The length classification of threefold flops via noncommutative algebras. (English) Zbl 1411.14010
Starting in the late 1980’s, techniques of (commutative) algebraic geometry have been used to study noncommutative algebra. See for example [M. Artin, Lond. Math. Soc. Lect. Note Ser. 238, 1–19 (1997; Zbl 0888.16025)] for an introduction to these ideas. More recently, noncommutative algebra has been used to study algebraic geometry. For instance, the paper under review uses quivers to refine a classification of threefold flops.
A flop is a type of birational transformation. In the seminal paper [J. Kollár, Nagoya Math. J. 113, 15–36 (1989; Zbl 0645.14004)], it was shown that if \(X\) and \(Y\) are threefolds, with minor singularities and nef canonical bundles, then any birational transformation between \(X\) and \(Y\) can be factored into flops. Thus flops are an important part of the minimal model program for threefolds. In [S. Katz and D. R. Morrison, J. Algebraic Geom. 1, No. 3, 449–530 (1992; Zbl 0788.14036)], flops were classified by a length invariant. The length \(\ell\) can take on the integer values 1–6, and for each \(\ell\) there exists a universal flop of length \(\ell\). Any flop of length \(\ell\) can then be constructed from this universal flop, at least in theory.
From a practical standpoint, these constructions are difficult. The paper under review simplifies these calculations using noncommutative algebra. A universal flopping algebra of length \(\ell\) is introduced, realized as the path algebra of certain quivers. The main theorem (Theorem 1.3) gives explicit, compact descriptions of the quivers involved.
The construction works for all possible lengths 1–6. Previously, [C. Curto and D. R. Morrison, J. Algebr. Geom. 22, No. 4, 599–627 (2013; Zbl 1360.14053)] used matrix factorization to study flops, with theorems for lengths 1 and 2, and conjectures for higher lengths. The current paper recovers those theorems and calculates factorizations that had been conjectured for lengths 3–6. Contraction algebras, introduced by W. Donovan and M. Wemyss [Duke Math. J. 165, No. 8, 1397–1474 (2016; Zbl 1346.14031)], are also calculated.
Several examples of universal flops and universal flopping algebras are also given, as well as noncommutative crepant resolutions [M. van den Bergh, in: The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3–8, 2002. Berlin: Springer. 749–770 (2004; Zbl 1082.14005)]. An appendix gives Magma code for calculating examples that would be too unwieldy to include in the paper.
This paper is a nice example of noncommutative algebra giving back to algebraic geometry. Such cross-pollination will undoubtedly continue.
A flop is a type of birational transformation. In the seminal paper [J. Kollár, Nagoya Math. J. 113, 15–36 (1989; Zbl 0645.14004)], it was shown that if \(X\) and \(Y\) are threefolds, with minor singularities and nef canonical bundles, then any birational transformation between \(X\) and \(Y\) can be factored into flops. Thus flops are an important part of the minimal model program for threefolds. In [S. Katz and D. R. Morrison, J. Algebraic Geom. 1, No. 3, 449–530 (1992; Zbl 0788.14036)], flops were classified by a length invariant. The length \(\ell\) can take on the integer values 1–6, and for each \(\ell\) there exists a universal flop of length \(\ell\). Any flop of length \(\ell\) can then be constructed from this universal flop, at least in theory.
From a practical standpoint, these constructions are difficult. The paper under review simplifies these calculations using noncommutative algebra. A universal flopping algebra of length \(\ell\) is introduced, realized as the path algebra of certain quivers. The main theorem (Theorem 1.3) gives explicit, compact descriptions of the quivers involved.
The construction works for all possible lengths 1–6. Previously, [C. Curto and D. R. Morrison, J. Algebr. Geom. 22, No. 4, 599–627 (2013; Zbl 1360.14053)] used matrix factorization to study flops, with theorems for lengths 1 and 2, and conjectures for higher lengths. The current paper recovers those theorems and calculates factorizations that had been conjectured for lengths 3–6. Contraction algebras, introduced by W. Donovan and M. Wemyss [Duke Math. J. 165, No. 8, 1397–1474 (2016; Zbl 1346.14031)], are also calculated.
Several examples of universal flops and universal flopping algebras are also given, as well as noncommutative crepant resolutions [M. van den Bergh, in: The legacy of Niels Henrik Abel. Papers from the Abel bicentennial conference, University of Oslo, Oslo, Norway, June 3–8, 2002. Berlin: Springer. 749–770 (2004; Zbl 1082.14005)]. An appendix gives Magma code for calculating examples that would be too unwieldy to include in the paper.
This paper is a nice example of noncommutative algebra giving back to algebraic geometry. Such cross-pollination will undoubtedly continue.
