The Geometry of Rings of Components of Hurwitz Spaces Béranger Seguin

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The Geometry of Rings of Components

of Hurwitz Spaces

Béranger Seguin∗

Abstract. We consider a variant of the ring of components of Hurwitz spaces introduced by

Ellenberg, Venkatesh and Westerland. By focusing on Hurwitz spaces classifying covers of the

projective line, the resulting ring of components is commutative, which lets us study it from the

point of view of algebraic geometry and relate its geometric properties to numerical invariants

involved in our previously obtained asymptotic counts. Speciﬁcally, we describe a stratiﬁcation

of the prime spectrum of the ring of components, and we compute the dimensions and degrees

of the strata. Using the stratiﬁcation, we give a complete description of the spectrum in some

cases.

Keywords: Hurwitz spaces ·Prime spectra of monoid rings

MSC 2010: 14A10 ·13A02 ·16S34

Contents

1. Introduction and main results 1

2. Deﬁnitions and preliminaries 3

3. The subgroup stratiﬁcation of the variety of components 6

4. Nilpotent elements of the ring of components 10

5. Dimensions and degrees of the strata 13

6. Explicit description of the variety of components 15

1. Introduction and main results

For the whole article, we ﬁx a ﬁnite group G, a nonempty set Dof nontrivial conjugacy classes of G,

a map ξ:D→Z>0(attributing a multiplicity to each conjugacy class γ∈D), and a ﬁeld kwhose

characteristic does not divide the order |G|of the group G.

1.1. Context

In [EVW16], Ellenberg, Venkatesh and Westerland introduced the ring of components of Hurwitz

spaces, a graded algebra whose elements are linear combinations of connected components of Hurwitz

spaces parametrizing marked G-covers1of the aﬃne line. The grading of that ring reﬂects the number

of branch points of the covers parametrized by each component, and the multiplicative structure is

induced by a geometric “concatenation” operation.

∗Universität Paderborn, Fakultät EIM, Institut für Mathematik, Warburger Str. 100, 33098 Paderborn, Germany.

Email: bseguin@math.upb.de.

1Here, a marked G-cover is a ﬁnite branched cover (not necessarily connected) with a marked point in an unramiﬁed

ﬁber, equipped with an action of Gon the cover inducing simply transitive actions of Gon each unramiﬁed ﬁber.

arXiv:2210.12793v2 [math.NT] 2 Oct 2024

The deﬁnition of that ring is motivated by the fact that its Hilbert function is tightly related to

the asymptotic behavior of the cohomology of Hurwitz spaces, which is in turn related (using the

Grothendieck-Lefschetz trace formula and Deligne’s bounds on the eigenvalues of Frobenius endo-

morphisms) to the count of Fq-points of Hurwitz spaces and hence to the distribution of extensions

F|Fq(T)with Galois group isomorphic to G, when qis large and coprime to |G|. In [ETW17], this

approach was used to obtain an upper bound consistent with the variant of Malle’s conjecture for

function ﬁelds over ﬁnite ﬁelds.

In [Seg24], we have extended some of the counting results of [EVW16]. For instance, we have

studied the analogous ring of components of Hurwitz spaces of marked G-covers of the projective line.

This ring is a commutative graded ﬁnitely generated algebra, and the growth of its Hilbert function

is related to geometric invariants of its spectrum. This observation was the starting point for a more

systematic study of the ring of components from the point of view of algebraic geometry.

1.2. Main results

In Section 2, we deﬁne the ring of components R(Deﬁnition 2.7), which is a ﬁnitely generated com-

mutative graded k-algebra. We then introduce its prime spectrum Spec R, which we call the variety of

components (Deﬁnition 2.8). In Section 3, we deﬁne subsets γ(H)of Spec R(Deﬁnition 3.4), indexed

by subgroups Hof G, and we prove that they form a stratiﬁcation of the variety of components:

Theorem 1.1. The locally closed subsets γ(H)form a stratiﬁcation of Spec R:

Spec R=G

H⊆G

γ(H).

This result, which is a particular case of the more general Theorem 3.9, has the following conse-

quence: in order to describe the variety of components fully, it suﬃces to describe each stratum γ(H).

