Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter May 25, 2022

Components and singularities of Quot schemes and varieties of commuting matrices

  • Joachim Jelisiejew ORCID logo EMAIL logo and Klemen Šivic ORCID logo

Abstract

We investigate the variety of commuting matrices. We classify its components for any number of matrices of size at most 7. We prove that starting from quadruples of 8×8 matrices, this scheme has generically nonreduced components, while up to degree 7 it is generically reduced. Our approach is to recast the problem as deformations of modules and generalize an array of methods: apolarity, duality and Białynicki–Birula decompositions to this setup. We include a thorough review of our methods to make the paper self-contained and accessible to both algebraic and linear-algebraic communities. Our results give the corresponding statements for the Quot schemes of points, in particular we classify the components of Quotd(𝒪𝔸nr) for d7 and all r, n.

Funding statement: Joachim Jelisiejew was supported by Polish National Science Center, project 2017/26/D/ST1/00755 and by the START fellowship of the Foundation for Polish Science. Klemen Šivic is partially supported by Slovenian Research Agency (ARRS), grant numbers N1-0103 and P1-0222.

A Functorial approach to comparison between Cn(𝕄d) and Quotrd

The bijection on points obtained in Lemma 3.4 is not enough to compare singularities of Cn(𝕄d) and Quotrd or to prove that the map 𝒰stQuotrd is a morphism. However, the same idea can be extended to give such a comparison, using the language of the functor of points [11, Section VI.1].

For a 𝕜-algebra A, an A-point of a scheme X is just a morphism Spec(A)X of schemes over Spec(𝕜). In particular, when 𝕜 is algebraically closed and X is reasonable (for example, locally of finite type), the 𝕜-points of X and closed points of X agree. The set of A-points is denoted by X(A). For a morphism of affine schemes Spec(A)Spec(B) we get a map of sets X(B)X(A). In this way X(-) becomes a functor from 𝕜-algebras to sets.

The idea of the functor of points is that for every A the set of X(A) resembles the set X(𝕜) of closed points of X, so it is easier to deal with sets of A-points for every A than with X viewed as a locally ringed topological space.

Example A.1.

What is an A-point of 𝕄d? Since 𝕄d=Spec𝕜[zij:1i,jd], an A-point of 𝕄d is a homomorphism φ:𝕜[zij]A, i.e., a matrix [φ(zij)]i,j with entries in A. Conversely, having such a matrix [aij] we can uniquely define φ by φ(zij)=aij. The conclusion is that an A-point of 𝕄d is a d×d matrix with entries in A. Similarly, since GLd=Spec𝕜[zij,Δ:1i,jd]/(Δdet[zij]=1), an A-point of GLd is a matrix with entries in A and with invertible determinant which is exactly an element of GLd(A).

Example A.2.

The scheme Cn(𝕄d) is closed in

𝕄dn=Spec𝕜[zije:1i,jd,1en]

given by the quadratic equations as explained in Section 3.1. An A-point of Cn(𝕄d) is a homomorphism

φ:𝕜[zije:1i,jd,1en]/I(Cn(𝕄d))A.

This gives an n-tuple [φ(zije)]1i,jd for e=1,2,,n of d×d matrices and the fact that φ factors through the quotient by I(Cn(𝕄d)) implies exactly that those matrices commute. This shows that an A-point of Cn(𝕄d) is just a commuting n-tuple of d×d matrices with entries in A.

For simplicity of notation, let SA:=S𝕜A, FA:=F𝕜A and VA:=V𝕜A, where F:=Sr is a free module. The Quot scheme is defined as a functor of points.

Example A.3 ([12, Chapter 5]).

We define the scheme Quotrd by declaring that for every 𝕜-algebra A the set of A-points of Quotrd is

Quotrd(A):={FAK:KFA is an SA-submodule, the A-module FAK is locally free and
for every maximal 𝔪A the (A/𝔪)-vector space F𝕜(A/𝔪)K¯
has dimension d}.

For example, Quotrd(𝕜)={F/K:KF an S-submodule,dim𝕜(F/K)=d}. An important observation is that K is a kernel of a surjection of locally free A-modules, so it is a locally free A-module as well. Warning: the SA-module FA/K is not locally free: it is not even torsion-free. For a map AA of algebras we declare that the map Quotrd(A)Quotrd(A) sends FAK to FAKAAFAKAA. This defines a functor Quotrd. It is nontrivial theorem that this functor is represented by a scheme, see [12, Chapter 5].

