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Journal articles on the topic 'Toric algebra'

Author: Grafiati

Published: 4 June 2021

Last updated: 4 February 2022

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1

Herzog, Jürgen, Raheleh Jafari, and Abbas Nasrollah Nejad. "On the Gauss algebra of toric algebras." Journal of Algebraic Combinatorics 51, no. 1 (January 2, 2019): 1–17. http://dx.doi.org/10.1007/s10801-018-0865-8.

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2

Petrović, Sonja, and Despina Stasi. "Toric algebra of hypergraphs." Journal of Algebraic Combinatorics 39, no. 1 (April 20, 2013): 187–208. http://dx.doi.org/10.1007/s10801-013-0444-y.

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3

Petrović, Sonja, Apostolos Thoma, and Marius Vladoiu. "Bouquet algebra of toric ideals." Journal of Algebra 512 (October 2018): 493–525. http://dx.doi.org/10.1016/j.jalgebra.2018.05.016.

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4

Chirivì, Rocco. "On some properties of LS algebras." Communications in Contemporary Mathematics 22, no. 02 (December 3, 2018): 1850085. http://dx.doi.org/10.1142/s0219199718500852.

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The discrete LS algebra over a totally ordered set is the homogeneous coordinate ring of an irreducible projective (normal) toric variety. We prove that this algebra is the ring of invariants of a finite abelian group containing no pseudo-reflection acting on a polynomial ring. This is used to study the Gorenstein property for LS algebras. Further we show that any LS algebra is Koszul.

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5

Kim, Jin Hong. "On the integral cohomology of toric varieties." Journal of Algebra and Its Applications 15, no. 02 (October 6, 2015): 1650032. http://dx.doi.org/10.1142/s0219498816500328.

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It is known that the integral cohomology algebra of any smooth compact toric variety XΣ associated to a complete regular fan Σ is isomorphic to the Stanley–Reisner algebra ℤ[Σ] modulo the ideal JΣ generated by linear relations determined by Σ. The aim of this paper is to show how to determine the integral cohomology algebra of a toric variety (in particular, a projective toric variety) associated to a certain simplicial fan. As a consequence, we confirm our expectation that for a certain simplicial fan the integral cohomology algebra is also given by the same formula as in a complete regular fan.

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6

Katsabekis, Anargyros, and Apostolos Thoma. "Toric sets and orbits on toric varieties." Journal of Pure and Applied Algebra 181, no. 1 (June 2003): 75–83. http://dx.doi.org/10.1016/s0022-4049(02)00305-5.

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7

Kaveh, Kiumars. "Vector Fields and the Cohomology Ring of Toric Varieties." Canadian Mathematical Bulletin 48, no. 3 (September 1, 2005): 414–27. http://dx.doi.org/10.4153/cmb-2005-039-1.

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AbstractLetXbe a smooth complex projective variety with a holomorphic vector field with isolated zero setZ. From the results of Carrell and Lieberman there exists a filtrationF0⊂F1⊂ · · · ofA(Z), the ring of ℂ-valued functions onZ, such thatas graded algebras. In this note, for a smooth projective toric variety and a vector field generated by the action of a 1-parameter subgroup of the torus, we work out this filtration. Our main result is an explicit connection between this filtration and the polytope algebra ofX.

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8

Geiger, Dan, Christopher Meek, and Bernd Sturmfels. "On the toric algebra of graphical models." Annals of Statistics 34, no. 3 (June 2006): 1463–92. http://dx.doi.org/10.1214/009053606000000263.

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9

Klyachko, A. A. "Toric bundles and problems of linear algebra." Functional Analysis and Its Applications 23, no. 2 (1989): 135–37. http://dx.doi.org/10.1007/bf01078785.

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10

Callegaro, Filippo, and Emanuele Delucchi. "The integer cohomology algebra of toric arrangements." Advances in Mathematics 313 (June 2017): 746–802. http://dx.doi.org/10.1016/j.aim.2017.04.017.

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11

De Loera, J. A., D. Haws, R. Hemmecke, P. Huggins, B. Sturmfels, and R. Yoshida. "Short rational functions for toric algebra and applications." Journal of Symbolic Computation 38, no. 2 (August 2004): 959–73. http://dx.doi.org/10.1016/j.jsc.2004.02.001.

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12

Soret, Marc, and Marina Ville. "Lissajous-toric knots." Journal of Knot Theory and Its Ramifications 29, no. 01 (January 2020): 2050003. http://dx.doi.org/10.1142/s0218216520500030.