Using the counting results of [Seg24], we compute in Section 5 the Krull dimension of the stratum γ(H)

corresponding to a subgroup Hof G. More precisely, we relate it to a numerical invariant deﬁned in

[Seg24], the splitting number Ω(DH)(Deﬁnition 2.12):

Theorem 1.2. We have dimKrull γ(H) = Ω(DH)+1.

In Subsection 5.3, we discuss further connections between group-theoretic and geometric invariants

by relating the degree of the stratum γ(H), seen as embedded in projective space, to (a quotient of)

the second homology group of H.

In Section 6, we approach the variety of components more “directly” by describing the strata fully

in Theorem 6.16 and its coordinate-based variant Theorem 6.20. However, our description relies on

strong assumptions on the ring of components. We do not reproduce the statement here as it uses a

lot of terminology. This result applies in particular to the classical situation where Gis a symmetric

group and Dcontains only the conjugacy class of transpositions. In that case, Theorem 6.21 gives a

full description of the variety of components.

1.3. Outline

This article is organized as follows:

•In Section 2, we deﬁne notation and terminology used throughout the article. Notably, we deﬁne

the ring of components (Deﬁnition 2.7) and its associated variety (Deﬁnition 2.8), which are our

main objects of study.

•In Section 3, we associate to each subgroup Hof Ga subring RHand four ideals IH, I∗

H, JH, J∗

Hof

the ring of components (Deﬁnition 3.1). We use these to deﬁne the strata γ(H)(Deﬁnition 3.4).

We then prove Theorem 3.9, which is the general form of the stratiﬁcation of the variety of

components (Theorem 1.1).

•In Section 4, we prove Theorem 4.1. This technical result, which is a weak asymptotic form of

reducedness for the ring of components R, is needed for the proof of Theorem 1.2.

•In Section 5, we compute the Krull dimension of each stratum γ(H)(Theorem 1.2). In Subsec-

tion 5.3, we also compute the degree of γ(H)in some cases. The proofs rely on the asymptotic

counting results from [Seg24].

•In Section 6, we prove Theorems 6.16 and 6.20, which give complete descriptions of the variety

of components in some cases. We apply these results to the classical case of symmetric groups

in Subsection 6.5.

1.4. Acknowledgments

This work was funded by the French ministry of research through a CDSN grant, and by the Deutsche

Forschungsgemeinschaft (DFG, German Research Foundation) — Project-ID 491392403 — TRR 358.

I thank my advisors Pierre Dèbes and Ariane Mézard for their support and their precious advice

during my time as a PhD student, and the reporters of my thesis Jean-Marc Couveignes and Craig

Westerland for providing helpful feedback.

2. Definitions and preliminaries

Recall that we have ﬁxed a ﬁnite group G, a set Dof nontrivial conjugacy classes of G, and a map

ξ:D→Z>0. Additionally, we deﬁne the set c=Fγ∈Dγand the integer |ξ|=Pγ∈Dξ(γ).

2.1. The monoid of components

We brieﬂy recall the deﬁnition of the monoid of components CompP1(C)(G, D, ξ), which was already

deﬁned in [Seg23, Deﬁnition 3.4.4] and [Seg24, Deﬁnition 2.6]. First, we deﬁne the braid group Bnby

its presentation:

Deﬁnition 2.1. The Artin braid group Bnon nstrands is deﬁned by the following presentation:

Bndef

=*σ1, σ2, . . . , σn−1

σiσj=σjσiif |i−j|>1

σiσi+1σi=σi+1σiσi+1 if i∈ {1, . . . , n −2}+.

Deﬁnition 2.2. The Hurwitz action of Bnon the set Gnof n-tuples of elements of Gis the (well-

deﬁned) action for which the generator σi∈Bnacts on a tuple g= (g1, . . . , gn)∈Gnas follows:

σi.(g1, . . . , gi−1, gi, gi+1, gi+2, . . . , gn)=(g1, . . . , gi−1, gigi+1g−1

i, gi, gi+2, . . . , gn).

Deﬁnition 2.3. Let g= (g1, . . . , gn)∈Gnbe a tuple of elements of G. The group of gis the

subgroup DgEof Ggenerated by g1, . . . , gn, and the product of gis the element πg def

=g1···gn∈G.

