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LIFTING FREE DIVISORS

RAGNAR-OLAF BUCHWEITZ AND BRIAN PIKE

Abstract. Let ϕ:X→Sbe a morphism between smooth complex analytic

spaces, and let f= 0 deﬁne a free divisor on S. We prove that if the deforma-

tion space T1

X/S of ϕis a Cohen-Macaulay OX–module of codimension 2, and

all of the logarithmic vector ﬁelds for f= 0 lift via ϕ, then f◦ϕ= 0 deﬁnes

a free divisor on X; this is generalized in several directions.

Among applications we recover a result of Mond–van Straten, generalize a

construction of Buchweitz–Conca, and show that a map ϕ:Cn+1 →Cnwith

critical set of codimension 2 has a T1

X/S with the desired properties. Finally,

if Xis a representation of a reductive complex algebraic group Gand ϕis the

algebraic quotient X→S=X//G with X//G smooth, we describe suﬃcient

conditions for T1

X/S to be Cohen–Macaulay of codimension 2. In one such

case, a free divisor on Cn+1 lifts under the operation of “castling” to a free

divisor on Cn(n+1), partially generalizing work of Granger–Mond–Schulze on

linear free divisors. We give several other examples of such representations.

Contents

1. Introduction 1

2. Deformation theory and free divisors 3

3. The Main Results 7

4. Lifting Euler vector ﬁelds 10

5. Adding components and dimensions 12

6. The case of maps ϕ:Cn+1 →Cn14

7. Coregular and Cofree Group Actions 15

8. Examples of group actions 21

References 28

1. Introduction

Let f:S→(C,0) be the germ of a holomorphic function deﬁning a reduced

hypersurface germ D=f−1(0) in a smooth complex analytic germ S= (Cm,0).

The OS–module DerS(−log f) of logarithmic vector ﬁelds consists of all germs of

holomorphic vector ﬁelds on Sthat are tangent to the smooth points of D. Then D

is called a free divisor when DerS(−log f) is a free OS–module, necessarily of rank

m, or equivalently, when DerS(−log f) requires only mgenerators, the smallest

number possible. A free divisor is either a smooth hypersurface or singular in

codimension one. The (Saito or discriminant) matrix that describes the inclusion

Date: December 16, 2013.

2010 Mathematics Subject Classiﬁcation. Primary 32S25; Secondary 17B66, 14L30.

Key words and phrases. free divisors, logarithmic vector ﬁelds, discriminants, coregular group

actions, invariants, tangent cohomology, Kodaira-Spencer map.

The ﬁrst author was partly supported by NSERC grant 3-642-114-80.

1

arXiv:1310.7873v1 [math.AG] 29 Oct 2013

2 RAGNAR-OLAF BUCHWEITZ AND BRIAN PIKE

of the logarithmic vector ﬁelds into all vector ﬁelds on the ambient space is square

and its determinant is an equation of the free divisor, thus, providing a compact

representation of an otherwise usually dense polynomial or power series.

Free divisors are often ‘discriminants’, broadly interpreted, and then describe

the locus of some type of degenerate behavior. For instance, free divisors classi-

cally arose as discriminants of versal unfoldings of isolated hypersurface ([Sai80])

and isolated complete intersection singularities ([Loo84]). As well, the locus in a

Frobenius manifold where the Euler vector ﬁeld is not invertible is a free divisor

([Her02]). More recently, other discriminants have been shown to be free divisors

(e.g., [Dam98,MvS01,Dam01,BEGvB09]).

Many hyperplane arrangements are classically known to be free divisors and it

is a long outstanding question whether freeness is a combinatorial property in this

case ([OT92]).

When DerS(−log f) has a free basis of linear vector ﬁelds, then Dis a linear free

divisor; these may be thought of as the discriminant of a prehomogeneous vector

space, a representation on Sof a linear algebraic group that has a Zariski open orbit.

While the above list of examples is meant to highlight that free divisors are

everywhere, and, for example, the assignment from isolated complete intersection

singularities to their discriminants in the base of a semi–universal deformation is

essentially injective by [Wir80], we still have very few methods to construct such

divisors explicitly in a given dimension.

Here we give one approach to such construction. Let ϕ:X= (Cn,0) →S=

(Cm,0) be a holomorphic map between smooth complex spaces, and let D=V(f)

be a free divisor in S. In this paper we ask:

When is ϕ−1(D)≡ {fϕ = 0} ⊆ Xagain a free divisor?

We give suﬃcient conditions for ϕ−1(D) to be a free divisor, and describe a num-

ber of situations in which these conditions hold. This gives a ﬂexible method to

construct new free divisors, and gives some insight into the behavior of logarithmic

vector ﬁelds under this pullback operation.

The structure of the paper is the following. Our suﬃcient conditions are stated

in terms of modules describing the deformations of ϕ, and the module of vector

ﬁelds on Sthat lift across ϕto vector ﬁelds on X. Hence, in §2we introduce some

deformation theory, the Kodaira-Spencer map, and also free divisors.

§3contains our two main results. Theorems 3.4 and 3.5 each give conditions for

ϕ−1(D) to be free; Theorem 3.5 is a consequence of Theorem 3.4 that has more

restrictive hypotheses—that are easier to check—and a stronger conclusion. All

but one of our applications use Theorem 3.5. Our ﬁrst example generalizes a result

of Mond and van Straten [MvS01].

Both Theorems require that all vector ﬁelds η∈DerS(−log f) lift across ϕ.

In §4, we relax this condition, at least for the Euler vector ﬁeld of a weighted-

homogeneous f. The motivation for this consequence of Theorem 3.5 is the case of

D={0} ⊂ S= (C,0), where the conditions for ϕ−1(D) = V(ϕ) to deﬁne a free

divisor are known and require no lifting of vector ﬁelds.

