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Components and singularities of Quot schemes and varieties of commuting matrices

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


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


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


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.


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


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


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


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


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).


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).


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). ∎


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.


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

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