Geodesic
Local Normal Forms of Noncommutative Functions
Forum of Mathematics, Pi, Tome 13 (2025) no. 1, p. e8

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This article describes local normal forms of functions in noncommuting variables, up to equivalence generated by isomorphism of noncommutative Jacobi algebras, extending singularity theory in the style of Arnold’s commutative local normal forms into the noncommutative realm. This generalisation unveils many new phenomena, including an ADE classification when the Jacobi ring has dimension zero and, by taking suitable limits, a further ADE classification in dimension one. These are natural generalisations of the simple singularities and those with infinite multiplicity in Arnold’s classification. We obtain normal forms away from some exceptional Type E cases. Remarkably, these normal forms have no continuous parameters, and the key new feature is that the noncommutative world affords larger families.This theory has a range of immediate consequences to the birational geometry of 3-folds. The normal forms of dimension zero are the analytic classification of smooth 3-fold flops, and one outcome of NC singularity theory is the first list of all Type D flopping germs, generalising Reid’s famous pagoda classification of Type A, with variants covering Type E. The normal forms of dimension one have further applications to divisorial contractions to a curve. In addition, the general techniques also give strong evidence towards new contractibility criteria for rational curves. 
Brown, Gavin; Wemyss, Michael. Local Normal Forms of Noncommutative Functions. Forum of Mathematics, Pi, Tome 13 (2025) no. 1, p. e8. doi: 10.1017/fmp.2025.2
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[A1] Artin, M., ‘Some numerical criteria for contractability of curves on algebraic surfaces’, Amer. J. Math. 84 (1962), 485–496. Google Scholar | DOI

[AM] Atiyah, M. F. and Macdonald, I. G., Introduction to Commutative Algebra (Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969). Google Scholar

[AB] Ardakov, K. and Brown, K. A., ‘Ring-theoretic properties of Iwasawa algebras: a survey’, Doc. Math. 2006, Extra Vol., 7–33. Google Scholar

[A2] Arnold, V. I., ‘Local normal forms of functions’, Invent. Math. 35 (1976), 87–109. Google Scholar | DOI

[A3] August, J., ‘The tilting theory of contraction algebras’, Adv. Math. 374 (2020), 107372. Google Scholar | DOI

[BCP] Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24 (1997), 235–265. http://www.magma.usyd.edu Google Scholar | DOI

[BW1] Brown, G. and Wemyss, M., ‘Gopakumar–Vafa invariants do not determine flops’, Comm. Math. Phys. 361(1) (2018), 143–154. Google Scholar | DOI

[BW2] Brown, G. and Wemyss, M., ‘Derived deformation theory of crepant curves’, J. Topol. 17(4) (2024), 42 pp. Google Scholar | DOI

[BW4] Brown, G. and Wemyss, M., ‘The classification of smooth 3-fold flops’, in preparation. Google Scholar

[BIRS] Buan, A. B., Iyama, O., Reiten, I. and Smith, D., ‘Mutation of cluster-tilting objects and potentials’, Amer. J. Math. 133(4) (2011), 835–887. Google Scholar | DOI

[BGS] Buchweitz, R.-O., Greuel, G.-M. and Schreyer, F.-O. ‘Cohen-Macaulay modules on hypersurface singularities. II’, Invent. Math. 88(1) (1987), 165–182. Google Scholar | DOI

[CB] Crawley-Boevey, W., ‘Noncommutative algebra 1’, Bielefeld, Summer Semester 2019, https://www.math.uni-bielefeld.de/ wcrawley/19noncommalg1/NA1-final.pdfpdf. Google Scholar

[CBH] Crawley-Boevey, W. and Holland, M. P., ‘Noncommutative deformations of Kleinian singularities’, Duke Math. J. 92(3) (1998), 605–635. Google Scholar | DOI

[D] Davison, B., ‘Refined invariants of flopping curves and finite-dimensional Jacobi algebras’, Algebr. Geom. 11(6) (2024), 757–795. Google Scholar | DOI

[DGPS] Decker, W., Greuel, G.-M., Pfister, G. and Schönemann, H., ‘Singular 4-1-0 — A computer algebra system for polynomial computations’, 2017, http://www.singular.uni-kl.de. Google Scholar

[DWZ] Derksen, H., Weyman, J. and Zelevinsky, A., ‘Quivers with potentials and their representations. I. Mutations’, Selecta Math. (N.S.) 14(1) (2008), 59–119. Google Scholar | DOI