In §5we describe a construction that, given a free divisor Din Xand an appro-

priate ideal I⊂ OX, constructs a free divisor on X×Ythat contains D×Yand has

additional nontrivial components. This application of Theorem 3.5 generalizes a

LIFTING FREE DIVISORS 3

construction of Buchweitz–Conca [BC13], and a construction for linear free divisors

[DP12,Pik10].

The rest of the paper describes situations where the deformation condition on

ϕis satisﬁed. Then the requirement for all η∈DerS(−log f) to lift is generally

satisﬁed for certain free divisors in S. For instance, in §6we show that maps

ϕ:Cn+1 →Cnwith critical set of codimension 2 satisfy the deformation condition.

We begin §7by describing our original interest in this problem. Granger–Mond–

Schulze [GMS11] showed that the set of prehomogeneous vector spaces that deﬁne

linear free divisors is invariant under ‘castling’, an operation on prehomogeneous

vector spaces. Thus, the corresponding transformation on a linear free divisor

produces another linear free divisor. In the simplest case, this transformation is

a lift across the map ϕ:X=Mn,n+1 →Cn+1, where Mn,n+1 is the space of

n×(n+ 1) matrices and each component of ϕis a signed determinant. In Theorem

7.2, we use Theorem 3.5 to show that pulling back an arbitrary free divisor via this

ϕproduces another free divisor. This ϕis the algebraic quotient of SL(n, C) acting

on Mn,n+1.

The rest of §7generalizes this result by studying algebraic quotients ϕ:X→S=

X//G of a reductive linear algebraic group Gacting on X. Since we require that S

is smooth, so the action of Gis coregular, the components of ϕgenerate the subring

of G-invariant polynomials on X. Proposition 7.12 gives suﬃcient conditions for

the deformation condition on ϕto be satisﬁed, and Lemma 7.16 suggests a method

to prove that certain vector ﬁelds are liftable. As an aside, we point out that the

invariants of lowest degree may be easily computed.

Finally, §8describes many—but not all—examples of group actions with quo-

tients satisfying the deformation condition; for each, we identify those fsatisfying

the lifting condition. This is a very productive method for producing free divisors.

Acknowledgements: Discussions with David Mond inspired us to look further at

castling, and Eleonore Faber gave helpful comments on the paper.

2. Deformation theory and free divisors

2.1.If ϕ:X→Sis any morphism of complex analytic germs, we write ϕ[:

OS→ OX, f 7→ f ϕ for the corresponding morphism of local analytic algebras. By

common abuse of notation, we often write ϕfor ϕ[. Furthermore, we throughout

denote by m∗the maximal ideal in O∗of germs of functions that vanish at 0, the

distinguished point of the germ.

Deformations. We begin with some background on deformation theory and tan-

gent cohomology.

2.2.If ϕ:X→Sis still any morphism of analytic germs and Man OX–module,

we denote Ti

X/S (M) = Hi(HomOX(LX/S,M)), the ith tangent cohomology of X

over Swith values in M. Here LX/S is a cotangent complex for ϕ, well deﬁned

up to isomorphism in the derived category of coherent OX–modules (e.g., [GLS07,

Appendix C]).

As usual, we abbreviate Ti

X/S =Ti

X/S (OX), and write simply Ti

Xif ϕis the

constant map to a point.

2.3.Note that T0

X/S (M) = DerS(OX,M) is the OX–module of ϕ−1(OS)–linear

vector ﬁelds on Xwith values in M, or, shorter, the OX–module of vertical vector

4 RAGNAR-OLAF BUCHWEITZ AND BRIAN PIKE

ﬁelds along ϕwith values in M. If M=OX, we simply speak of the module of

vertical vector ﬁelds along ϕ.

2.4.If ϕis smooth, then Ti

X/S (M) = 0 for all i6= 0, and all M. As tangent coho-

mology localizes on X, the OX–modules Ti

X/S (M), for i > 0, are thus supported

on the critical locus of ϕ, the closed subgerm C(ϕ)⊆X, where ϕfails to be smooth

(e.g., [GLS07]).

2.5.If ϕ:X→Sis any morphism of analytic germs, it induces the (dual) Zariski–

Jacobi sequence in tangent cohomology, the long exact sequence of OX–modules

0//T0

X/S //T0

X

Jac(ϕ)

//T0

S(OX)δ//T1

X/S //T1

X//· · · ,

where Jac(ϕ) is the OX–dual to dϕ :ϕ∗Ω1

S∼

=OX⊗OSΩ1

S→Ω1

Xthat in turn sends

1⊗OSds to d(sϕ) for any function germ s∈ OS.

If x1, ..., xnare local coordinates on Xand s1, ..., smare local coordinates on S,

then a vector ﬁeld

Z=

n

X

i=1

gi

∂

∂xi

∈T0

X⊆

n

M

i=1

OX

∂

∂xi

,(1)

with coeﬃcients gi∈ OX, maps to the vector ﬁeld

Jac(ϕ)(Z) =

m

X

j=1

n

X

i=1

gi

∂(sj◦ϕ)

∂xi

∂

∂sj

∈T0

S(OX)⊆

m

M

j=1

OX

∂

∂sj

.(2)

2.6.Of particular importance is the OS–linear Kodaira–Spencer map deﬁned by ϕ.

It is the composition

δKS =δϕ

KS =δ◦T0

S(ϕ[) : T0

S(OS)T0

S(ϕ[)

−−−−→ T0

S(OX)δ

−−→ T1

X/S

that sends a vector ﬁeld D=Pm

j=1 fj∂

∂sj∈T0

Sto the class

δKS (D) = δ m

X

j=1

fjϕ∂

∂sj!∈T1

X/S .

Thus we have a commutative diagram

T0

S

T0

S(ϕ[)

δKS

$$

0//T0

X/S //T0

X

Jac(ϕ)

//T0

S(OX)δ//T1

X/S //T1

X//· · · .

