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Journal articles on the topic "Divisors (Algebraic Geometry)"
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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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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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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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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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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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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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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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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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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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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.
Dissertations / Theses on the topic "Divisors (Algebraic Geometry)"
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Staal, Andrew Philippe. "On the existence of jet schemes logarithmic along families of divisors." Thesis, University of British Columbia, 2008. http://hdl.handle.net/2429/3327.
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A section of the total tangent space of a scheme X of finite type over a field k, i.e. a vector field on X, corresponds to an X-valued 1-jet on X. In the language of jets the notion of a vector field becomes functorial, and the total tangent space constitutes one of an infinite family of jet schemes Jm(X) for m ≥ 0. We prove that there exist families of “logarithmic” jet schemes JDm(X) for m ≥ 0, in the category of k-schemes of finite type, associated to any given X and its family of divisors D = (D₁, . . . ,Dr). The sections of JD₁(X) correspond to so-called vector fields on X with logarithmic poles along the family of divisors D = (D₁, . . . ,Dr). To prove this, we first introduce the categories of pairs (X,D) where D is as mentioned, an r-tuple of (effective Cartier) divisors on the scheme X. The categories of pairs provide a convenient framework for working with only those jets that pull back families of divisors.Science, Faculty of
Mathematics, Department of
Graduate
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Staal, Andrew Phillipe. "On the existence of jet schemes logarithmic along families of divisors." Thesis, University of British Columbia, 2008. http://hdl.handle.net/2429/3327.
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A section of the total tangent space of a scheme X of finite type over a field k, i.e. a vector field on X, corresponds to an X-valued 1-jet on X. In the language of jets the notion of a vector field becomes functorial, and the total tangent space constitutes one of an infinite family of jet schemes Jm(X) for m ≥ 0. We prove that there exist families of “logarithmic” jet schemes JDm(X) for m ≥ 0, in the category of k-schemes of finite type, associated to any given X and its family of divisors D = (D₁, . . . ,Dr). The sections of JD₁(X) correspond to so-called vector fields on X with logarithmic poles along the family of divisors D = (D₁, . . . ,Dr). To prove this, we first introduce the categories of pairs (X,D) where D is as mentioned, an r-tuple of (effective Cartier) divisors on the scheme X. The categories of pairs provide a convenient framework for working with only those jets that pull back families of divisors.
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Porto, Anderson Corrêa. "Divisores sobre curvas e o Teorema de Riemann-Roch." Universidade Federal de Juiz de Fora (UFJF), 2018. https://repositorio.ufjf.br/jspui/handle/ufjf/6612.
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O objetivo desse trabalho é o estudo de conceitos básicos da Geometria Algébrica sob o ponto de vista clássico. O foco central do trabalho é o estudo do Teorema de Riemann- Roch e algumas de suas aplicações. Esse teorema constitui uma importante ferramenta no estudo da Geometria Algébrica clássica uma vez que possibilita, por exemplo, o cáculo do gênero de uma curva projetiva não singular no espaço projetivo de dimensão dois. Para o desenvolvimento do estudo do Teorema de Riemann-Roch e suas aplicações serão estudados conceitos tais como: variedades, dimensão, diferenciais de Weil, divisores, divisores sobre curvas e o anel topológico Adèle.
The goal of this work is the study of basic concepts of Algebraic Geometry from the classical point of view. The central focus of the paper is the study of Riemann-Roch Theorem and some of its applications. This theorem constitutes an important tool in the study of classical Algebraic Geometry since it allows, for example, the calculation of the genus of a non-singular projective curve in the projective space of dimension two. For the development of the study of the Riemann-Roch Theorem and its applications we will study concepts such as: varieties, dimension, Weil differentials, divisors, divisors on curves and the Adèle topological ring.
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Bruns, Gregor. "Divisors on moduli spaces of level curves." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2017. http://dx.doi.org/10.18452/17674.