A technically less demanding way of proving that Quotrd is a scheme is to exhibit an open cover by affine schemes and use [11, Theorem VI.14]. We present this approach, without much detail, below.

Example A.4.

Fix d elements of the module F that are “monomials”, i.e., that have the form y1a1ynanej for some ai0, j{1,2,,r}, and denote their set by λ. Inside Quotrd we consider the locus Uλ of quotients F/K such that the image of λ is a 𝕜-linear basis of F/K. More precisely, for each A, we define Uλ(A) as the set of SA-submodules KFA such that the images of λ form an A-linear base of FA/K. This is an open condition and the resulting open subscheme UλQuotrd is affine by the argument repeating the one done for the Hilbert scheme in [41, Section 18.1]. The subschemes {Uλ}λ form an open cover of the Quot scheme. To prove representability, apply [11, Theorem VI.14].

Having discussed the existence of Quotrd, we discuss the analogues of the maps defined in Section 3.2.

Lemma A.5.

Let A be a 𝕜-algebra. The map (x1,,xn,v1,,vr)(FAkerπM,πM¯) is a bijection between the A-points of Ust and the set

{(FAK,φ):[FA/K] is an A-point of Quotrd,
φ:FA/KVA is an isomorphism of A-𝑚𝑜𝑑𝑢𝑙𝑒𝑠}.

This bijection gives rise to an isomorphism of functors.

Proof.

The proof works exactly as in the case A=𝕜 proven in Lemma 3.4. ∎

Arguing as in Lemma 3.6 but for A-points, we get a map of functors GL(V)×𝒰st𝒰st so a GL(V)-action on 𝒰st. For an algebraic group G, a (Zariski local) principal G-bundlef:PT is a morphism of schemes with G-action fiberwise that locally trivializes: for every point tT there exists an open neighborhood U of t and a G-equivariant isomorphism f-1(U)G×U of schemes over U.

Corollary A.6.

There is a morphism of schemes p:UstQuotrd defined on A-points by the formula

(x1,,xn,v1,,vr)[FAkerπM].

This map makes Ust a principal GL(V)-bundle over Quotrd.

Proof.

By Lemma A.5, the map above is a map of functors, so by Yoneda’s Lemma [11, Lemma VI.1] it gives a morphism of schemes p:𝒰stQuotrd. To prove that p is a principal GL(V)-bundle, we can argue locally on Quotrd. Choose a point of this scheme and its open neighborhood Z=Spec(B). The corresponding submodule 𝒦FB has a quotient 𝒬=FB/𝒦, which is a locally free B-module of rank d. Shrink Z so that 𝒬 becomes a free B-module and choose an isomorphism φ0:𝒬VB of B-modules. The preimage p-1(Z) is the fiber product Z×Quotrd𝒰st, so an A-point of this preimage is a morphism

j:Spec(A)Spec(B)

and an A-point of 𝒰st. By Lemma A.5, this A-point gives an SA-submodule KFA together with an isomorphism of A-modules φ:FA/KVA. The product is fibered over Quotrd which means that the submodules K and 𝒦BA of FA are equal.

Summing up, an A-point of p-1(Z) is an isomorphism of A-modules

φ:FA/(𝒦BA)VA.

Let φ¯0:𝒬BAVBBA=VA be obtained from the isomorphism φ0. Then we get an automorphism φφ¯0-1:VAVA of the A-module VA, hence an A-point of GL(V). Conversely, such an A-point gives an isomorphism of 𝒬BA with VA. This shows that the functor of points of p-1(Z) is isomorphic to the functor of points of GL(V)×Z, so by Yoneda’s lemma we get the claim. ∎

We can repeat the argument of Lemma A.5 for Cn(𝕄d).

Lemma A.7.

Let A be a 𝕜-algebra. The map (x1,,xn)(M,id) is a bijection between the A-points of Cn(Md) and the set

{(M,φ):M a locally free A-module, φ:MVA is an A-linear isomorphism}/iso.