2.7.The signiﬁcance of the Kodaira–Spencer map is twofold: a vector ﬁeld D∈

T0

Sis liftable to X, if, and only if, δK S (D) = 0. Indeed, the exactness of the

Zariski–Jacobi tangent cohomology sequence shows that the image T0

S(ϕ[)(D) =

Pm

j=1 fjϕ∂

∂sjin T0

S(OX) of the vector ﬁeld Dis induced from a vector ﬁeld Eon

X, in that T0

S(ϕ[)(D) = Jac(ϕ)(E), for some Eif, and only if, δK S (D) = 0. One

therefore calls the kernel of the Kodaira–Spencer map also the OS–submodule of

liftable vector ﬁelds in T0

S.

LIFTING FREE DIVISORS 5

0

0

OS

in2

OS

·f

0//DerS(−log f)˜σf

//T0

S⊕ OS

(Jac(f),f)

//

pr1

OS//

OΣ//0 (†)

0//DerS(−log f)σf

//T0

S

jac(f)

//

OS/(f)//

OΣ//0

0 0

Figure 1. The commutative diagram exhibiting Σ and

DerS(−log f), described in 2.10–2.12.

A deformation-theoretic interpretation is that such a lift trivializes the inﬁni-

tesimal ﬁrst-order deformation of X/S along D, whence we also say that X/S is

(inﬁnitesimally) trivial along Das soon as δKS (D) = 0.

2.8.On the other hand, if ϕis a ﬂat morphism, then the Kodaira–Spencer map

is surjective, if, and only if, ϕrepresents a versal deformation of the ﬁbre X0=

ϕ−1(0) ⊆Xof ϕover the origin ([Fle81]).

2.9.If Xis smooth, then T1

X= 0 and the inclusion (1) is an equality, while the

inclusion (2) becomes an equality if Sis smooth.

In particular, if both Xand Sare smooth, the dual Zariski–Jacobi sequence

truncates to a resolution of T1

X/S , with T0

X/S as a second syzygy module.

In the language of the Thom–Mather theory of the singularities of diﬀerentiable

maps, T1

X/S is isomorphic as a vector space to the extended normal space of ϕunder

right equivalence, while the cokernel of δKS :T0

S→T1

X/S is isomorphic as a vector

space to the extended normal space of ϕunder left-right equivalence (see [GL08]).

Free divisors. After this short excursion into the general theory of the (co-)tangent

complex and its cohomology, we recall the pertinent facts about free divisors.

2.10.Let f∈ OSbe the germ of a nonzero function on a smooth germ Swith zero

locus the divisor, or hypersurface germ V(f)≡ {f= 0} ⊆ S. Diﬀerentiating f

yields the commutative diagram in Figure 1of OS–modules with exact rows and

exact columns, the rows exhibiting, one may say, deﬁning, the singular locus Σ

of V(f) as well as the OS–module DerS(−log f) of logarithmic vector ﬁelds on S

along V(f), as cokernel, respectively kernel, of the OS–linear maps in the middle.

Here Jac(f)(D) = D(f) for any vector ﬁeld or derivation D∈T0

S, and jac(f)(D)

is the class of D(f) modulo f.

2.11. Deﬁnition. Recall that fdeﬁnes a free divisor in S, if fis reduced and

DerS(−log f) is a free OS–module, necessarily of rank m= dim S(see [Sai80]).

We recall the basic notions of the theory.

6 RAGNAR-OLAF BUCHWEITZ AND BRIAN PIKE

2.12.If f∈ OSdeﬁnes a free divisor, {ej}m

j=1 is a choice of an OS–basis of

DerS(−log f), and {∂/∂sj}m

j=1 is the canonical basis of T0

Sdetermined by local

coordinates sjon S(as in 2.5), then the matrix of the inclusion σfin Figure 1with

respect to these bases is a Saito or discriminant matrix for f.

The matrix of ˜σf, where we extend the basis {∂/∂sj}m

j=1 of T0

Sby the canonical

basis 1 ∈ OSof that free OS–module of rank 1, yields then the extended Saito or

discriminant matrix for f, in that

˜σf(ej) = m

X

i=1

aij

∂

∂si

,−hj!

records that the vector ﬁeld Dj=Pm

i=1 aij ∂

∂siis logarithmic along f, as

0 = (Jac(f), f )˜σf(ej) =

m

X

i=1

aij

∂f

∂si

−hjf=Dj(f)−hjf,

whence

Dj(log f) := Dj(f)

f=hj∈ OS.

Moreover, the minor ∆jobtained by removing the column corresponding to ∂/∂sj

and taking the determinant of the remaining square matrix equals ∂f /∂sjup to

multiplication by a unit, while the matrix of σfwith respect to the chosen bases

returns ftimes a unit.

In these terms, the vector ﬁelds D1, ..., Dmform a basis of the logarithmic vector

ﬁelds as a submodule of T0

S.

The commutative diagram in Figure 1yields as well Aleksandrov’s characteriza-

tion of free divisors.

2.13. Proposition ([Ale90]).A hypersurface germ V(f)⊂Sis a free divisor, if,

and only if, the singular1locus Σis Cohen–Macaulay of codimension 2in S.

Proof. Indeed, the codimension of Σ in Sis at least 2 if, and only if, fis squarefree,

that is, V(f) is reduced. On the other hand, DerS(−log f) is free if, and only if,

OΣis of projective dimension, and thus, of codimension at most 2.

To prepare for our main result, we record how Figure 1behaves with respect to

base change.

2.14. Lemma. Assume V(f)⊂Sis a free divisor and let ϕ:X→Sbe a morphism

from an analytic germ Xto S. If Xis Cohen–Macaulay and the inverse image

ϕ−1(Σ) of the singular locus Σof V(f)is still2of codimension 2, then the exact

row (†)in Figure 1pulls back to an exact sequence

0//ϕ∗DerS(−log f)ϕ∗(˜σf)

//T0

S(OX)⊕ OXα//OX//Oϕ−1(Σ) //0,

where α=ϕ[(Jac(f), f) = (Jac(f)ϕ, f ϕ).