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In dieser Arbeit untersuchen wir drei Fragestellungen. Zwei beschäftigen sich mit Divisoren auf Modulräumen von Kurven mit Levelstruktur, die dritte handelt von Stabilitätseigenschaften der Normalenbündel von kanonischen Kurven. Die erste Frage, die in Kapitel 2 studiert wird, beschäftigt sich mit der Kodairadimension des Modulraums R15,2 von Prym-Varietäten vom Geschlecht 15. Wir studieren einen neuen Divisor auf diesem Modulraum und berechnen seine Klasse in der Standardbasis der Picardgruppe. Mit Hilfe dieser Klasse können wir schlussfolgern, dass R15,2 von allgemeinem Typ ist. In Kapitel 3 setzen wir unsere Untersuchung von Kurven mit Levelstruktur fort und untersuchen für jede Primzahl l Theta-Divisoren auf den Modulräumen R6,l und R8,l. Die Divisoren werden mit Hilfe der Mukai-Bündel von Kurven vom Geschlecht 6 beziehungsweise 8 definiert. Diese Bündel liefern kanonische Einbettungen unserer Kurven in Grassmann-Varietäten und beschreiben fundamentale geometrische Aspekte von Kurven dieser Geschlechter. Indem wir die Klasse des Divisors für g = 8 und l = 3 berechnen, können wir zeigen, dass R8,3 ebenfalls von allgemeinem Typ ist. Schließlich studieren wir in Kapitel 4 die Stabilität des Normalenbündels kanonischer Kurven vom Geschlecht 8 und beweisen, dass das Bündel auf einer generischen Kurve stabil ist. Für kanonische Kurven vom Geschlecht 9 beweisen wir die Stabilität zumindest im Bezug auf Unterbündel von niedrigem Rang. Ebenfalls liefern wir zusätzliche Hinweise für die Vermutung von M. Aprodu, G. Farkas und A. Ortega, die besagt, dass eine generische kanonische Kurve jedes Geschlechts g >= 7 ein stabiles Normalenbündel besitzt.In this thesis we investigate three questions. Two are about divisors on moduli spaces of level curves, and about the consequences for the birational geometry of these spaces. The third asks about the stability properties of normal bundles of canonical curves. The first question, to be studied in Chapter 2, is about the Kodaira dimension of the moduli space R15,2 of Prym varieties of genus 15. We study a new divisor on this space and calculate its class in terms of the standard basis of the Picard group. This allows us to conclude that R15,2 is of general type. Continuing the study of level curves in Chapter 3, we investigate, for every l, theta divisors on R6,l and R8,l defined in terms of the Mukai bundle of genus 6 and 8 curves, respectively. These bundles provide canonical embeddings of our curves in Grassmann varieties and describe fundamental aspects of the geometry of curves of these genera. Using the class of the divisor for g = 8 and l = 3, we are able to prove that R8,3 is of general type as well. Finally, in Chapter 4 we study the stability of the normal bundle of canonical genus 8 curves and prove that on a general curve the bundle is stable. For canonical genus 9 curves we prove stability at least with respect to subbundles of low ranks. We also provide some more evidence for the conjecture of M. Aprodu, G. Farkas, and A. Ortega that a a general canonical curve of every genus g >= 7 has stable normal bundle.
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Müller, Fabian. "Effective divisors on moduli spaces of pointed stable curves." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2013. http://dx.doi.org/10.18452/16866.