Both the group and product of a tuple are invariant under the Hurwitz action, and thus we extend

the deﬁnition of these invariants and the notations ⟨m⟩, πm when mis an orbit for the Hurwitz action.

Deﬁnition 2.4. Acomponent (of degree n) is the orbit, under the Hurwitz action of the braid

group Bn|ξ|, of a tuple g= (g1, . . . , gn|ξ|)∈Gn|ξ|satisfying πg = 1 and such that exactly n·ξ(γ)entries

of gbelong to each conjugacy class γ∈D. The monoid of components CompP1(C)(G, D, ξ)is the

(nonnegatively) graded set whose elements of degree nare the components of degree n, equipped with

the multiplication induced by the concatenation of tuples:

(g1, . . . , gn|ξ|)(g′

1, . . . , g′

n′|ξ|)=(g1, . . . , gn|ξ|, g′

1, . . . , g′

n|ξ|).

The monoid of components is well-deﬁned and commutative ([Seg23, Proposition 3.3.8], [Seg23,

Proposition 3.3.11]). Components of degree nare named this way because they correspond bijectively

to connected components of the Hurwitz space classifying marked G-covers of the projective line

branched at n· |ξ|points, among which n·ξ(γ)have their monodromy elements in each class γ∈D.

This connection is explained more carefully in [Seg23, Subsection 3.3.2]. The identity element of

the monoid CompP1(C)(G, D, ξ)is the orbit of the empty tuple, which corresponds to the connected

component containing only the trivial G-cover (with no branch points).

Deﬁnition 2.5. A nontrivial element of CompP1(C)(G, D, ξ)is a non-factorizable component2if it

does not equal any product of two nontrivial components.

A simple pigeonhole argument (carried out in [Seg23, Lemma 3.4.17]) shows that there are ﬁnitely

many non-factorizable components. Therefore, the monoid of components is a ﬁnitely generated

commutative graded monoid.

Remark 2.6.The non-factorizable components do not necessarily all have the same degree; see [Seg23,

Remark 3.4.20] for a counterexample.

2.2. The ring of components

2.2.1. Deﬁnition. We now deﬁne the ring of components:

Deﬁnition 2.7. The ring of components Ris the graded k-algebra k[CompP1(C)(G, D, ξ)] obtained

as the monoid ring (over k) of the monoid of components. The irrelevant ideal ϖis the (maximal)

ideal of Rgenerated by components of positive degree.

The ring Rof Deﬁnition 2.7 corresponds to RP1(C)(G, D, ξ)in the notation of [Seg23, Deﬁni-

tion 3.4.12]. The properties mentioned in Subsection 2.1 imply that the ring Ris a commutative

graded k-algebra of ﬁnite type, generated by the non-factorizable components.

2.2.2. Variety of components. We now deﬁne our main object of study, the variety of components:

Deﬁnition 2.8. The variety of components is the set Spec Rof prime ideals p⊊R, equipped with

the Zariski topology. If Iis an ideal of R, we denote by V(I)the closed subset of Spec Rconsisting of

all prime ideals containing I.

2.2.3. Aﬃne embedding. Assume that kis algebraically closed, and let Spm Rbe the subset of

Spec Rconsisting of closed points, i.e., of maximal ideals m⊊R. Then, the set Spm Rcan be

identiﬁed with the set of morphisms of k-algebras from Rto k(identifying a maximal ideal mwith

the projection R↠R/m≃k), or equivalently with the set of k-points of the scheme Spec R. Let Σ

be the ﬁnite set of non-factorizable components. Then, we can identify Spm Rwith a classical variety

by embedding it in kΣas follows: a point (xm)m∈Σbelongs to Spm Rif and only if the equality

xm1···xmu=xm′

1···xm′

vholds whenever the equality m1···mu=m′

1···m′

vholds in the monoid of

components. Note that, as the monoid of components is commutative and ﬁnitely generated, Dickson’s

lemma implies that it is presented by ﬁnitely many equalities of that type.