If furthermore fϕ remains a non-zero-divisor in OX, then the pull back of Figure

1by ϕgives a diagram with exact rows and columns.

1The empty set has any codimension.

2Note that the codimension cannot go up under pullback.

LIFTING FREE DIVISORS 7

Proof. Let I=I(Σ) ⊂ OS.

Apply OX⊗OS– to the exact sequence (†) in Figure 1to get the pullback OX–

complex C. The last nonzero term of Cis ϕ∗(OΣ)∼

=OX/J for J=OX·ϕ[(I), and

this is the pullback of the scheme OΣ, as claimed. It is straightforward to check that

under the identiﬁcation of the two middle free modules in Cwith T0

S(OX)⊕ OX

and OXrespectively, the complex Cis the sequence given in the statement.

For i≥0, we have Hi(C) = TorOS

i(OX,OΣ), and so each homology module is

supported on ϕ−1(Σ). In particular, Jannihilates each Hi(C).

Since Xis Cohen–Macaulay, depth(J, OX) = codim(J) = 2, and similarly

depth(J, (OX)k) = 2 for k≥1. Thus, if C0=OX, C1, . . . are the free modules

in C, then depth(J, Ci)> i −1 for all i≥1. This fact, and the earlier observation

that J·Hi(C) = 0 for i≥1, are enough to ensure that Hi(C) = 0 for all i≥1 (see

[Bou07,§1.2, Corollaire 1]). Thus Cis exact.

The second assertion then follows.

3. The Main Results

3.1.Fix as before a smooth germ Sand let f∈ OSdeﬁne a free divisor V(f)⊂S

with singular locus Σ ⊂V(f). By our deﬁnition, fis reduced. In fact, reducedness

does not matter when computing the module of logarithmic vector ﬁelds for a

hypersurface.

3.2. Lemma ([HM93, p. 313], [GS06, Lemma 3.4]).If Xis smooth and h1, h2∈ OX

deﬁne the same zero loci as sets in X, then DerX(−log h1) = DerX(−log h2).

Proof. Let g∈ OXfactor into distinct irreducible components as g=gk1

1· · · gk`

`.

By an easy argument using the product rule and the fact that OXis a unique

factorization domain, DerX(−log g) = ∩iDerX(−log gi). The result follows.

3.3. Example. For g=gk1

1· · · gk`

`as in the proof, the logarithmic vector ﬁelds

satisfy DerX(−log g) = DerX(−log g1· · · g`).

We now give our main result, a suﬃcient condition for the reduction of fϕ to

deﬁne a free divisor in X.

3.4. Theorem. Let ϕ:X→Sbe a morphism of smooth germs and let f∈

OSdeﬁne a free divisor V(f)⊂Swith singular locus Σ⊂V(f). Assume that

Image(ϕ)*V(f), i.e., fϕ is not zero. Let gbe a reduction of fϕ in OX, a reduced

function deﬁning the same zero locus as fϕ. If

(a) the module of vertical vector ﬁelds T0

X/S is free,

(b) the Kodaira–Spencer map δK S :T0

S→T1

X/S vanishes on DerS(−log f), that is,

δKS ◦σf= 0, and

(c) the inverse image ϕ−1(Σ) of the singular locus is still of codimension 2in X,

then gdeﬁnes a free divisor in Xand its OX–module of logarithmic vector ﬁelds

satisﬁes

DerX(−log g) = DerX(−log fϕ)∼

=T0

X/S ⊕ϕ∗DerS(−log f).(3)

If Σ = ∅, then by our convention on the codimension of the empty set, (c) is

satisﬁed.

Proof. The three OX–linear maps:

8 RAGNAR-OLAF BUCHWEITZ AND BRIAN PIKE

•α=ϕ[(Jac(f), f ) : T0

S(OX)⊕ OX→ OXas above,

•β= Jac(ϕ)⊕idOX:T0

X⊕ OX→T0

S(OX)⊕ OX, and

•γ= (Jac(fϕ), f ϕ) : T0

X⊕ OX→ OX

satisfy γ=αβ and give rise to the following diagram relating kernels and cokernels

of these maps, where Iis the ideal generated by fϕ and its partial derivatives.

0

0

ϕ∗DerS(−log f)

ϕ∗˜σf

ω//T1

X/S

==

T0

S(OX)⊕ OX

α

δ+0

88

0//DerX(−log fϕ)˜σfϕ

//

77

T0

X⊕ OX

β

77

γ////OX//

OX/I //

xx

0

T0

X/S

OO77

Oϕ−1(Σ)

vv

0

99

0

OO

0 0

The horizontal exact sequence involving γis the one described in 2.10 and 2.12

that deﬁnes DerX(−log fϕ) and the singular locus of V(f ϕ) as a scheme.

The vertical sequence involving αis exact by (c) as explained in Lemma 2.14

above. As Xis smooth, T1

X= 0, and the diagonal exact sequence including βis

the direct sum of the identity on OXand (the initial segment of) the exact dual

Zariski–Jacobi sequence for ϕas recalled in 2.5 above.

Now observe that ω, the OX–linear map connecting ker(α) to coker(β), satisﬁes

ω=δ◦ϕ[(˜σf). Hence if D∈DerS(−log f)⊆T0

Sand 1⊗D∈ OX⊗OSDerS(−log f)

is the pulled-back vector ﬁeld in ϕ∗DerS(−log f), then ω(1 ⊗D) = δ◦ϕ[(˜σf)(1 ⊗

D) = δKS (D). Our assumption (b) is hence equivalent to ω= 0. Thus, the Ker–

Coker exact sequence deﬁned by γ=αβ splits into two short exact sequences for

the kernels, respectively cokernels,

0//T0

X/S //DerX(−log fϕ)//ϕ∗DerS(−log f)//0

and

0//T1

X/S //OX/I //Oϕ−1(Σ) //0.

