×

zbMATH — the first resource for mathematics

Khovanskii bases, higher rank valuations, and tropical geometry. (English) Zbl 1423.13145

MSC:
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14T05 Tropical geometry (MSC2010)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
13A18 Valuations and their generalizations for commutative rings
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] V. Alexeev and M. Brion, Toric degenerations of spherical varieties, Selecta Math. (N.S.), 10 (2004), pp. 453–478. · Zbl 1078.14075
[2] D. Anderson, A. Küronya, and V. Lozovanu, Okounkov bodies of finitely generated divisors, Int. Math. Res. Not. IMRN, 9 (2014), pp. 2343–2355. · Zbl 1316.14013
[3] D. Anderson, Okounkov bodies and toric degenerations, Math. Ann., 356 (2013), pp. 1183–1202. · Zbl 1273.14104
[4] T. Boege, A. D’Alì, T. Kahle, and B. Sturmfels, The Geometry of Gaussoids, preprint, https://arxiv.org/abs/1710.07175, 2017.
[5] L. Bossinger, X. Fang, G. Fourier, M. Hering, and M. Lanini, Toric Degenerations of Gr(2,n) and Gr(3,6) via Plabic Graphs, preprint, https://arxiv.org/abs/1612.03838, 2016. · Zbl 06948249
[6] W. Bruns and J. Gubeladze, Polytopes, Rings, and \(K\)-Theory, Springer Monogr. Math., Springer, Dordrecht, 2009.
[7] M. Baker, S. Payne, and J. Rabinoff, On the structure of non-Archimedean analytic curves, in Tropical and Non-Archimedean Geometry, Contemp. Math. 605, AMS, Providence, RI, 2013, pp. 93–121. · Zbl 1320.14040
[8] M. Brion, The total coordinate ring of a wonderful variety, J. Algebra, 313 (2007), pp. 61–99. · Zbl 1123.14024
[9] P. Caldero, Toric degenerations of Schubert varieties, Transform. Groups, 7 (2002), pp. 51–60. · Zbl 1050.14040
[10] D. A. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, 4th ed., Undergrad. Texts Math., Springer, Cham, 2015.
[11] M. A. Cueto, Implicitization of Surfaces via Geometric Tropicalization, preprint, https://arxiv.org/abs/1105.0509, 2011.
[12] T. Duff and F. Sottile, Certification for Polynomial Systems via Square Subsystems, preprint, https://arxiv.org/abs/1812.02851, 2018.
[13] D. Eisenbud, Commutative Algebra. With a View toward Algebraic Geometry, Grad. Texts in Math. 150, Springer-Verlag, New York, 1995.
[14] E. Feigin, G. Fourier, and P. Littelmann, Favourable modules: Filtrations, polytopes, Newton-Okounkov bodies and flat degenerations, Transform. Groups, 22 (2017), pp. 321–352. · Zbl 06793800
[15] T. Foster and D. Ranganathan, Hahn analytification and connectivity of higher rank tropical varieties, Manuscripta Math., 151 (2016), pp. 353–374. · Zbl 1379.14031
[16] M. Gross, P. Hacking, S. Keel, and M. Kontsevich, Canonical bases for cluster algebras, J. Amer. Math. Soc., 31 (2018), pp. 497–608.
[17] M. Göbel, Computing bases for rings of permutation-invariant polynomials, J. Symbolic Comput., 19 (1995), pp. 285–291. · Zbl 0832.13006
[18] W. M. Goldman, An Exposition of Results of Fricke and Vogt, preprint, https://arxiv.org/abs/math/0402103v1, 2004.
[19] K. A. H. Gravett, Ordered abelian groups, Quart. J. Math. Oxford Ser. (2), 7 (1956), pp. 57–63.
[20] W. Gubler, J. Rabinoff, and A. Werner, Skeletons and tropicalizations, Adv. Math., 294 (2016), pp. 150–215.
[21] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, Ann. of Math. (2), 79 (1964), pp. 109–203. · Zbl 0122.38603
[22] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. II, Ann. of Math. (2), 79 (1964), pp. 205–326. · Zbl 0122.38603
[23] P. Hacking, S. Keel, and J. Tevelev, Compactification of the moduli space of hyperplane arrangements, J. Algebraic Geom., 15 (2006), pp. 657–680. · Zbl 1117.14036
[24] C. Huneke and I. Swanson, Integral Closure of Ideals, Rings, and Modules, London Math. Soc. Lecture Note Ser. 336, Cambridge University Press, Cambridge, UK, 2006.
[25] B. Huber, F. Sottile, and B. Sturmfels, Numerical Schubert calculus: Symbolic numeric algebra for polynomials, J. Symbolic Comput., 26 (1998), pp. 767–788. · Zbl 1064.14508
[26] N. Ilten and M. Wrobel, Khovanskii-Finite Valuations, Rational Curves, and Torus Actions, preprint, https://arxiv.org/abs/1807.08780, 2018.
