Kyoto Journal of Mathematics

On 81 symplectic resolutions of a 4-dimensional quotient by a group of order 32

Maria Donten-Bury and Jarosław A. Wiśniewski

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We provide a construction of 81 symplectic resolutions of a 4-dimensional quotient singularity obtained by an action of a group of order 32. The existence of such resolutions is known by a result of Bellamy and Schedler. Our explicit construction is obtained via geometric invariant theory (GIT) quotients of the spectrum of a ring graded in the Picard group generated by the divisors associated to the conjugacy classes of symplectic reflections of the group in question. As a result we infer the geometric structure of these resolutions and their flops. Moreover, we represent the group in question as a group of automorphisms of an abelian 4-fold so that the resulting quotient has singularities with symplectic resolutions. This yields a new Kummer-type symplectic 4-fold.

Article information

Source
Kyoto J. Math., Volume 57, Number 2 (2017), 395-434.

Dates
Received: 24 December 2015
Revised: 17 March 2016
Accepted: 23 March 2016
First available in Project Euclid: 9 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1494295224

Digital Object Identifier
doi:10.1215/21562261-3821846

Mathematical Reviews number (MathSciNet)
MR3648055

Zentralblatt MATH identifier
06736607

Subjects
Primary: 14E15: Global theory and resolution of singularities [See also 14B05, 32S20, 32S45]
Secondary: 14E30: Minimal model program (Mori theory, extremal rays) 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 14L24: Geometric invariant theory [See also 13A50] 14C20: Divisors, linear systems, invertible sheaves 53C26: Hyper-Kähler and quaternionic Kähler geometry, "special" geometry

Keywords
Quotient singularity symplectic resolution Cox ring hyper-Kähler manifold

Citation

Donten-Bury, Maria; Wiśniewski, Jarosław A. On $81$ symplectic resolutions of a $4$ -dimensional quotient by a group of order $32$. Kyoto J. Math. 57 (2017), no. 2, 395--434. doi:10.1215/21562261-3821846. https://projecteuclid.org/euclid.kjm/1494295224