Since DerS(−log f) is a free OS–module by assumption, and thus ϕ∗DerS(−log f)

is a free OX–module, it follows that the ﬁrst exact sequence splits, giving the

decomposition of DerX(−log f ϕ) in (3). By (a) and Lemma 3.2, DerX(−log g) =

DerX(−log fϕ) is a free OX–module and hence gdeﬁnes a free divisor.

As condition (a) of Theorem 3.4 can be diﬃcult to prove directly, it is often

easier to verify the following stronger hypotheses; in fact, only Example 8.10 applies

Theorem 3.4.

LIFTING FREE DIVISORS 9

3.5. Theorem. Let ϕ:X→Sbe a morphism of smooth germs and let f∈ OS

deﬁne a free divisor V(f)⊂Swith singular locus Σ⊂V(f). If both

(b) the Kodaira–Spencer map δK S :T0

S→T1

X/S vanishes on DerS(−log f), that is,

δKS ◦σf= 0, and

(d) T1

X/S is Cohen–Macaulay of codimension 2,

then fϕ is reduced and deﬁnes a free divisor, and DerX(−log f ϕ)has the decom-

position as in (3)of Theorem 3.4.

Proof. We check the conditions of Theorem 3.4. Condition (b) is assumed.

(d) implies (a), that T0

X/S is free, as it is a second syzygy module of T1

X/S via

the dual Zariski–Jacobi sequence for ϕ.

Since T1

X/S is supported on the critical locus C(ϕ) of the map, ϕis smooth oﬀ a

set of codimension 2. In particular, this implies that fϕ is nonzero: if fϕ = 0, so

Image(ϕ)⊆V(f), then ϕis nowhere smooth.

For (c), ﬁrst note that the codimension of ϕ−1(Σ) is ≤2, as the codimension

cannot go up under pullback. Let ϕ0and ϕ00 be the restriction of ϕto C(ϕ) and

its complement in X. Then ϕ−1(Σ) = (ϕ0)−1(Σ) ∪(ϕ00)−1(Σ), both of which have

codimension ≥2 in X: the ﬁrst is contained in C(ϕ), and the second because Σ

has codimension 2 in Sand ϕ00 is smooth. Thus we have (c).

By Theorem 3.4 and its proof, DerX(−log f ϕ) is free, with the decomposition

as in (3) and the exact sequence

0//T1

X/S //OX/I //Oϕ−1(Σ) //0,

where Iis generated by fϕ and its partial derivatives, so that OX/I =OSing(V(f ϕ)).

The outer terms T1

X/S and Oϕ−1(Σ) are Cohen–Macaulay OX–modules of codimen-

sion 2 by assumption (d) and (c), whence OX/I is also a Cohen–Macaulay OX–

module of codimension 2. Since codim(OX/I) = 2, fϕ is necessarily reduced and

hence deﬁnes a free divisor.

3.6. Remark. If Theorem 3.5 applies with S∼

=C2and f∈ OS, then the Theorem

produces many examples of free divisors in Xbecause any reduced plane curve in S

is a free divisor, and any such curve which has famong its components will satisfy

condition (b).

As a ﬁrst application we obtain a result originally observed by Mond and van

Straten [MvS01, Remark 1.5].

3.7. Theorem. Let Cbe the germ of an isolated complete intersection curve singu-

larity. If ϕ:X→Sis any versal deformation of C, then the union of the singular

ﬁbres of ϕ, that is, the pullback along ϕof the discriminant ∆⊂Sin the base, is

a free divisor.

More generally, if f= 0 deﬁnes a free divisor in Sthat contains the discriminant

as a component, then its pre-image f◦ϕ= 0 deﬁnes a free divisor in X.

Proof. It is well known (see [Loo84, 6.13, 6.12]) that ∆ is a free divisor in a smooth

germ S, that Xis smooth as well, and that T1

X/S is a Cohen–Macaulay OX–module

of codimension two. Finally, in this case the kernel of the Kodaira–Spencer map

δKS :T0

S→T1

X/S consists precisely of the logarithmic vector ﬁelds along ∆ (see

[BEGvB09]) and so all the assumptions of Theorem 3.5 are satisﬁed for ∆ itself

and then also for any free divisor in Sthat contains ∆ as a component.

10 RAGNAR-OLAF BUCHWEITZ AND BRIAN PIKE

3.8. Example. A versal deformation of the plane curve deﬁned by x3

1+x2

2is the

map ϕ:X=C3→S=C2deﬁned by ϕ(x1, x2, s1)=(s1, x3

1+x2

2+s1x1).

With coordinates (s1, s2) on S, the discriminant of ϕis the free divisor deﬁned by

∆=4s3

1+ 27s2

2. The module of liftable vector ﬁelds is DerS(−log ∆). By Theorem

3.7,

∆ϕ= 27x6

1+ 54x3

1x2

2+ 54x4

1s1+ 27x4

2+ 54x1x2

2s1+ 27x2

1s2

1+ 4s3

1

deﬁnes a free divisor on X, and the same is true for the lift of any reduced plane

curve containing ∆ as a component, for example (∆ ·s2)ϕ. Note that ∆ϕis equiv-

alent to the classical swallowtail.

3.9. Remark. Note that Theorem 3.7 can only hold for versal deformations of iso-

lated complete intersection singularities on curves. Indeed, for a versal deformation

of any isolated complete intersection singularity the corresponding module T1

X/S is

Cohen–Macaulay, but of codimension equal to the dimension of the singularity plus

one ([Loo84, 6.12]).

4. Lifting Euler vector fields

Theorem 3.5 requires that all elements of DerS(−log f) lift. This hypothesis

may be relaxed, at least for the Euler vector ﬁeld of a weighted-homogeneous free

divisor. We ﬁrst examine how general Theorem 3.5 is in a well-understood situation.