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A point in the [Formula: see text]-torus knot in [Formula: see text] goes [Formula: see text] times along a vertical circle while this circle rotates [Formula: see text] times around the vertical axis. In the Lissajous-toric knot [Formula: see text], the point goes along a vertical Lissajous curve (parametrized by [Formula: see text] while this curve rotates [Formula: see text] times around the vertical axis. Such a knot has a natural braid representation [Formula: see text] which we investigate here. If [Formula: see text], [Formula: see text] is ribbon; if [Formula: see text], [Formula: see text] is the [Formula: see text]th power of a braid which closes in a ribbon knot. We give an upper bound for the [Formula: see text]-genus of [Formula: see text] in the spirit of the genus of torus knots; we also give examples of [Formula: see text]’s which are trivial knots.

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13

Thompson, Howard M. "Toric singularities revisited." Journal of Algebra 299, no. 2 (May 2006): 503–34. http://dx.doi.org/10.1016/j.jalgebra.2005.05.029.

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14

Sullivant, Seth. "Toric fiber products." Journal of Algebra 316, no. 2 (October 2007): 560–77. http://dx.doi.org/10.1016/j.jalgebra.2006.10.004.

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15

Altmann, Klaus, and Frederik Witt. "Toric co-Higgs sheaves." Journal of Pure and Applied Algebra 225, no. 8 (August 2021): 106634. http://dx.doi.org/10.1016/j.jpaa.2020.106634.

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16

Ichim, Bogdan, and Tim Römer. "On toric face rings." Journal of Pure and Applied Algebra 210, no. 1 (July 2007): 249–66. http://dx.doi.org/10.1016/j.jpaa.2006.09.010.

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17

Borisov, A. A., and L. A. Borisov. "SINGULAR TORIC FANO VARIETIES." Russian Academy of Sciences. Sbornik Mathematics 75, no. 1 (February 28, 1993): 277–83. http://dx.doi.org/10.1070/sm1993v075n01abeh003385.

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18

Ishii, Shihoko. "ON TORIC IMAGE DIVISORS." Communications in Algebra 29, no. 3 (February 28, 2001): 1165–73. http://dx.doi.org/10.1081/agb-100001674.

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19

Birkar, C., and Y. Chen. "Singularities on toric fibrations." Sbornik: Mathematics 212, no. 3 (March 1, 2021): 288–304. http://dx.doi.org/10.1070/sm9446.

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20

Favacchio, Giuseppe, Johannes Hofscheier, Graham Keiper, and Adam Van Tuyl. "Splittings of toric ideals." Journal of Algebra 574 (May 2021): 409–33. http://dx.doi.org/10.1016/j.jalgebra.2021.01.012.

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21

Blose, Alexander, Patricia Klein, Owen Mcgrath, and A. N. D. Jackson Morris. "Toric double determinantal varieties." Communications in Algebra 49, no. 7 (February 24, 2021): 3085–93. http://dx.doi.org/10.1080/00927872.2021.1887882.

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22

Ostrover, Yaron, and Ilya Tyomkin. "On the quantum homology algebra of toric Fano manifolds." Selecta Mathematica 15, no. 1 (May 16, 2009): 121–49. http://dx.doi.org/10.1007/s00029-009-0526-9.

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23

Bigatti, A. M., R. La Scala, and L. Robbiano. "Computing Toric Ideals." Journal of Symbolic Computation 27, no. 4 (April 1999): 351–65. http://dx.doi.org/10.1006/jsco.1998.0256.

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24

NAAIJKENS, PIETER. "LOCALIZED ENDOMORPHISMS IN KITAEV'S TORIC CODE ON THE PLANE." Reviews in Mathematical Physics 23, no. 04 (May 2011): 347–73. http://dx.doi.org/10.1142/s0129055x1100431x.

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We consider various aspects of Kitaev's toric code model on a plane in the C*-algebraic approach to quantum spin systems on a lattice. In particular, we show that elementary excitations of the ground state can be described by localized endomorphisms of the observable algebra. The structure of these endomorphisms is analyzed in the spirit of the Doplicher–Haag–Roberts program (specifically, through its generalization to infinite regions as considered by Buchholz and Fredenhagen). Most notably, the statistics of excitations can be calculated in this way. The excitations can equivalently be described by the representation theory of [Formula: see text], i.e. Drinfel'd's quantum double of the group algebra of ℤ2.

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25

Dasgupta, Jyoti, Bivas Khan, and Vikraman Uma. "Cohomology of torus manifold bundles." Mathematica Slovaca 69, no. 3 (June 26, 2019): 685–98. http://dx.doi.org/10.1515/ms-2017-0257.