Reviewer: Dennis Keeler (Oxford, OH)
MSC:
| 14B07 | Deformations of singularities |
| 14E30 | Minimal model program (Mori theory, extremal rays) |
| 16S38 | Rings arising from noncommutative algebraic geometry |
| 18E30 | Derived categories, triangulated categories (MSC2010) |
Keywords:
noncommutative algebra; minimal model program; threefold flopping contractions; simultaneous resolution of singularities; Kleinian singularities; contraction algebrasCitations:
Zbl 0888.16025; Zbl 0645.14004; Zbl 0788.14036; Zbl 1360.14053; Zbl 1346.14031; Zbl 1082.14005References:
| [1] | Ando, T., Some examples of simple small singularities, Comm. Algebra, 41, 6, 2193-2204 (2013) · Zbl 1286.14050 |
| [2] | Artin, M.; Verdier, J.-L., Reflexive modules over rational double points, Math. Ann., 270, 1, 79-82 (1985) · Zbl 0553.14001 |
| [3] | Aspinwall, P. S.; Morrison, D. R., Quivers from matrix factorizations, Comm. Math. Phys., 313, 3, 607-633 (2012) · Zbl 1250.81076 |
| [4] | Bodzenta, A.; Bondal, A., Flops and spherical functors (November 2015) |
| [5] | Bosma, W.; Cannon, J.; Playoust, C., The Magma algebra system. I. The user language, Computational Algebra and Number Theory. Computational Algebra and Number Theory, London, 1993. Computational Algebra and Number Theory. Computational Algebra and Number Theory, London, 1993, J. Symbolic Comput., 24, 3-4, 235-265 (1997) · Zbl 0898.68039 |
| [6] | Bridgeland, T., Flops and derived categories, Invent. Math., 147, 3, 613-632 (2002) · Zbl 1085.14017 |
| [7] | Brieskorn, E., Über die Auflösung gewisser Singularitäten von holomorphen Abbildungen, Math. Ann., 166, 76-102 (1966) · Zbl 0145.09402 |
| [8] | Brieskorn, E., Die Auflösung der rationalen Singularitäten holomorpher Abbildungen, Math. Ann., 178, 255-270 (1968) · Zbl 0159.37703 |
| [9] | Brieskorn, E., Singular elements of semi-simple algebraic groups, Actes, Congrès Intern. Math., 2, 279-284 (1970) · Zbl 0223.22012 |
| [10] | Brown, G.; Wemyss, M., Gopakumar-Vafa invariants do not determine flops, Comm. Math. Phys., 361, 143 (2018) · Zbl 1423.14311 |
| [11] | Buan, A. B.; Iyama, O.; Reiten, I.; Smith, D., Mutation of cluster-tilting objects and potentials, Amer. J. Math., 133, 4, 835-887 (2011) · Zbl 1285.16012 |
| [12] | Butler, M. C.R.; King, A. D., Minimal resolutions of algebras, J. Algebra, 212, 1, 323-362 (1999) · Zbl 0926.16006 |
| [13] | Cafuta, K.; Klep, I.; Povh, J., NCSOStools: a computer algebra system for symbolic and numerical computation with noncommutative polynomials, Optim. Methods Softw., 26, 3, 363-380 (2011) · Zbl 1226.90063 |
| [14] | Clemens, H.; Kollár, J.; Mori, S., Higher-dimensional complex geometry, Astérisque, 166, Article 1989 pp. (1988), 144 pp. · Zbl 0689.14016 |
| [15] | Crawley-Boevey, W.; Holland, M. P., Noncommutative deformations of Kleinian singularities, Duke Math. J., 92, 3, 605-635 (1998) · Zbl 0974.16007 |
| [16] | Curto, C., Matrix model superpotentials and ADE singularities, Adv. Theor. Math. Phys., 12, 2, 353-404 (2008) · Zbl 1144.81500 |
| [17] | Curto, C.; Morrison, D. R., Threefold flops via matrix factorization, J. Algebraic Geom., 22, 4, 599-627 (2013) · Zbl 1360.14053 |
| [18] | Donovan, W.; Wemyss, M., Noncommutative deformations and flops, Duke Math. J., 165, 8, 1397-1474 (2016) · Zbl 1346.14031 |
| [19] | Eisenbud, D., Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc., 260, 1, 35-64 (1980) · Zbl 0444.13006 |
| [20] | Hua, Z., On a conjecture of Donovan and Wemyss (October 2016) |
| [21] | Hua, Z.; Toda, Y., Contraction algebra and invariants of singularities, Int. Math. Res. Not., 10, 3173-3198 (2017) · Zbl 1423.14074 |
| [22] | Ito, Y.; Nakamura, I., McKay correspondence and Hilbert schemes, Proc. Japan Acad. Ser. A Math. Sci., 72, 7, 135-138 (1996) · Zbl 0881.14002 |