4.1. Example. Suppose that ϕ:X=Cn→S=C(and hence f◦ϕfor f=s1)

already deﬁnes a free divisor. What is the content of Theorem 3.5 in this case?

Here, T1

X/S ∼

=coker Jac(ϕ)∼

=OX/Jϕ, where Jϕis the Jacobian ideal generated

by the partial derivatives of ϕ. If ϕ∈Jϕ, equivalently, there exists an “Euler-like”

vector ﬁeld ηsuch that η(ϕ) = ϕ, then T1

X/S is Cohen-Macaulay of codimension

2 by Proposition 2.13 as ϕdeﬁnes a free divisor. Moreover, the vector ﬁeld s1∂

∂s1

that generates DerS(−log s1) lifts if and only if ϕ∈Jϕ. Hence, the hypotheses of

Theorem 3.5 are satisﬁed exactly when ϕ∈Jϕ, in which case the conclusion says

that DerX(−log ϕ) is the direct sum of OX·ηand the (vertical) vector ﬁelds that

annihilate ϕ.

A free divisor without an Euler-like vector ﬁeld does not have this direct sum

decomposition. Hence, as this Example suggests, we may weaken the lifting condi-

tion of Theorem 3.5, modify the algebraic condition, and obtain a conclusion that

lacks the direct sum decomposition as in (3) of Theorem 3.4.

4.2. Corollary. Let ϕ:X→Sbe a morphism of smooth germs with module

L= ker(δKS )⊆T0

Sof liftable vector ﬁelds. Let f∈ OSdeﬁne a free divisor

with singular locus Σ⊂V(f). Let (w1, . . . , wm)be a set of nonnegative integral

weights for the coordinates (s1, . . . , sm)on S. Let E=Pm

i=1 wisi∂

∂si∈T0

Sbe the

corresponding Euler vector ﬁeld, so that T0

S(ϕ[)(E) = Pm

i=1 wi(si◦ϕ)∂

∂si∈T0

S(OX).

If fis weighted homogeneous of degree dwith respect to these weights, if

(4) N=T0

S(OX)/(Image(Jac(ϕ)) + OX·T0

S(ϕ[)(E))

is a Cohen-Macaulay OX–module of codimension 2, and if DerS(−log f)⊆L+

OS·E, then f◦ϕdeﬁnes a free divisor.

LIFTING FREE DIVISORS 11

Proof. Let tbe a coordinate on C, and let ϕ= (ϕ1, . . . , ϕm). Deﬁne θ:Y=

X×C→Sby θ(x, t) = (ew1t·ϕ1(x), . . . , ewmt·ϕm(x)). Since

θ[(f)(x, t) = f(ew1t·ϕ1(x), . . . , ewmt·ϕm(x))

=edt ·ϕ[(f)(x),

and edt is a unit in OY, if Theorem 3.5 applies, then the lift of fvia θwill give a

free divisor V(f◦ϕ)×Cin Y. It follows that f◦ϕdeﬁnes a free divisor in X. It

remains only to check the hypotheses of the Theorem.

A matrix representation of Jac(θ) is

(5)

ew1t∂ϕ1

∂x1· · · ew1t∂ϕ1

∂xnw1ew1tϕ1

.

.

.....

.

..

.

.

ewmt∂ϕm

∂x1· · · ewmt∂ϕm

∂xnwmewmtϕm

,

with values in T0

S(OY). The isomorphism ψ:T0

S(OY)→T0

S(OY) with ψ(∂

∂si) =

e−wit∂

∂sishows that deleting the exponential coeﬃcients in (5) gives an isomorphic

cokernel. Thus, T1

Y/S ∼

=coker Jac(θ) is isomorphic to N⊗OXOY, and hence a

Cohen–Macaulay OY–module of codimension 2. This establishes condition (d) of

Theorem 3.5.

Now let η=Pm

i=1 ai∂

∂si∈T0

Sbe homogeneous of degree λ, in that λ= deg(ai)−

wifor i= 1, . . . , m. Suppose that ηlifts under ϕto some ξ=Pn

j=1 bj∂

∂xj∈T0

X,

so that ai◦ϕ=Pn

j=1 bj∂ϕi

∂xjfor i= 1, . . . , m. Let ξ0∈T0

Yhave the same deﬁning

equation. Then

Jac(θ)eλt ·ξ0=

m

X

i=1

e(λ+wi)t

n

X

j=1

bj

∂ϕi

∂xj

∂

∂si

=

m

X

i=1

edeg(ai)t·(ai◦ϕ)∂

∂si

=

m

X

i=1

ai◦ew1t·ϕ1, . . . , ewmt·ϕm∂

∂si

=T0

S(θ[)(η).

Thus, homogeneous elements of Llift via θ. The Euler vector ﬁeld Ealso lifts,

as Jac(θ)( ∂

∂t ) = T0

S(θ[)(E). It follows that the module generated by homogeneous

elements of L+OSElifts via θ. Since fis weighted homogeneous, DerS(−log f)

has a homogeneous generating set and hence elements of DerS(−log f) lift via θ,

verifying condition (b) of Theorem 3.5.

4.3.This corollary may create free divisors without an Euler–like vector ﬁeld, and

may be applied to maps between spaces of the same dimension.

4.4. Example. Let ϕ:X=C3→S=C2be deﬁned by ϕ(x1, x2, x3)=(x2

1+

x3

2, x2

2+x1x3), and let f=s1s2(s1+s2). Let Lbe the module of vector ﬁelds liftable

through ϕ. Although T1

X/S is Cohen-Macaulay of codimension 2, DerS(−log f)*

L. For weights w1=w2= 1, we have DerS(−log f)⊆L+OS·E, and the module

12 RAGNAR-OLAF BUCHWEITZ AND BRIAN PIKE

of (4) is also Cohen–Macaulay of codimension 2. By Corollary 4.2,

f◦ϕ= (x2

1+x3

2)(x2

2+x1x3)(x2

1+x3

2+x2

2+x1x3)

deﬁnes a free divisor; it has no Euler-like vector ﬁeld.