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Diese Arbeit untersucht verschiedene Fragen hinsichtlich der birationalen Geometrie der Modulräume $\Mbar_g$ und $\Mbar_{g,n}$, mit besonderem Augenmerk auf der Berechnung effektiver Divisorklassen. In Kapitel 2 definieren wir für jedes $n$-Tupel ganzer Zahlen $\d$, die sich zu $g-1$ summieren, einen geometrisch bedeutsamen Divisor auf $\Mbar_{g,n}$, der durch Zurückziehen des Thetadivisors einer universellen Jacobi-Varietät mittels einer Abel-Jacobi-Abbildung erhalten wird. Er ist eine Verallgemeinerung verschiedener in der Literatur verwendeten Arten von Divisoren. Wir berechnen die Klasse dieses Divisors und zeigen, dass er für bestimmte $\d$ irreduzibel und extremal im effektiven Kegel von $\Mbar_{g,n}$ ist. Kapitel 3 beschäftigt sich mit einem birationalen Modell $X_6$ von $\Mbar_6$, das durch quadrische Hyperebenenschnitte auf der del-Pezzo-Fläche vom Grad $5$ erhalten wird. Wir berechnen die Klasse des großen Divisors, der die birationale Abbildung $\Mbar_6 \dashrightarrow X_6$ induziert, und erhalten so eine obere Schranke an die bewegliche Steigung von $\Mbar_6$. Wir zeigen, dass $X_6$ der letzte nicht-triviale Raum im log-minimalen Modellprogramm für $\Mbar_6$ ist. Weiterhin geben wir einige Resultate bezüglich der Unirationalität der Weierstraßorte auf $\Mbar_{g,1}$. Für $g = 6$ hängen diese mit der del-Pezzo-Konstruktion zusammen, die benutzt wurde, um das Modell $X_6$ zu konstruieren. Kapitel 4 konzentriert sich auf den Fall $g = 0$. Castravet and Tevelev führten auf $\Mbar_{0,n}$ kombinatorisch definierte Hyperbaumdivisoren ein, die für $n = 6$ zusammen mit den Randdivisoren den effektiven Kegel erzeugen. Wir berechnen die Klasse des Hyperbaumdivisors auf $\Mbar_{0,7}$, der bis auf Permutation der markierten Punkte eindeutig ist. Wir geben eine geometrische Charakterisierung für ihn an, die zu der von Keel und Vermeire für den Fall $n = 6$ gegebenen analog ist.This thesis investigates various questions concerning the birational geometry of the moduli spaces $\Mbar_g$ and $\Mbar_{g,n}$, with a focus on the computation of effective divisor classes. In Chapter 2 we define, for any $n$-tuple $\d$ of integers summing up to $g-1$, a geometrically meaningful divisor on $\Mbar_{g,n}$ that is essentially the pullback of the theta divisor on a universal Jacobian variety under an Abel-Jacobi map. It is a generalization of various kinds of divisors used in the literature, for example by Logan to show that $\Mbar_{g,n}$ is of general type for all $g \geq 4$ as soon as $n$ is big enough. We compute the class of this divisor and show that for certain choices of $\d$ it is irreducible and extremal in the effective cone of $\Mbar_{g,n}$. Chapter 3 deals with a birational model $X_6$ of $\Mbar_6$ that is obtained by taking quadric hyperplane sections of the degree $5$ del Pezzo surface. We compute the class of the big divisor inducing the birational map $\Mbar_6 \dashrightarrow X_6$ and use it to derive an upper bound on the moving slope of $\Mbar_6$. Furthermore we show that $X_6$ is the final non-trivial space in the log minimal model program for $\Mbar_6$. We also give a few results on the unirationality of Weierstraß loci on $\Mbar_{g,1}$, which for $g = 6$ are related to the del Pezzo construction used to construct the model $X_6$. Finally, Chapter 4 focuses on the case $g = 0$. Castravet and Tevelev introduced combinatorially defined hypertree divisors on $\Mbar_{0,n}$ that for $n = 6$ generate the effective cone together with boundary divisors. We compute the class of the hypertree divisor on $\Mbar_{0,7}$, which is unique up to permutation of the marked points. We also give a geometric characterization of it that is analogous to the one given by Keel and Vermeire in the $n = 6$ case.
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Backman, Spencer Christopher Foster. "Combinatorial divisor theory for graphs." Diss., Georgia Institute of Technology, 2014. http://hdl.handle.net/1853/51908.