[DW1] Donovan, W. and Wemyss, M., ‘Noncommutative deformations and flops’, Duke Math. J. 165(8) (2016), 1397–1474. Google Scholar | DOI

[DW2] Donovan, W. and Wemyss, M., ‘Contractions and deformations’, Amer. J. Math. 141(3) (2019), 563–592. Google Scholar | DOI

[DW3] Donovan, W. and Wemyss, M., ‘Twists and braids for general 3-fold flops’, J. Eur. Math. Soc. (JEMS) 21(6) (2019), 1641–1701. Google Scholar | DOI

[DW4] Donovan, W. and Wemyss, M., ‘Noncommutative enhancements of contractions’, Adv. Math. 344 (2019), 99–136. Google Scholar | DOI

[Du] Ducat, T., ‘Divisorial extractions from singular curves in smooth 3-folds’, Internat. J. Math. 27(1) (2016), 1650005, 23 pp. Google Scholar | DOI

[E1] Eisenbud, D., Algebra, Commutative. With a View Toward Algebraic Geometry (Graduate Texts in Mathematics) vol. 150 (Springer-Verlag, New York, 1995). Google Scholar

[E2] Ershov, M., ‘Golod–Shafarevich groups: a survey’, Internat. J. Algebra Comput. 22(5) (2012), 1230001, 68 pp. Google Scholar | DOI

[GH] Gerritzen, L. and Holtkamp, R., ‘On Gröbner bases of noncommutative power series’, Indag. Mathem., N.S. 9(4), 1998, 503–519. Google Scholar | DOI

[GS] Golod, E. S. and Šafarevič, I. R., ‘On the class field tower’, Izv. Akad. Nauk SSSR Ser. Mat. 28, (2964), 261–272. Google Scholar

[H] Hironaka, H., ‘Resolution of singularities of an algebraic variety over a field of characteristic zero: I and II’, Ann. Math. 79(1–2) (1964), 109–203 and 205–326. Google Scholar | DOI

[HT] Hua, Z. and Toda, Y., ‘Contraction algebra and invariants of singularities’, Int. Math. Res. Not. IMRN 2018(10), 3173–3198. Google Scholar | DOI

[HZ] Hua, Z., and Zhou, G., ‘Quasi-homogeneity of potentials’, J. Noncommut. Geom. 15(2) (2021), 399–422. Google Scholar | DOI

[IW1] Iyama, O. and Wemyss, M., ‘Singular derived Categories of Q -factorial terminalizations and maximal modification algebras’, Adv. Math. 261 (2014), 85–121. Google Scholar | DOI

[IW2] Iyama, O. and Wemyss, M., ‘Reduction of triangulated categories and Maximal Modification Algebras for singularities’, J. Reine Angew. Math. 738 (2018), 149–202. Google Scholar | DOI

[I] Iyudu, N., ‘Classification of contraction algebras and pre-Lie algebras associated to braces and trusses’, Preprint, 2020, . Google Scholar | arXiv

[IS1] Iyudu, N and Shkarin, S., ‘Potential algebras with few generators’, Preprint, 2018, v1. Google Scholar | arXiv

[IS2] Iyudu, N and Shkarin, S., ‘Golod-Shafarevic-Vinberg type theorems and finiteness conditions for potential algebras’, Preprint, 2022, . Google Scholar | arXiv

[ISm] Iyudu, N and Smoktunowicz, A., ‘Golod-Shafarevich-type theorems and potential algebras’, Int. Math. Res. Not. IMRN 2019(15), 4822–4844. Google Scholar | DOI

[JKM] Jasso, G., Keller, B. and Muro, F., ‘The derived Auslander-Iyama correspondence’, Preprint, 2023, . Google Scholar | arXiv

[J] Jiménez, J., ‘Contraction of nonsingular curves’, Duke Math. J. 65(2) (1992), 313–332. Google Scholar | DOI

[KM] Katz, S. and Morrison, D. R., ‘Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups’, J. Algebraic Geom. 1 (1992) 449–530. Google Scholar

[K] Karmazyn, J., ‘The length classification of threefold flops via noncommutative algebras’, Adv. Math. 343 (2019), 393–447. Google Scholar | DOI

[Ka] Kawamata, Y., ‘Non-commutative deformations of perverse coherent sheaves and rational curves’, J. Algebr. Geom. 32(1) (2023), 59–91. Google Scholar | DOI