4.5. Example. Let ϕ:X=C3→S=C3be deﬁned by ϕ(x1, x2, x3)=(x1x3+

x2

2, x2, x3). For w1=w2=w3= 1 the module of (4) is Cohen–Macaulay of

codimension 2, although T1

X/S is not. As L+OSEcontains DerS(−log f) for, e.g.,

f=s1s2s3or f=s1s3(s1s3−s2

2), by Corollary 4.2 each such f◦ϕdeﬁnes a free

divisor in X.

4.6. Remark. If fis multi-weighted homogeneous, that is, weighted homogeneous

of degree dkwith respect to weights (w1k, . . . , wmk) for k= 1, . . . , p (or, f= 0

is invariant under the action of an algebraic p-torus), then a version of Corollary

4.2 holds, with Ereplaced by the pEuler vector ﬁelds. To adapt the proof, let

θ:X×Cp→Sbe deﬁned by

θ(x, t) = ePp

k=1 w1ktk·ϕ1(x),· · · , ePp

k=1 wmktk·ϕm(x),

for ϕ= (ϕ1, . . . , ϕm), and show that multi-weighted homogeneous vector ﬁelds lift

and generate DerS(−log f).

For instance, if f=s1· · · smis the normal crossings divisor in S=Cmwith m

weightings of the form (0,· · · ,1,· · · ,0), then the analog of the module Nof (4) is

the cokernel of

A=

∂ϕ1

∂x1· · · ∂ϕ1

∂xnϕ10· · · 0

∂ϕ2

∂x1· · · ∂ϕ2

∂xn0ϕ2· · · 0

.

.

.....

.

..

.

..

.

.....

.

.

∂ϕm

∂x1· · · ∂ ϕm

∂xn0 0 · · · ϕm

.

When each ϕiis nonzero, then ker(A)∼

=∩iDerX(−log ϕi) = DerX(−log ϕ1· · · ϕm) =

DerX(−log ϕ[(f)); the conditions on Nensure that ker(A) is free, and ϕ1· · · ϕmis

reduced and nonzero.

4.7. Remark. A result similar to Corollary 4.2 may be obtained by applying The-

orem 3.4 instead of Theorem 3.5.

5. Adding components and dimensions

We now examine a way to add components to a free divisor on Cmto produce a

free divisor on Cm×Cn. Use coordinates (x1, . . . , xm) and (y1, . . . , yn) on Cmand

Cnrespectively.

For an OX–ideal Ion a smooth germ X, deﬁne the OX–module of logarithmic

vector ﬁelds by

DerX(−log I) = {η∈DerX:η(I)⊆I}.

This agrees with our earlier deﬁnition for hypersurfaces.

5.1. Proposition. Let I= (g1, . . . , gn)be a OCm–ideal such that OCm/I is Cohen-

Macaulay of codimension 2. If h∈ OCmdeﬁnes a free divisor on Cmwith

(6) DerCm(−log h)⊆DerCm(−log I),

then h·(Pn

i=1 giyi)deﬁnes a free divisor on X=Cm×Cn.

LIFTING FREE DIVISORS 13

Proof. Let S=Cm×Chave coordinates (z1, . . . , zm, t) and view giand has

elements of OS. Deﬁne ϕ:X→Sby ϕ(x, y)=(x, Pn

i=1 gi(x)·yi). Let f(z, t) =

h(z)·tdeﬁne the free divisor in Swhich is the “product-union” of V(h)⊂Cmand

{0} ⊂ C. The statement will then follow from Theorem 3.5 by lifting fvia ϕ.

To check condition (d) of the Theorem, observe that with respect to the coordi-

nates given, the matrix form of the Jacobian is

Jac(ϕ) = Im,m 0m,n

∗g1· · · gn,

where the subscripts on Iand 0 denote the sizes of identity and zero blocks re-

spectively. In particular, T1

X/S ∼

=coker Jac(ϕ) is isomorphic to OX/(I⊗OCmOX)∼

=

(OCm/I)⊗OCmOX. Since OCm/I is a Cohen-Macaulay OCm-module of codimension

2, by ﬂatness it follows that T1

X/S is a Cohen-Macaulay OX-module of the same

codimension.

For (b), Der(−log f) is generated by elements of Der(−log h) extended to S

with 0 as the coeﬃcient of ∂

∂t , together with t∂

∂t . The latter lifts:

Jac(ϕ) n

X

i=1

yi

∂

∂yi!= n

X

i=1

giyi!∂

∂t =T0

S(ϕ[)t∂

∂t .

Now, if η=Pm

i=1 ai∂

∂zi∈DerCmis logarithmic for I, then there exist γj,k ∈ OCm

such that η(gj) = Pn

k=1 γj,k ·gkfor all j. Then ηextended to Slifts as well:

Jac(ϕ)

m

X

i=1

ai

∂

∂xi

−

n

X

j,k=1

γj,kyj

∂

∂yk

=

m

X

i=1

ai

∂

∂zi

+

n

X

j=1

∂gj

∂zi

yj

∂

∂t

−

n

X

j,k=1

γj,kgkyj

∂

∂t

=

m

X

i=1

ai

∂

∂zi

+

m

X

i=1

n

X

j=1

ai

∂gj

∂zi

yj

∂

∂t −

n

X

j=1

η(gj)yj

∂

∂t

=

m

X

i=1

ai

∂

∂zi

+

n

X

j=1

η(gj)yj

∂

∂t −

n

X

j=1

η(gj)yj

∂

∂t

=T0

S(ϕ[)(η).

In view of assumption (6), thus all generators of Der(−log f) lift.