Remark 2.9.The ring Ris a graded k-algebra, and thus it may be more natural to consider the projec-

tive variety that it deﬁnes (i.e., the set of homogeneous ideals which are maximal among those properly

contained in ϖ) instead of the aﬃne variety Spm R. However, since non-factorizable components need

not all have the same degree (cf. Remark 2.6), the space in which the variety naturally embeds is

a “weighted” projective space, namely the set of orbits of kΣ\ {0}under the action of k×for which

a scalar λ∈k×acts on a point zby multiplying its coordinate zm(associated to a non-factorizable

2Non-factorizable components are simply the irreducible elements of the monoid CompP1(C)(G, D, ξ), but we avoid

using the ambiguous term “irreducible component”.

component m∈Σ) by λdeg m. Weighted projective spaces do embed in ordinary projective spaces of

higher dimension [Hos20, Theorem 3.4.9], but we mostly work with the aﬃne variety associated to R

to avoid dealing with these subtleties.

2.3. D-generated subgroups

We brieﬂy recall the notion of D-generated subgroups from [Seg24, Deﬁnition 1.1]:

Deﬁnition 2.10. A subgroup Hof Gis D-generated if the sets γ∩Hfor γ∈Dare all nonempty

and collectively generate H. We denote by SubG,D the set of subgroups of Gwhich are either trivial

or D-generated.

The relevance of this deﬁnition comes from the following proposition, which is proved in [Seg23,

Proposition 3.2.22]:

Proposition 2.11. A subgroup Hof Gbelongs to SubG,D if and only if there is a component m∈

CompP1(C)(G, D, ξ)whose group is H.

If His a D-generated subgroup of G, we deﬁne its splitting number as in [Seg24, Deﬁnition 1.2]:

Deﬁnition 2.12. Let H∈SubG,D. Let DHdef

={γ∩H|γ∈D}, let cHdef

=c∩H(which is also

Fγ′∈DHγ′), and denote by D∗

Hthe set of conjugacy classes of Hwhich are contained in cH. The

splitting number of His the integer Ω(DH)def

=|D∗

H|−|DH|.

Deﬁnition 2.13. AD-generated subgroup Hof Gis a non-splitter if Ω(DH)=0, i.e., if DHconsists

of conjugacy classes of H.

The splitting number of Hplays a central role in the asymptotic count of components with group

H, cf. [Seg24, Theorem 1.4].

2.4. Chart of notations

For quick reference, the chart below indicates where the deﬁnitions introduced in this section can be

found. A short description is also given.

Notation Reference Short description

G, D, ξ, k Top of Section 1 setup

c, |ξ|Top of Section 2

BnDeﬁnition 2.1 Artin braid group

DgE, πg Deﬁnition 2.3 invariants of a tuple (or component)

CompP1(C)(G, D, ξ)Deﬁnition 2.4 monoid of components

RDeﬁnition 2.7 ring of components

Spec R, V (I)Deﬁnition 2.8 variety of components and its closed subsets

SubG,D Deﬁnition 2.10 set of D-generated (or trivial) subgroups

Ω(DH)Deﬁnition 2.12 splitting number of H

We also include a chart of notation introduced in later sections:

Notation Reference Short description

IH, I∗

H, JH, J∗

HDeﬁnition 3.1 ideals of Rassociated to a subgroup H

RHDeﬁnition 3.1 subring of Rassociated to a subgroup H

γ(H)Deﬁnition 3.4 stratum associated to a subgroup H

ΓHDeﬁnition 3.6 ideal quotient (pI∗

H:√IH)

Rn,H , Nn,H Top of Section 4 space spanned by components of degree nand

group H(resp. subspace of nilpotent elements)

µH, H2(H, cH)Subsection 4.1 notation related to the lifting invariant

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TheGeometryofRingsofComponentsofHurwitzSpacesBérangerSeguin∗Abstract.WeconsideravariantoftheringofcomponentsofHurwitzspacesintroducedbyEllenberg,VenkateshandWesterland.ByfocusingonHurwitzspacesclassifyingcoversoftheprojectiveline,theresultingringofcomponentsiscommutative,whichletsusstudyitfromthepoi...

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The Geometry of Rings of Components of Hurwitz Spaces Béranger Seguin

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