa quartic with the Segre cubic, the symmetric configuration of 15 lines and 15 points, the Weddle quartic surface and the Kummer surface.

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30

Pandharipande, R. "Descendent bounds for effective divisors on $\overline {M}_{g}$." Journal of Algebraic Geometry 21, no. 2 (January 3, 2011): 299–303. http://dx.doi.org/10.1090/s1056-3911-2010-00554-1.

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31

Fedorchuk, Maksym, and David Ishii Smyth. "Ample divisors on moduli spaces of pointed rational curves." Journal of Algebraic Geometry 20, no. 4 (2011): 599–629. http://dx.doi.org/10.1090/s1056-3911-2011-00547-x.

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32

Whang, Junho Peter. "Global geometry on moduli of local systems for surfaces with boundary." Compositio Mathematica 156, no. 8 (August 2020): 1517–59. http://dx.doi.org/10.1112/s0010437x20007241.

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AbstractWe show that every coarse moduli space, parametrizing complex special linear rank-2 local systems with fixed boundary traces on a surface with nonempty boundary, is log Calabi–Yau in that it has a normal projective compactification with trivial log canonical divisor. We connect this to a novel symmetry of generating series for counts of essential multicurves on the surface.

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33

Boucksom, Sébastien, Charles Favre, and Mattias Jonsson. "Differentiability of volumes of divisors and a problem of Teissier." Journal of Algebraic Geometry 18, no. 2 (May 1, 2009): 279–308. http://dx.doi.org/10.1090/s1056-3911-08-00490-6.

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34

Méo, Michel. "A Dual of the Chow Transformation." Complex Manifolds 5, no. 1 (September 1, 2018): 158–94. http://dx.doi.org/10.1515/coma-2018-0011.

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AbstractWe define a dual of the Chow transformation of currents on the complex projective space. This transformation factorizes a left inverse of the Chow transformation and its composition with the Chow transformation is a right inverse of a linear diferential operator. In such a way we complete the general scheme of integral geometry for the Chow transformation. On another hand we prove the existence of a well defined closed positive conormal current associated to every closed positive current on the projective space. This is a consequence of the existence of a dual current, defined on the dual projective space. This allows us to extend to the case of a closed positive current the known inversion formula for the conormal of the Chow divisor of an effective algebraic cycle.

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35

Farkas, Gavril. "Rational maps between moduli spaces of curves and Gieseker-Petri divisors." Journal of Algebraic Geometry 19, no. 2 (May 1, 2010): 243–84. http://dx.doi.org/10.1090/s1056-3911-09-00510-4.

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36

Coppens, Marc. "Generalized inflection points of very general effective divisors on smooth curves." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 52, no. 1 (March 30, 2011): 125–32. http://dx.doi.org/10.1007/s13366-011-0016-z.

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37

Erdenberger, C., S. Grushevsky, and K. Hulek. "Some intersection numbers of divisors on toroidal compactifications of $\mathcal {A}_g$." Journal of Algebraic Geometry 19, no. 1 (January 1, 2010): 99–132. http://dx.doi.org/10.1090/s1056-3911-09-00512-8.

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38

Bertram, Aaron. "Moduli of rank-2 vector bundles, theta divisors, and the geometry of curves in projective space." Journal of Differential Geometry 35, no. 2 (1992): 429–69. http://dx.doi.org/10.4310/jdg/1214448083.

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39

Ccapa, Javier, and Ricardo L. Soto. "On elementary divisors perturbation of nonnegative matrices." Linear Algebra and its Applications 432, no. 2-3 (January 2010): 546–55. http://dx.doi.org/10.1016/j.laa.2009.09.001.

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40

LAKSHMIBAI, V., and R. SINGH. "COTANGENT BUNDLES OF PARTIAL FLAG VARIETIES AND CONORMAL VARIETIES OF THEIR SCHUBERT DIVISORS." Transformation Groups 25, no. 1 (April 11, 2019): 127–48. http://dx.doi.org/10.1007/s00031-019-09523-w.

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41

Arizmendi, H., and V. Müller. "On algebras without generalized topological divisors of zero." Linear Algebra and its Applications 223-224 (July 1995): 65–71. http://dx.doi.org/10.1016/0024-3795(94)00355-h.

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42

Pérez-Díaz, Sonia, and Li-Yong Shen. "The μ-Basis of Improper Rational Parametric Surface and Its Application." Mathematics 9, no. 6 (March 17, 2021): 640. http://dx.doi.org/10.3390/math9060640.