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Chip-firing is a deceptively simple game played on the vertices of a graph, which was independently discovered in probability theory, poset theory, graph theory, and statistical physics. In recent years, chip-firing has been employed in the development of a theory of divisors on graphs analogous to the classical theory for Riemann surfaces. In particular, Baker and Norin were able to use this set up to prove a combinatorial Riemann-Roch formula, whose classical counterpart is one of the cornerstones of modern algebraic geometry. It is now understood that the relationship between divisor theory for graphs and algebraic curves goes beyond pure analogy, and the primary operation for making this connection precise is tropicalization, a certain type of degeneration which allows us to treat graphs as “combinatorial shadows” of curves. The development of this tropical relationship between graphs and algebraic curves has allowed for beautiful applications of chip-firing to both algebraic geometry and number theory. In this thesis we continue the combinatorial development of divisor theory for graphs. In Chapter 1 we give an overview of the history of chip-firing and its connections to algebraic geometry. In Chapter 2 we describe a reinterpretation of chip-firing in the language of partial graph orientations and apply this setup to give a new proof of the Riemann-Roch formula. We introduce and investigate transfinite chip-firing, and chip-firing with respect to open covers in Chapters 3 and 4 respectively. Chapter 5 represents joint work with Arash Asadi, where we investigate Riemann-Roch theory for directed graphs and arithmetical graphs, the latter of which are a special class of balanced vertex weighted graphs arising naturally in arithmetic geometry.
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Arruda, Rafael Lucas de [UNESP]. "Teorema de Riemann-Roch e aplicações." Universidade Estadual Paulista (UNESP), 2011. http://hdl.handle.net/11449/86493.
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O objetivo principal deste trabalho é estudar o Teorema de Riemann-Roch, um dos resultados fundamentais na teoria de curvas algébricas, e apresentar algumas de suas aplicações. Este teorema é uma importante ferramenta para a classificação das curvas algébricas, pois relaciona propriedades algébricas e topológicas. Daremos uma descrição das curvas algébricas de gênero g, 1≤ g ≤ 5, e faremos um breve estudo dos pontos de inflexão de um sistema linear sobre uma curva algébrica
The main purpose of this work is to discuss The Riemann-Roch Theorem, wich is one of the most important results of the theory algebraic curves, and to present some applications. This theorem is an important tool of the classification of algebraic curves, sinces relates algebraic and topological properties. We will describle the algebraic curves of genus g, 1≤ g ≤ 5, and also study inflection points of a linear system on an algebraic curve
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Arruda, Rafael Lucas de. "Teorema de Riemann-Roch e aplicações /." São José do Rio Preto : [s.n.], 2011. http://hdl.handle.net/11449/86493.
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Orientador: Parham SalehyanBanca: Eduardo de Sequeira Esteves
Banca: Jéfferson Luiz Rocha Bastos
Resumo: O objetivo principal deste trabalho é estudar o Teorema de Riemann-Roch, um dos resultados fundamentais na teoria de curvas algébricas, e apresentar algumas de suas aplicações. Este teorema é uma importante ferramenta para a classificação das curvas algébricas, pois relaciona propriedades algébricas e topológicas. Daremos uma descrição das curvas algébricas de gênero g, 1≤ g ≤ 5, e faremos um breve estudo dos pontos de inflexão de um sistema linear sobre uma curva algébrica
Abstract: The main purpose of this work is to discuss The Riemann-Roch Theorem, wich is one of the most important results of the theory algebraic curves, and to present some applications. This theorem is an important tool of the classification of algebraic curves, sinces relates algebraic and topological properties. We will describle the algebraic curves of genus g, 1≤ g ≤ 5, and also study inflection points of a linear system on an algebraic curve
Mestre
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Rajeev, B. "Riemann Roch Theorem For Algebraic Curves." Thesis, 2005. https://etd.iisc.ac.in/handle/2005/1448.
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Rajeev, B. "Riemann Roch Theorem For Algebraic Curves." Thesis, 2005. http://etd.iisc.ernet.in/handle/2005/1448.
Full textBooks on the topic "Divisors (Algebraic Geometry)"
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Marubayashi, Hidetoshi. Prime Divisors and Noncommutative Valuation Theory. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012.