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Abstract Let X be a 2n-dimensional torus manifold with a locally standard T ≅ (S1)n action whose orbit space is a homology polytope. Smooth complete complex toric varieties and quasitoric manifolds are examples of torus manifolds. Consider a principal T-bundle p : E → B and let π : E(X) → B be the associated torus manifold bundle. We give a presentation of the singular cohomology ring of E(X) as a H*(B)-algebra and the topological K-ring of E(X) as a K*(B)-algebra with generators and relations. These generalize the results in [17] and [19] when the base B = pt. These also extend the results in [20], obtained in the case of a smooth projective toric variety, to any smooth complete toric variety.

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26

Camerlo, Riccardo, Giovanni Pistone, and Fabio Rapallo. "Modal operators and toric ideals." Journal of Logic and Computation 29, no. 5 (February 27, 2019): 577–93. http://dx.doi.org/10.1093/logcom/exz003.

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Abstract In the present paper, we consider modal propositional logic and look for the constraints that are imposed to the propositions of the special type $\operatorname{\Box } a$ by the structure of the relevant finite Kripke frame. We translate the usual language of modal propositional logic in terms of notions of commutative algebra, namely polynomial rings, ideals and bases of ideals. We use extensively the perspective obtained in previous works in algebraic statistics. We prove that the constraints on $\operatorname{\Box } a$ can be derived through a binomial ideal containing a toric ideal and we give sufficient conditions under which the toric ideal, together with the fact that the truth values are in $\left \{0,1\right \} $, fully describes the constraints.

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27

Jones, A. G. "Rings of differential operators on toric varieties." Proceedings of the Edinburgh Mathematical Society 37, no. 1 (February 1994): 143–60. http://dx.doi.org/10.1017/s0013091500018770.

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Let be a finite dimensional toric variety over an algebraically closed field of characteristic zero, k. Let be the sheaf of differential operators on . We show that the ring of global sections, is a finitely generated Noetherian k-algebra and that its generators can be explicitly found. We prove a similar result for the sheaf of differential operators with coefficients in a line bundle.

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28

Buczynski, Jaroslaw. "Toric Legendrian Subvarieties." Transformation Groups 12, no. 4 (November 28, 2007): 631–46. http://dx.doi.org/10.1007/s00031-007-0064-5.

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29

Legendre, Eveline. "Existence and non-uniqueness of constant scalar curvature toric Sasaki metrics." Compositio Mathematica 147, no. 5 (July 27, 2011): 1613–34. http://dx.doi.org/10.1112/s0010437x1100529x.

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AbstractWe study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least five. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using an existence result of E. Legendre [Toric geometry of convex quadrilaterals, J. Symplectic Geom. 9 (2011), 343–385], we show that a co-oriented compact toric contact 5-manifold whose moment cone has four facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on S2×S3 admitting two non-isometric and non-transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.

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30

Baralic, Djordje, Jelena Grbic, Ivan Limonchenko, and Aleksandar Vucic. "Toric objects associated with the dodecahedron." Filomat 34, no. 7 (2020): 2329–56. http://dx.doi.org/10.2298/fil2007329b.

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In this paper we illustrate a tight interplay between homotopy theory and combinatorics within toric topology by explicitly calculating homotopy and combinatorial invariants of toric objects associated with the dodecahedron. In particular, we calculate the cohomology ring of the (complex and real) moment-angle manifolds over the dodecahedron, and of a certain quasitoric manifold and of a related small cover. We finish by studying Massey products in the cohomology ring of moment-angle manifolds over the dodecahedron and how the existence of nontrivial Massey products influences the behaviour of the Poincar? series of the corresponding Pontryagin algebra.

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31

Payne, Sam. "Frobenius splittings of toric varieties." Algebra & Number Theory 3, no. 1 (February 1, 2009): 107–19. http://dx.doi.org/10.2140/ant.2009.3.107.

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32

Payne, Sam. "Moduli of toric vector bundles." Compositio Mathematica 144, no. 5 (September 2008): 1199–213. http://dx.doi.org/10.1112/s0010437x08003461.

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AbstractWe give a presentation of the moduli stack of toric vector bundles with fixed equivariant total Chern class as a quotient of a fine moduli scheme of framed bundles by a linear group action. This fine moduli scheme is described explicitly as a locally closed subscheme of a product of partial flag varieties cut out by combinatorially specified rank conditions. We use this description to show that the moduli of rank three toric vector bundles satisfy Murphy’s law, in the sense of Vakil. The preliminary sections of the paper give a self-contained introduction to Klyachko’s classification of toric vector bundles.