Export citation

References

  • [1] M. Andreatta and J. A. Wiśniewski, On the Kummer construction, Rev. Mat. Complut. 23 (2010), 191–215.
  • [2] M. Andreatta and J. A. Wiśniewski, $4$-dimensional symplectic contractions, Geom. Dedicata 168 (2014), 311–337.
  • [3] M. Artebani, J. Hausen, and A. Laface, On Cox rings of K3 surfaces, Compos. Math. 146 (2010), 964–998.
  • [4] I. V. Arzhantsev, U. Derenthal, J. Hausen, and A. Laface, Cox Rings, Cambridge Stud. Adv. Math. 144, Cambridge Univ. Press, Cambridge, 2015.
  • [5] I. V. Arzhantsev and S. A. Gaĭfullin, Cox rings, semigroups, and automorphisms of affine varieties, Mat. Sb. 201 (2010), 3–24.
  • [6] T. Bauer, M. Funke, and S. Neumann, Counting Zariski chambers on del Pezzo surfaces, J. Algebra 324 (2010), 92–101.
  • [7] A. Beauville, Symplectic singularities, Invent. Math. 139 (2000), 541–549.
  • [8] G. Bellamy, Counting resolutions of symplectic quotient singularities, Compos. Math. 152 (2016), 99–114.
  • [9] G. Bellamy and T. Schedler, A new linear quotient of $\textbf{C}^{4}$ admitting a symplectic resolution, Math. Z. 273 (2013), 753–769.
  • [10] D. J. Benson, Polynomial Invariants of Finite Groups, London Math. Soc. Lecture Note Ser. 190, Cambridge Univ. Press, Cambridge, 1993.
  • [11] F. Berchtold and J. Hausen, GIT equivalence beyond the ample cone, Michigan Math. J. 54 (2006), 483–515.
  • [12] A.-M. Castravet and J. Tevelev, Hilbert’s 14th problem and Cox rings, Compos. Math. 142 (2006), 1479–1498.
  • [13] D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995), 17–50.
  • [14] D. A. Cox, J. B. Little, and H. K. Schenck, Toric Varieties, Grad. Stud. Math. 124, Amer. Math. Soc., Providence, 2011.
  • [15] W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann, Singular Version 3-1-6, Univ. Kaiserslautern, Kaiserslautern, Germany, 2012.
  • [16] I. Dolgachev, Lectures on Invariant Theory, London Math. Soc. Lecture Note Ser. 296, Cambridge Univ. Press, Cambridge, 2003.
  • [17] M. Donten, On Kummer $3$-folds, Rev. Mat. Complut. 24 (2011), 465–492.
  • [18] M. Donten-Bury, Cox rings of minimal resolutions of surface quotient singularities, Glasg. Math. J. 58 (2016), 325–355.
  • [19] L. Facchini, V. González-Alonso, and M. Lasoń, Cox rings of Du Val singularities, Matematiche (Catania) 66 (2011), 115–136.
  • [20] W. Fulton and J. Harris, Representation Theory, Grad. Texts in Math. 129, Springer, New York, 1991.
  • [21] V. Ginzburg and D. Kaledin, Poisson deformations of symplectic quotient singularities, Adv. Math. 186 (2004), 1–57.
  • [22] D. R. Grayson and M. E. Stillman, Macaulay2, 2013, http://www.math.uiuc.edu/Macaulay2.
  • [23] D. Guan, On the Betti numbers of irreducible compact hyperkähler manifolds of complex dimension four, Math. Res. Lett. 8 (2001), 663–669.
  • [24] J. Hausen, Cox rings and combinatorics, II, Mosc. Math. J. 8 (2008), 711–757, 847.
  • [25] J. Hausen and S. Keicher, A software package for Mori dream spaces, LMS J. Comput. Math. 18 (2015), 647–659.
  • [26] J. Hausen, S. Keicher, and A. Laface, Computing Cox rings, Math. Comp. 85 (2016), 467–502.
  • [27] Y. Hu and S. Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331–348.
  • [28] Y. Ito and M. Reid, “The McKay correspondence for finite subgroups of $\mathrm{SL}(3,\mathbf{C})$,” in Higher-Dimensional Complex Varieties (Trento, 1994), de Gruyter, Berlin, 1996, 221–240.
  • [29] D. Kaledin, McKay correspondence for symplectic quotient singularities, Invent. Math. 148 (2002), 151–175.
  • [30] S. Keicher, Computing the GIT-fan, Internat. J. Algebra Comput. 22 (2012), no. 1250064.
  • [31] J. Kollár, Shafarevich maps and plurigenera of algebraic varieties, Invent. Math. 113 (1993), 177–215.
  • [32] J. Kollár, Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3) 32, Springer, Berlin, 1996.
  • [33] Y. I. Manin, Cubic Forms, 2nd ed., North-Holland Math. Libr. 4, North-Holland, Amsterdam, 1986.
  • [34] D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, 3rd ed., Ergeb. Math. Grenzgeb. (2) 34, Springer, Berlin, 1994.
  • [35] Y. Namikawa, Poisson deformations of affine symplectic varieties, II, Kyoto J. Math. 50 (2010), 727–752.
  • [36] Y. Namikawa, Poisson deformations of affine symplectic varieties, Duke Math. J. 156 (2011), 51–85.
  • [37] M. Newman, Integral Matrices, Pure Appl. Math. 45, Academic Press, New York, 1972.
  • [38] M. Reid, What is a flip?, preprint, 1992. http://homepages.warwick.ac.uk/staff/Miles.Reid/3folds.
  • [39] M. Reid, McKay correspondence, preprint, arXiv:alg-geom/9702016v3.
  • [40] R. P. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.) 1 (1979), 475–511.
  • [41] W. A. Stein, et al., Sage Mathematics Software Version 5.12, 2013. http://www.sagemath.org.
  • [42] M. Stillman, D. Testa, and M. Velasco, Gröbner bases, monomial group actions, and the Cox rings of del Pezzo surfaces, J. Algebra 316 (2007), 777–801.
  • [43] G. Temple, The group properties of Dirac’s operators, Proc. R. Soc. Lond. A 127 (1930), 339–348.
  • [44] J. Wierzba, Contractions of symplectic varieties, J. Algebraic Geom. 12 (2003), 507–534.
  • [45] J. Wierzba and J. A. Wiśniewski, Small contractions of symplectic $4$-folds, Duke Math. J. 120 (2003), 65–95.