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Milnor, John W. Dynamical systems (1984-2012). Edited by Bonifant Araceli 1963-. Providence, Rhode Island: American Mathematical Society, 2014.
Find full textBook chapters on the topic "Divisors (Algebraic Geometry)"
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Shafarevich, Igor R. "Divisors and Differential Forms." In Basic Algebraic Geometry 1, 151–222. Berlin, Heidelberg: Springer Berlin Heidelberg, 1994. http://dx.doi.org/10.1007/978-3-642-57908-0_3.
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Shafarevich, Igor R. "Divisors and Differential Forms." In Basic Algebraic Geometry 1, 147–232. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-37956-7_3.
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Falb, Peter. "Projective Algebraic Geometry VII: Divisors." In Methods of Algebraic Geometry in Control Theory: Part II, 259–70. Boston, MA: Birkhäuser Boston, 1999. http://dx.doi.org/10.1007/978-1-4612-1564-6_18.
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Falb, Peter. "Projective Algebraic Geometry VII: Divisors." In Methods of Algebraic Geometry in Control Theory: Part II, 259–70. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-96574-1_17.
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van Lint, Jacobus H., and Gerard van der Geer. "Divisors on algebraic curves." In Introduction to Coding Theory and Algebraic Geometry, 45–54. Basel: Birkhäuser Basel, 1988. http://dx.doi.org/10.1007/978-3-0348-9286-5_11.
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Diaz, Steven, and Joe Harris. "Geometry of severi varieties, II: Independence of divisor classes and examples." In Algebraic Geometry Sundance 1986, 23–50. Berlin, Heidelberg: Springer Berlin Heidelberg, 1988. http://dx.doi.org/10.1007/bfb0082907.
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"Valuations and Divisors." In Algebraic Geometry for Associative Algebras, 153–202. CRC Press, 2000. http://dx.doi.org/10.1201/9781482270525-10.
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Farkas, Gavril. "Effective Divisors on Hurwitz Spaces." In Facets of Algebraic Geometry, 221–40. Cambridge University Press, 2022. http://dx.doi.org/10.1017/9781108877831.009.
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"Frobenius Split Anticanonical Divisors." In Integrable Systems and Algebraic Geometry, 92–101. Cambridge University Press, 2020. http://dx.doi.org/10.1017/9781108773355.004.
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Morey, S., and W. Vasconcelos. "Special Divisors of Blowup Algebras." In Ring Theory And Algebraic Geometry. CRC Press, 2001. http://dx.doi.org/10.1201/9780203907962.ch16.
Full textConference papers on the topic "Divisors (Algebraic Geometry)"
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Benger, Werner. "Illustrating Geometric Algebra and Differential Geometry in 5D Color Space." In WSCG 2023 – 31. International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision. University of West Bohemia, Czech Republic, 2023. http://dx.doi.org/10.24132/csrn.3301.1.