There is no scheme X whose A-points correspond to locally free A-modules. However, there is such an algebraic stack (see [46] for introduction to stacks) and it is called Modd(𝔸n).

Corollary A.8.

The variety Cn(Md) is an GL(V)-bundle over Modd(An).

Proof.

This follows from Lemma A.7 along the same lines as Corollary A.6. ∎

Corollary A.9.

The variety Cn(Md) has an obstruction theory, where a given point (x1,,xn) with corresponding module M has obstruction group Ext2(M,M).

Proof.

The map Cn(𝕄d)Modd(𝔸n) is smooth by Corollary A.8, so the obstruction theory for Modd(𝔸n), see [12, Proposition 6.5.1], lifts to an obstruction theory for Cn(𝕄d). ∎

Acknowledgements

We very much thank Nathan Ilten for coding and sharing an experimental version of his VersalDeformations package for Macaulay2 which allows one to compute deformations of modules. We thank Joseph Landsberg, Maciej Gałązka, Hang Huang, and the referee for suggesting several improvements to the text.

References

[1] M. F. Atiyah, N. J. Hitchin, V. G. Drinfel’d and Y. I. Manin, Construction of instantons, Phys. Lett. A 65 (1978), no. 3, 185–187. 10.1142/9789812794345_0018Search in Google Scholar

[2] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading 1969. Search in Google Scholar

[3] M. D. Atkinson and S. Lloyd, Primitive spaces of matrices of bounded rank, J. Aust. Math. Soc. Ser. A 30 (1980/81), no. 4, 473–482. 10.1017/S144678870001795XSearch in Google Scholar

[4] V. Baranovsky, Moduli of sheaves on surfaces and action of the oscillator algebra, J. Differential Geom. 55 (2000), no. 2, 193–227. 10.4310/jdg/1090340878Search in Google Scholar

[5] W. Bruns and J. Herzog, Cohen–Macaulay rings, Cambridge Stud. Adv. Math. 39, Cambridge University, Cambridge 1993. Search in Google Scholar

[6] D. A. Cartwright, D. Erman, M. Velasco and B. Viray, Hilbert schemes of 8 points, Algebra Number Theory 3 (2009), no. 7, 763–795. 10.2140/ant.2009.3.763Search in Google Scholar

[7] W. Crawley-Boevey and J. Schröer, Irreducible components of varieties of modules, J. Reine Angew. Math. 553 (2002), 201–220. 10.1515/crll.2002.100Search in Google Scholar

[8] D. Eisenbud, Commutative algebra, Grad. Texts in Math. 150, Springer, New York 1995. 10.1007/978-1-4612-5350-1Search in Google Scholar

[9] D. Eisenbud, The geometry of syzygies, Grad. Texts in Math. 229, Springer, New York 2005. Search in Google Scholar

[10] D. Eisenbud and J. Harris, Vector spaces of matrices of low rank, Adv. in Math. 70 (1988), no. 2, 135–155. 10.1016/0001-8708(88)90054-0Search in Google Scholar

[11] D. Eisenbud and J. Harris, The geometry of schemes, Grad. Texts in Math. 197, Springer, New York 2000. Search in Google Scholar

[12] B. Fantechi, L. Göttsche, L. Illusie, S. L. Kleiman, N. Nitsure and A. Vistoli, Fundamental algebraic geometry, Math. Surveys Monogr. 123, American Mathematical Society, Providence 2005. 10.1090/surv/123Search in Google Scholar

[13] B. Fantechi and M. Manetti, Obstruction calculus for functors of Artin rings. I, J. Algebra 202 (1998), no. 2, 541–576. 10.1006/jabr.1997.7239Search in Google Scholar

[14] D. Fiorenza, D. Iacono and E. Martinengo, Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 2, 521–540. 10.4171/JEMS/310Search in Google Scholar

[15] M. Gerstenhaber, On dominance and varieties of commuting matrices, Ann. of Math. (2) 73 (1961), 324–348. 10.2307/1970336Search in Google Scholar

[16] R. M. Guralnick, A note on commuting pairs of matrices, Linear Multilinear Algebra 31 (1992), no. 1–4, 71–75. 10.1080/03081089208818123Search in Google Scholar

