## Kyoto Journal of Mathematics

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

#### 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
Revised: 17 March 2016
Accepted: 23 March 2016
First available in Project Euclid: 9 May 2017

https://projecteuclid.org/euclid.kjm/1494295224

Digital Object Identifier
doi:10.1215/21562261-3821846

Mathematical Reviews number (MathSciNet)
MR3648055

Zentralblatt MATH identifier
06736607

#### 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

#### 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.