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The μ-basis is a newly developed algebraic tool in curve and surface representations and it is used to analyze some essential geometric properties of curves and surfaces. However, the theoretical frame of μ-bases is still developing, especially of surfaces. We study the μ-basis of a rational surface V defined parametrically by P(t¯),t¯=(t1,t2) not being necessarily proper (or invertible). For applications using the μ-basis, an inversion formula for a given proper parametrization P(t¯) is obtained. In addition, the degree of the rational map ϕP associated with any P(t¯) is computed. If P(t¯) is improper, we give some partial results in finding a proper reparametrization of V. Finally, the implicitization formula is derived from P (not being necessarily proper). The discussions only need to compute the greatest common divisors and univariate resultants of polynomials constructed from the μ-basis. Examples are given to illustrate the computational processes of the presented results.

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43

Farkas, Gavril, and Mihnea Popa. "Effective divisors on $\overline{\mathcal{M}}_g$, curves on $K3$ surfaces, and the slope conjecture." Journal of Algebraic Geometry 14, no. 2 (May 1, 2005): 241–67. http://dx.doi.org/10.1090/s1056-3911-04-00392-3.

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44

Afkhami, Mojgan, Kazem Khashyarmanesh, and Khosro Nafar. "Zero divisor graph of a lattice with respect to an ideal." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 56, no. 1 (October 22, 2013): 217–25. http://dx.doi.org/10.1007/s13366-013-0172-4.

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45

Ensenbach, Marc. "Determinantal divisors of products of matrices over Dedekind domains." Linear Algebra and its Applications 432, no. 11 (June 2010): 2739–44. http://dx.doi.org/10.1016/j.laa.2009.12.039.

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46

Griggs, Jerrold R., and Chuanzhong Zhu. "Applications of the symmetric chain decomposition of the lattice of divisors." Order 11, no. 1 (March 1994): 41–46. http://dx.doi.org/10.1007/bf01462228.

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47

Fuhrmann, P. A., and U. Helmke. "On the elementary divisors of the Sylvester and Lyapunov maps." Linear Algebra and its Applications 432, no. 10 (May 2010): 2572–88. http://dx.doi.org/10.1016/j.laa.2009.12.003.

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48

Merca, Mircea. "Families of Ramanujan-Type Congruences Modulo 4 for the Number of Divisors." Axioms 11, no. 7 (July 18, 2022): 342. http://dx.doi.org/10.3390/axioms11070342.

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In this paper, we explore Ramanujan-type congruences modulo 4 for the function σ0(n), counting the positive divisors of n. We consider relations of the form σ08(αn+β)+r≡0(mod4), with (α,β)∈N2 and r∈{1,3,5,7}. In this context, some conjectures are made and some Ramanujan-type congruences involving overpartitions are obtained.

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Acharyya, Amrita, Sudip Kumar Acharyya, Sagarmoy Bag, and Joshua Sack. "Intermediate rings of complex-valued continuous functions." Applied General Topology 22, no. 1 (April 1, 2021): 47. http://dx.doi.org/10.4995/agt.2021.13165.

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<p>For a completely regular Hausdorff topological space X, let C(X, C) be the ring of complex-valued continuous functions on X, let C ∗ (X, C) be its subring of bounded functions, and let Σ(X, C) denote the collection of all the rings that lie between C ∗ (X, C) and C(X, C). We show that there is a natural correlation between the absolutely convex ideals/ prime ideals/maximal ideals/z-ideals/z ◦ -ideals in the rings P(X, C) in Σ(X, C) and in their real-valued counterparts P(X, C) ∩ C(X). These correlations culminate to the fact that the structure space of any such P(X, C) is βX. For any ideal I in C(X, C), we observe that C ∗ (X, C)+I is a member of Σ(X, C), which is further isomorphic to a ring of the type C(Y, C). Incidentally these are the only C-type intermediate rings in Σ(X, C) if and only if X is pseudocompact. We show that for any maximal ideal M in C(X, C), C(X, C)/M is an algebraically closed field, which is furthermore the algebraic closure of C(X)/M ∩C(X). We give a necessary and sufficient condition for the ideal CP (X, C) of C(X, C), which consists of all those functions whose support lie on an ideal P of closed sets in X, to be a prime ideal, and we examine a few special cases thereafter. At the end of the article, we find estimates for a few standard parameters concerning the zero-divisor graphs of a P(X, C) in Σ(X, C).</p>

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Hou, Xiang-Dong. "Elementary divisors of tensor products and p-ranks of binomial matrices." Linear Algebra and its Applications 374 (November 2003): 255–74. http://dx.doi.org/10.1016/s0024-3795(03)00576-7.

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