[Ka2] Kawamata, Y., ‘On non-commutative formal deformations of coherent sheaves on an algebraic variety’, EMS Surv. Math. Sci. 8(1–2) (2021), 237–263. Google Scholar | DOI

[KMM] Kawamata, Y., Matsuda, K. and Matsuki, K., ‘Introduction to the minimal model problem’, in Algebraic Geometry (Adv. Stud. Pure Math.) vol. 10 (North-Holland, Amsterdam, 1987), 283–360. Google Scholar

[KL] Krause, G. R. and Lenagan, T. H., Growth of Algebras and Gelfand-Kirillov Dimension (Graduate Studies in Mathematics) vol. 22, revised edn. (American Mathematical Society, Providence, RI, 2000). Google Scholar

[L1] Lam, T. Y., A First Course in Noncommutative Rings (Graduate Texts in Mathematics) vol. 131, second edn. (Springer- Verlag, New York, 2001). Google Scholar | DOI

[L2] Laufer, Henry B., ‘On as an exceptional set’, in Recent Developments in Several Complex Variables (Proc. Conf., Princeton Univ., Princeton, N. J., 1979) (Ann. of Math. Stud.) vol. 100 (Princeton Univ. Press, Princeton, NJ, 1981), 261–275. Google Scholar

[M1] Matsumura, H., Commutative Ring Theory (Cambridge Studies in Advanced Mathematics) vol. 8, second edn. (Cambridge University Press, Cambridge, 1989). Google Scholar

[M2] Mellit, A., ‘Kleinian singularities and algebras generated by elements that have given spectra and satisfy a scalar sum relation’, Algebra Discrete Math. 2004(3), 89–110. Google Scholar

[M3] Mora, T., ‘An introduction to commutative and noncommutative Gröbner bases’, Theoretical Computer Science 134 (1994) 131–173. Google Scholar | DOI

[R1] Reid, M., ‘Canonical 3-folds’, in Journées de Géometrie Algébrique d’Angers, Juillet 1979 (Sijthoff & Noordhoff, Alphen aan den Rijn—Germantown, Md., 1980), 273–310. Google Scholar

[R2] Reid, M., ‘Minimal models of canonical 3-folds’, in Algebraic Varieties and Analytic Varieties (Tokyo, 1981) (Adv. Stud. Pure Math.) vol. 1 (North-Holland, Amsterdam, 1983), 131–180. Google Scholar | DOI

[T] Toda, Y., ‘Non-commutative width and Gopakumar–Vafa invariants’, Manuscripta Math. 148(3–4) (2015), 521–533. Google Scholar | DOI

[T2] Tziolas, N., ‘Terminal 3-fold divisorial contractions of a surface to a curve. I’, Compos. Math. 139(3) (2003), 239–261. Google Scholar | DOI

[T3] Tziolas, N., ‘Three-fold divisorial extremal neighborhoods over cE7 and cE6 compound DuVal singularities’, Internat. J. Math. 21(1) (2010), 1–23. Google Scholar | DOI

[V1] Van Den Bergh, M., ‘Calabi-Yau algebras and superpotentials’, Selecta Math. (N.S.) 21(2) (2015), 555–603. Google Scholar | DOI

[V2] Van Den Bergh, M., ‘Three-dimensional flops and noncommutative rings’, Duke Math. J. 122(3) (2004), 423–455. Google Scholar | DOI

[vG] Van Garderen, O., ‘Donaldson–Thomas invariants of length 2 flops’, Preprint, 2022, . Google Scholar | arXiv

[vG2] Van Garderen, O., ‘Donaldson–Thomas invariants of threefold flops’, PhD thesis, 2021. Google Scholar

[V3] Vinberg, E. B., ‘On the theorem concerning the infinite-dimensionality of an associative algebra’, Izv. Akad. Nauk SSSR Ser. Mat. 29 (1965), 209–214. Google Scholar

[Wa] Warner, S., Topological Rings (North-Holland Mathematics Studies) vol. 178 (North-Holland Publishing Co., Amsterdam, 1993). Google Scholar

[W] Wilson, P. M. H., ‘The Kähler cone on Calabi-Yau threefolds’, Invent. Math. 107(3) (1992), 561–583. Erratum, Invent. Math. (1) (1993), 231–233. Google Scholar | DOI

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