[27] A. Jensen, Algorithmic Aspects of Gröbner Fans and Tropical Varieties, Ph.D. thesis, University of Aarhus, 2007.
[28] K. Kaveh, SAGBI bases and degeneration of spherical varieties to toric varieties, Michigan Math. J., 53 (2005), pp. 109–121. · Zbl 1076.14071
[29] K. Kaveh, Crystal bases and Newton-Okounkov bodies, Duke Math. J., 164 (2015), pp. 2461–2506.
[30] K. Kaveh and A. G. Khovanskii, Mixed volume and an extension of intersection theory of divisors, Mosc. Math. J., 10 (2010), pp. 343–375, 479. · Zbl 1287.14001
[31] K. Kaveh and A. G. Khovanskii, Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. (2), 176 (2012), pp. 925–978. · Zbl 1270.14022
[32] K. Kaveh and A. Khovanskii, Algebraic equations and convex bodies, in Perspectives in Analysis, Geometry, and Topology, Progr. Math. 296, Birkhäuser/Springer, New York, 2012, pp. 263–282. · Zbl 1316.52011
[33] K. Kaveh, C. Manon, and T. Murata, On degenerations of projective varieties to complexity-one T-varieties, in preparation.
[34] J. Kollár, Lectures on Resolution of Singularities, Ann. of Math. Stud. 166, Princeton University Press, Princeton, NJ, 2007.
[35] E. Katz and S. Urbinati, Newton–Okounkov bodies over discrete valuation rings and linear systems on graphs, Internat. Math. Res. Not., 2018, rnx248, https://doi.org/10.1093/imrn/rnx248.
[36] R. Lazarsfeld and M. Mustaţua, Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4), 42 (2009), pp. 783–835. · Zbl 1182.14004
[37] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc., 3 (1990), pp. 447–498. · Zbl 0703.17008
[38] I. Makhlin, Gelfand–Tsetlin Degenerations of Representations and Flag Varieties, preprint, https://arxiv.org/abs/1809.02258, 2018.
[39] C. Manon, Newton-Okounkov polyhedra for character varieties and configuration spaces, Trans. Amer. Math. Soc., 368 (2016), pp. 5979–6003. · Zbl 1366.14046
[40] C. Manon, Toric geometry of \(SL_2(\mathbb{C})\) free group character varieties from outer space, Canad. J. Math., 70 (2018), pp. 354–399. · Zbl 1408.14164
[41] E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Grad. Texts in Math. 227, Springer-Verlag, New York, 2005.
[42] D. Maclagan and B. Sturmfels, Introduction to Tropical Geometry, Grad. Stud. Math. 161, AMS, Providence, RI, 2015.
[43] C. Manon and Z. Zhou, Semigroups of \(sl_3(\mathbb{C})\) tensor product invariants, J. Algebra, 400 (2014), pp. 94–104. · Zbl 1298.17013
[44] A. Okounkov, Why would multiplicities be log-concave?, in The Orbit Method in Geometry and Physics (Marseille, 2000), Progr. Math. 213, Birkhäuser Boston, Boston, MA, 2003, pp. 329–347.
[45] S. Payne, Analytification is the limit of all tropicalizations, Math. Res. Lett., 16 (2009), pp. 543–556. · Zbl 1193.14077
[46] V. L. Popov, Contraction of the actions of reductive algebraic groups, Sb. Math., 2 (1987), pp. 311–335.
[47] L. Robbiano and M. Sweedler, Subalgebra bases, in Commutative Algebra (Salvador, 1988), Lecture Notes in Math. 1430, Springer, Berlin, 1990, pp. 61–87.
[48] K. Rietsch and L. Williams, Newton-Okounkov Bodies, Cluster Duality, and Mirror Symmetry for Grassmannians, preprint, https://arxiv.org/abs/1712.00447, 2017.
[49] H. Seppänen, Okounkov bodies for ample line bundles with applications to multiplicities for group representations, Beitr. Algebra Geom., 57 (2016), pp. 735–749.
[50] D. Speyer and B. Sturmfels, The tropical Grassmannian, Adv. Geom., 4 (2004), pp. 389–411.
[51] B. Sturmfels and J. Tevelev, Elimination theory for tropical varieties, Math. Res. Lett., 15 (2008), pp. 543–562. · Zbl 1157.14038
[52] B. Sturmfels, Gröbner Bases and Convex Polytopes, Univ. Lecture Ser. 8, AMS, Providence, RI, 1996.
[53] B. Sturmfels and Z. Xu, SAGBI bases of Cox-Nagata rings, J. Eur. Math. Soc. (JEMS), 12 (2010), pp. 429–459. · Zbl 1202.14053
[54] B. Teissier, Valuations, deformations, and toric geometry, in Valuation Theory and Its Applications, Vol. II (Saskatoon, SK, 1999), Fields Inst. Commun. 33, AMS, Providence, RI, 2003, pp. 361–459. · Zbl 1061.14016
[55] J. Tevelev, Compactifications of subvarieties of tori, Amer. J. Math., 129 (2007), pp. 1087–1104. · Zbl 1154.14039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.