| [23] | Kalck, M.; Iyama, O.; Wemyss, M.; Yang, D., Frobenius categories, Gorenstein algebras and rational surface singularities, Compos. Math., 151, 3, 502-534 (2015) · Zbl 1327.14172 |
| [24] | Kapranov, M.; Vasserot, E., Kleinian singularities, derived categories and Hall algebras, Math. Ann., 316, 3, 565-576 (2000) · Zbl 0997.14001 |
| [25] | Karmazyn, J., Quiver GIT for varieties with tilting bundles, Manuscripta Math., 154, 1-2, 91-128 (2017) · Zbl 1401.14198 |
| [26] | Katz, S.; Morrison, D. R., Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups, J. Algebraic Geom., 1, 3, 449-530 (1992) · Zbl 0788.14036 |
| [27] | Kawamata, Y., General hyperplane sections of nonsingular flops in dimension 3, Math. Res. Lett., 1, 1, 49-52 (1994) · Zbl 0834.32007 |
| [28] | King, A. D., Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2), 45, 180, 515-530 (1994) · Zbl 0837.16005 |
| [29] | Kollár, J., Flops, Nagoya Math. J., 113, 15-36 (1989) · Zbl 0645.14004 |
| [30] | Laufer, H. B., On \(C P^1\) as an exceptional set, (Recent Developments in Several Complex Variables. Recent Developments in Several Complex Variables, Proc. Conf., Princeton Univ., Princeton, N.J., 1979. Recent Developments in Several Complex Variables. Recent Developments in Several Complex Variables, Proc. Conf., Princeton Univ., Princeton, N.J., 1979, Ann. of Math. Stud., vol. 100 (1981), Princeton Univ. Press: Princeton Univ. Press Princeton, N.J.), 261-275 · Zbl 0523.32007 |
| [31] | Morrison, D. R., The birational geometry of surfaces with rational double points, Math. Ann., 271, 3, 415-438 (1985) · Zbl 0539.14008 |
| [32] | Namikawa, Y., Flops and Poisson deformations of symplectic varieties, Publ. Res. Inst. Math. Sci., 44, 2, 259-314 (2008) · Zbl 1148.14008 |
| [33] | Nolla de Celis, Á.; Sekiya, Y., Flops and mutations for crepant resolutions of polyhedral singularities, Asian J. Math., 21, 1, 1-45 (2017) · Zbl 1368.14024 |
| [34] | Pinkham, H., Deformations of normal surface singularities with \(C^\ast\) action, Math. Ann., 232, 1, 65-84 (1978) · Zbl 0351.14004 |
| [35] | Pinkham, H. C., Factorization of birational maps in dimension 3, (Singularities, Part 2. Singularities, Part 2, Arcata, Calif., 1981. Singularities, Part 2. Singularities, Part 2, Arcata, Calif., 1981, Proc. Sympos. Pure Math., vol. 40 (1983), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 343-371 · Zbl 0544.14005 |
| [36] | Reid, M., Minimal models of canonical 3-folds, (Algebraic Varieties and Analytic Varieties. Algebraic Varieties and Analytic Varieties, Tokyo, 1981. Algebraic Varieties and Analytic Varieties. Algebraic Varieties and Analytic Varieties, Tokyo, 1981, Adv. Stud. Pure Math., vol. 1 (1983), North-Holland: North-Holland Amsterdam), 131-180 · Zbl 0558.14028 |
| [37] | Slodowy, P., Simple Singularities and Simple Algebraic Groups, Lecture Notes in Mathematics, vol. 815 (1980), Springer: Springer Berlin · Zbl 0441.14002 |
| [38] | Tjurina, G. N., Resolution of singularities of flat deformations of double rational points, Funkc. Anal. Prilož., 4, 1, 77-83 (1970) · Zbl 0221.32008 |
| [39] | Toda, Y., Non-commutative width and Gopakumar-Vafa invariants, Manuscripta Math., 148, 3-4, 521-533 (2015) · Zbl 1348.14040 |
| [40] | Van den Bergh, M., Non-commutative crepant resolutions, (The Legacy of Niels Henrik Abel (2004), Springer: Springer Berlin), 749-770 · Zbl 1082.14005 |
| [41] | Van den Bergh, M., Three-dimensional flops and noncommutative rings, Duke Math. J., 122, 3, 423-455 (2004) · Zbl 1074.14013 |
| [42] | Wemyss, M., The \(GL(2, C)\) McKay correspondence, Math. Ann., 350, 3, 631-659 (2011) · Zbl 1233.14012 |
| [43] | Wemyss, M., Flops and clusters in the homological minimal model program, Invent. Math. (2017) |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