[17] R. M. Guralnick and B. A. Sethuraman, Commuting pairs and triples of matrices and related varieties, Linear Algebra Appl. 310 (2000), no. 1–3, 139–148. 10.1016/S0024-3795(00)00065-3Search in Google Scholar

[18] Y. Han, Commuting triples of matrices, Electron. J. Linear Algebra 13 (2005), 274–343. 10.13001/1081-3810.1166Search in Google Scholar

[19] T. Harima, T. Maeno, H. Morita, Y. Numata, A. Wachi and J. Watanabe, The Lefschetz properties, Lecture Notes in Math. 2080, Springer, Heidelberg 2013. 10.1007/978-3-642-38206-2Search in Google Scholar

[20] T. Harima, J. C. Migliore, U. Nagel and J. Watanabe, The weak and strong Lefschetz properties for Artinian K-algebras, J. Algebra 262 (2003), no. 1, 99–126. 10.1016/S0021-8693(03)00038-3Search in Google Scholar

[21] R. Hartshorne, Deformation theory, Grad. Texts in Math. 257, Springer, New York 2010. 10.1007/978-1-4419-1596-2Search in Google Scholar

[22] A. A. Henni and D. M. Guimarães, A note on the ADHM description of Quot schemes of points on affine spaces, Internat. J. Math. 32 (2021), no. 6, Paper No. 2150031. 10.1142/S0129167X21500312Search in Google Scholar

[23] A. A. Henni and M. Jardim, Commuting matrices and the Hilbert scheme of points on affine spaces, Adv. Geom. 18 (2018), no. 4, 467–482. 10.1515/advgeom-2018-0011Search in Google Scholar

[24] J. Holbrook and K. C. O’Meara, Some thoughts on Gerstenhaber’s theorem, Linear Algebra Appl. 466 (2015), 267–295. 10.1016/j.laa.2014.10.009Search in Google Scholar

[25] J. Holbrook and M. Omladič, Approximating commuting operators, Linear Algebra Appl. 327 (2001), no. 1–3, 131–149. 10.1016/S0024-3795(00)00286-XSearch in Google Scholar

[26] R. A. Horn and C. R. Johnson, Matrix analysis, Cambridge University, Cambridge 1985. 10.1017/CBO9780511810817Search in Google Scholar

[27] A. Iarrobino and J. Emsalem, Some zero-dimensional generic singularities; finite algebras having small tangent space, Compos. Math. 36 (1978), no. 2, 145–188. Search in Google Scholar

[28] A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Math. 1721, Springer, Berlin 1999. 10.1007/BFb0093426Search in Google Scholar

[29] A. Iarrobino, P. M. Marques and C. McDaniel, Artinian algebras and Jordan type, preprint (2020), https://arxiv.org/abs/1802.07383v5. 10.1216/jca.2022.14.365Search in Google Scholar

[30] J. Jelisiejew, Classifying local Artinian Gorenstein algebras, Collect. Math. 68 (2017), no. 1, 101–127. 10.1007/s13348-016-0183-1Search in Google Scholar

[31] J. Jelisiejew, Elementary components of Hilbert schemes of points, J. Lond. Math. Soc. (2) 100 (2019), no. 1, 249–272. 10.1112/jlms.12212Search in Google Scholar

[32] J. Jelisiejew, Pathologies on the Hilbert scheme of points, Invent. Math. 220 (2020), no. 2, 581–610. 10.1007/s00222-019-00939-5Search in Google Scholar

[33] J. Jelisiejew and Ł. U. Sienkiewicz, Białynicki–Birula decomposition for reductive groups, J. Math. Pures Appl. (9) 131 (2019), 290–325. 10.1016/j.matpur.2019.04.006Search in Google Scholar

[34] J. Jelisiejew and K. Šivic, Components and singularities of Quot schemes and varieties of commuting matrices, preprint (2021), https://arxiv.org/abs/2106.13137. 10.1515/crelle-2022-0018Search in Google Scholar

[35] J. L. Kass, The compactified jacobian can be nonreduced, Bull. Lond. Math. Soc. 47 (2015), no. 4, 686–692. 10.1112/blms/bdv036Search in Google Scholar