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33

Cox, David, and Jessica Sidman. "Secant varieties of toric varieties." Journal of Pure and Applied Algebra 209, no. 3 (June 2007): 651–69. http://dx.doi.org/10.1016/j.jpaa.2006.07.008.

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34

Iwanari, Isamu. "The category of toric stacks." Compositio Mathematica 145, no. 03 (May 2009): 718–46. http://dx.doi.org/10.1112/s0010437x09003911.

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AbstractIn this paper, we show that there is an equivalence between the 2-category of smooth Deligne–Mumford stacks with torus embeddings and actions and the 1-category of stacky fans. To this end, we prove two main results. The first is related to a combinatorial aspect of the 2-category of toric algebraic stacks defined by I. Iwanari [Logarithmic geometry, minimal free resolutions and toric algebraic stacks, Preprint (2007)]; we establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second result provides a geometric characterization of toric algebraic stacks. Logarithmic geometry in the sense of Fontaine–Illusie plays a central role in obtaining our results.

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35

Gaifullin, S. A. "Affine toric SL(2)-embeddings." Sbornik: Mathematics 199, no. 3 (April 30, 2008): 319–39. http://dx.doi.org/10.1070/sm2008v199n03abeh003922.

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36

Stapledon, A. "Motivic Integration on Toric Stacks." Communications in Algebra 37, no. 11 (November 11, 2009): 3943–65. http://dx.doi.org/10.1080/00927870902828819.

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37

Campillo, Antonio, and Philippe Gimenez. "Syzygies of affine toric varieties." Journal of Algebra 225, no. 1 (March 2000): 142–61. http://dx.doi.org/10.1006/jabr.1999.8102.

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38

Blickle, Manuel. "Cartier Isomorphism for Toric Varieties." Journal of Algebra 237, no. 1 (March 2001): 342–57. http://dx.doi.org/10.1006/jabr.2000.8569.

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39

Huang, Zhizhong. "Rational approximations on toric varieties." Algebra & Number Theory 15, no. 2 (April 7, 2021): 461–512. http://dx.doi.org/10.2140/ant.2021.15.461.

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40

Elkadi, M., A. Galligo, and M. Weimann. "Towards toric absolute factorization." Journal of Symbolic Computation 44, no. 9 (September 2009): 1194–211. http://dx.doi.org/10.1016/j.jsc.2008.03.007.

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41

Verrill, Helena, and David Joyner. "Computing with toric varieties." Journal of Symbolic Computation 42, no. 5 (May 2007): 511–32. http://dx.doi.org/10.1016/j.jsc.2006.08.005.

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42

Gualdi, Roberto. "Heights of hypersurfaces in toric varieties." Algebra & Number Theory 12, no. 10 (December 31, 2018): 2403–43. http://dx.doi.org/10.2140/ant.2018.12.2403.

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43

D'Andrea, Carlos, and Amit Khetan. "Macaulay style formulas for toric residues." Compositio Mathematica 141, no. 03 (April 21, 2005): 713–28. http://dx.doi.org/10.1112/s0010437x05001326.

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44

Hering, Milena, Hal Schenck, and Gregory G. Smith. "Syzygies, multigraded regularity and toric varieties." Compositio Mathematica 142, no. 06 (November 2006): 1499–506. http://dx.doi.org/10.1112/s0010437x0600251x.

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45

Crona, Kristina. "Gröbner bases for some toric rings." Communications in Algebra 27, no. 11 (January 1999): 5711–22. http://dx.doi.org/10.1080/00927879908826785.

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46

Dueck, Pierre, Serkan Hoşten, and Bernd Sturmfels. "Normal toric ideals of low codimension." Journal of Pure and Applied Algebra 213, no. 8 (August 2009): 1636–41. http://dx.doi.org/10.1016/j.jpaa.2008.11.045.

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47

Moci, Luca, and Simona Settepanella. "The homotopy type of toric arrangements." Journal of Pure and Applied Algebra 215, no. 8 (August 2011): 1980–89. http://dx.doi.org/10.1016/j.jpaa.2010.11.008.

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48

Nødland, Bernt Ivar Utstøl. "Local Euler obstructions of toric varieties." Journal of Pure and Applied Algebra 222, no. 3 (March 2018): 508–33. http://dx.doi.org/10.1016/j.jpaa.2017.04.016.

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49

Grieve, Nathan. "Generalized GCD for toric Fano varieties." Acta Arithmetica 195, no. 4 (2020): 415–28. http://dx.doi.org/10.4064/aa190430-5-12.

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50

Altmann, Klaus. "Torsion of differentials on toric varieties." Semigroup Forum 53, no. 1 (December 1996): 89–97. http://dx.doi.org/10.1007/bf02574124.

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