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Geometric Algebra (GA) is popular for its immediate geometric interpretations of algebraic objects and operations. It is based on Clifford Algebra on vector spaces and extends linear algebra of vectors by operations such as an invertible product, i.e. divisions by vectors. This formalism allows for a complete algebra on vectors same as for scalar or complex numbers. It is particularly suitable for rotations in arbitrary dimensions. In Euclidean 3D space quaternions are known to be numerically superior to rotation matrices and already widely used in computer graphics. However, their meaning beyond its numerical formalism often remains mysterious. GA allows for an intuitive interpretation in terms of planes of rotations and extends this concept to arbitrary dimensions by embedding vectors into a higher dimensional, but still intuitively graspable space of multi-vectors. However, out intuition of more than three spatial dimensions is deficient. The space of colors forms a vector space as well, though one of non-spatial nature, but spun by the primary colors red, green, blue. The GA formalism can be applied here as well, amalgamating surprisingly with the notion of vectors and co-vectors known from differential geometry: tangential vectors on a manifold correspond to additive colors red/green/blue, whereas co-vectors from the co-tangential space correspond to subtractive primary colors magenta, yellow, cyan. GA in turn considers vectors, bi-vectors and anti-vectors as part of its generalized multi-vector zoo of algebraic objects. In 3D space vectors, anti-vectors, bi-vectors and covectors are all three-dimensional objects that can be identified with each other, so their distinction is concealed. Confusions arise from notions such as “normal vectors” vs. “axial vectors”. Higher dimensional spaces exhibit the differences more clearly. Using colors instead of spatial dimensions we can expand our intuition by considering "transparency" as an independent, four-dimensional property of a color vector. We can thereby explore 4D GA alternatively to spacetime in special/general relativity. However, even in 4D possibly confusing ambiguities remain between vectors, co-vectors, bi-vectors and bi-co-vectors: bi-vectors and bi-co-vectors - both six-dimensional objects - are visually equivalent. They become unequivocal only in five or higher dimensions. Envisioning five-dimensional geometry is even more challenging to the human mind, but in color space we can add another property, "texture" to constitute a five-dimensional vector space. The properties of a bi-vector and a bi-co-vector becomes evident there: We can still study all possible combinations of colors/transparency/texture visually. This higher-dimensional yet intuitive approach demonstrates the need to distinguish among different kinds of vectors before identifying them in special situations, which also clarifies the meanings of algebraic objects in 3D Euclidean space and allows for better formulations of algorithms in 3D.
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Hettiger, Christof. "Applied Structural Simulation in Railcar Design." In 2017 Joint Rail Conference. American Society of Mechanical Engineers, 2017. http://dx.doi.org/10.1115/jrc2017-2330.
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Fifty years ago, the railcar industry relied entirely on classical analysis methods using fundamental solid mechanics theory to establish design and manufacturing protocols. While this method produced working designs, the assumptions required by this type of analysis often led to overdesigned railcars. In the 1950s, the generalized mathematical approach of Finite Element Analysis (FEA) was developed to model the structural behaviors of mechanical systems. FEA involves creating a numerical model by discretizing a continuous system into a finite system of grid divisions. Each grid division, or element, has an inherent geometric shape and each element is comprised of points which are referred to as nodes. The connected pattern of nodes and elements is called a mesh. A solver organizes the mesh into a matrix of differential equations and computes the displacements using linear algebraic operations from which strains and stresses are obtained. The rapid development of computing technology provided the catalyst to drive FEA from research into industry. FEA is currently the standard approach for improving product design cycle times that were previously achieved by trial and error. Moreover, simulation has improved design efficiency allowing for greater advances in weight, strength, and material optimization. While FEA had its roots planted in the aerospace industry, competitive market conditions have driven simulation into many other professional fields of engineering. For the last few decades, FEA has become essential to the submittal of new railcar designs for unrestricted interchange service across North America. All new railcar designs must be compliant to a list of structural requirements mandated by the Association of American Railroads (AAR), which are listed in its MSRP (Manual of Standards and Recommended Practices) in addition to recommended practices in Finite Element (FE) modeling procedures. The MSRP recognizes that these guidelines are not always feasible to completely simulate, allowing the analyst to justify situations where deviations are necessary. Benefits notwithstanding, FEA has inherent challenges. It is understood that FEA does not provide exact solutions, only approximations. While FEA can provide meaningful insight into actual physical behavior leading to shorter development times and lower costs, it can also create bogus solutions that lead to potential safety and engineering risks. Regardless of how appropriate the FEA assumptions may be, engineering judgment is required to interpret the accuracy and significance of the results. A constant balance is made between model fidelity and computational solve time. The purpose of this paper is to discuss the FEA approach to railcar analysis that is used by BNSF Logistics, LLC (BNSFL) in creating AAR compliant railcar designs. Additionally, this paper will discuss the challenges inherent to FEA using experiences from actual case studies in the railcar industry. These challenges originate from assumptions that are made for the analysis including element types, part connections, and constraint locations for the model. All FEA terminology discussed in this paper is written from the perspective of an ANSYS Mechanical user. Closing remarks will be given about where current advances in FEA technology may be able to further improve railcar industry standards.