[36] J. M. Landsberg, Tensors: Geometry and applications, Grad. Stud. Math. 128, American Mathematical Society, Providence 2012. Search in Google Scholar

[37] J. M. Landsberg and M. Michałek, Abelian tensors, J. Math. Pures Appl. (9) 108 (2017), no. 3, 333–371. 10.1016/j.matpur.2016.11.004Search in Google Scholar

[38] P. Levy, N. V. Ngo and K. Šivic, Commuting varieties and cohomological complexity theory, preprint (2021), https://arxiv.org/abs/2105.07918; to appear in J. Lond. Math. Soc. (2). 10.1112/jlms.12650Search in Google Scholar

[39] M. Manetti, Deformation theory via differential graded Lie algebras, Algebraic Geometry Seminars, 1998–1999 (Pisa), Scuola Normale Superiore, Pisa (1999), 21–48. Search in Google Scholar

[40] M. Manetti, Differential graded Lie algebras and formal deformation theory, Algebraic geometry—Seattle 2005, Proc. Sympos. Pure Math. 80 Part 2, American Mathematical Society, Providence (2009), 785–810. 10.1090/pspum/080.2/2483955Search in Google Scholar

[41] E. Miller and B. Sturmfels, Combinatorial commutative algebra, Grad. Texts in Math. 227, Springer, New York 2005. Search in Google Scholar

[42] R. Moschetti and A. T. Ricolfi, On coherent sheaves of small length on the affine plane, J. Algebra 516 (2018), 471–489. 10.1016/j.jalgebra.2018.09.028Search in Google Scholar

[43] T. S. Motzkin and O. Taussky, Pairs of matrices with property L. II, Trans. Amer. Math. Soc. 80 (1955), 387–401. 10.2307/1992996Search in Google Scholar

[44] H. Nakajima, Lectures on Hilbert schemes of points on surfaces, Univ. Lecture Ser. 18, American Mathematical Society, Providence 1999. 10.1090/ulect/018Search in Google Scholar

[45] N. V. Ngo and K. Šivic, On varieties of commuting nilpotent matrices, Linear Algebra Appl. 452 (2014), 237–262. 10.1016/j.laa.2014.03.032Search in Google Scholar

[46] M. Olsson, Algebraic spaces and stacks, Amer. Math. Soc. Colloq. Publ. 62, American Mathematical Society, Providence 2016. 10.1090/coll/062Search in Google Scholar

[47] A. T. Ricolfi, The Hilbert scheme of hyperelliptic Jacobians and moduli of Picard sheaves, Algebra Number Theory 14 (2020), no. 6, 1381–1397. 10.2140/ant.2020.14.1381Search in Google Scholar

[48] J. Schur, Zur Theorie der vertauschbaren Matrizen, J. Reine Angew. Math. 130 (1905), 66–76. 10.1007/978-3-642-61947-2_5Search in Google Scholar

[49] E. Sernesi, Deformations of algebraic schemes, Grundlehren Math. Wiss. 334, Springer, Berlin 2006. Search in Google Scholar

[50] K. Šivic, On varieties of commuting triples III, Linear Algebra Appl. 437 (2012), no. 2, 393–460. 10.1016/j.laa.2011.08.015Search in Google Scholar

[51] S. A. Strømme, Elementary introduction to representable functors and Hilbert schemes, Parameter spaces (Warsaw 1994), Banach Center Publ. 36, Polish Academy of Sciences, Warsaw (1996), 179–198. 10.4064/-36-1-179-198Search in Google Scholar

[52] M. Szachniewicz, Non-reducedness of the Hilbert schemes of points, preprint (2021), https://arxiv.org/abs/2109.11805. Search in Google Scholar

[53] R. Vakil, Murphy’s law in algebraic geometry: badly-behaved deformation spaces, Invent. Math. 164 (2006), no. 3, 569–590. 10.1007/s00222-005-0481-9Search in Google Scholar

[54] Stacks Project, http://math.columbia.edu/algebraic_geometry/stacks-git, 2017. Search in Google Scholar

Received: 2021-07-28
Revised: 2022-03-01
Published Online: 2022-05-25
Published in Print: 2022-07-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 2.3.2024 from https://www.degruyter.com/document/doi/10.1515/crelle-2022-0018/html
Scroll to top button