5.2. Remark. By the form of Jac(ϕ) in the proof, the vertical vector ﬁelds of ϕ

are generated by the OCm–syzygies of {g1, . . . , gn}, and thus form a free module as

OCm/I is Cohen–Macaulay of codimension 2.

5.3. Remark. There is no need for (g1, . . . , gn) to be a minimal generating set.

5.4. Remark. The conclusion of Proposition 5.1 also holds if I= (1). Then some

giis a unit in the local ring, and so a local change of coordinates of Xtakes

h·(Pn

i=1 giyi) to h·y1, which deﬁnes a “product-union” of free divisors.

14 RAGNAR-OLAF BUCHWEITZ AND BRIAN PIKE

5.5.To ﬁnd an hand Ithat satisfy assumption (6), a natural approach is to use the

ideal (Jh, h) deﬁning the singular locus Σ of V(h). In particular, we have the follow-

ing generalization of the “ff∗” construction of Buchweitz–Conca ([BC13, Theorem

8.1]), where we have removed the hypothesis that hbe weighted homogeneous.

5.6. Corollary. If h∈ OCmdeﬁnes a free divisor on Cmand g1, . . . , gngenerate

the OCm–ideal I= (Jh, h), then h·(Pn

i=1 giyi)deﬁnes a free divisor on Cm×Cn. In

particular, h·hym+1 +Pm

i=1 ∂h

∂xiyialways deﬁnes a free divisor on Cm×Cm+1,

and if h∈Jhthen h·Pm

i=1 ∂h

∂xiyideﬁnes a free divisor on Cm×Cm.

Proof. It is enough to prove the ﬁrst assertion, as the rest follows from it. Let Σ

be the singular locus of V(h), deﬁned by I. If V(h) is smooth, then I= (1) and we

may apply Remark 5.4. Otherwise, OCm/I is Cohen–Macaulay of codimension 2 by

Proposition 2.13. Any vector ﬁeld that is logarithmic to V(h) is also logarithmic

to I, as is easily seen from the chain rule. Now apply Proposition 5.1.

However, this is not the only way to ﬁnd a satisfactory hand I.

5.7. Example. Let M=Mn,n be the space of n×ncomplex matrices with co-

ordinates {xij }, let N=Mn−1,n, and let π:M→Nbe the projection that

deletes the last row. Diﬀerentiate ρ: GL(n−1,C)×GL(n, C)→GL(N) deﬁned

by ρ(A, B)(X) = AXB−1to obtain a ﬁnite-dimensional Lie algebra gof linear

vector ﬁelds on N. Let D⊂DerNbe the ON–submodule generated by g. Let f

deﬁne a free divisor on Nfor which DerN(−log f)⊆D; for instance, fcould be a

linear free divisor on Nobtained by restricting ρto an appropriate subgroup.

Now ρleaves invariant N0={X: rank(X)< n−1} ⊂ N, and hence all elements

of g,D, and DerN(log f) are tangent to the variety N0. Note that N0is Cohen-

Macaulay of codimension 2 and deﬁned by I= ((−1)n+1g1,...,(−1)n+ngn), where

gi:N→Cdeletes column iand takes the determinant. Since Pn

i=1(−1)n+igixni =

det on M, by Proposition 5.1, (f◦π)·det deﬁnes a free divisor on M. By the lifts in

the proof and the observation that the vertical vector ﬁelds are generated by linear

vector ﬁelds (e.g., by Hilbert–Burch), we see that if fdeﬁnes a linear free divisor

on Nthen (f◦π)·det deﬁnes a linear free divisor on M. (This linear free divisor

case partially recovers [Pik10, Prop. 5.3.7].)

As a concrete example, for the linear free divisor on M2,3deﬁned by

f=x11x12

x11 x12

x21 x22

x12 x13

x22 x23

,

(f◦π)·det deﬁnes a linear free divisor Don M3,3, part of the “modiﬁed LU”

series of [DP12,Pik10]. In fact, Dmay be constructed from {x11 = 0} ⊂ M1,1

by repeatedly applying Proposition 5.1, as, e.g., DerM2,2(−log(x11 x12(x11 x22 −

x12x21 ))) ⊆DerM2,2(−log(x12, x22 )).

6. The case of maps ϕ:Cn+1 →Cn

We now show that for germs ϕ:X=Cn+1 →S=Cnwith critical set of

codimension 2, the OX–module T1

X/S is Cohen–Macaulay of codimension 2. In

fact, this is the idea behind Theorem 3.7, about the versal deformations of isolated

complete intersection curve singularities.

LIFTING FREE DIVISORS 15

6.1. Proposition. Let X=Cn+1,S=Cn, and let ϕ:Cn+1 →Cnbe holomorphic

with critical set C(ϕ)⊆Cn+1. If C(ϕ)is nonempty and has codimension 2, then

T1

X/S is a Cohen-Macaulay OX-module of codimension 2. The vertical vector ﬁelds

form the free OX–module of rank 1generated by η=Pn+1

i=1 (−1)idi∂

∂xi, where diis

the determinant of Jac(ϕ)with column ideleted.

Proof. [Loo84, Proposition 6.12] uses the Buchsbaum–Rim complex to prove that

for g:Cp→Cr,p≥r, if C(g) has the expected dimension r−1, then coker(Jac(g))

is a Cohen–Macaulay OCp–module of dimension r−1.

Thus, in the case at hand, T1

X/S ∼

=coker(Jac(ϕ)) is Cohen–Macaulay of codi-

mension 2, and the Buchsbaum–Rim complex for V1Jac(ϕ) = Jac(ϕ) is exact and

of the form

(7) 0 //OX

·η //(OX)n+1 Jac(ϕ)

//(OX)n//T1

X/S //0,

where = (−1)(n+2

2). Hence T0

X/S is the free module generated by η.

6.2. Example. Let ϕ:C3→C2be deﬁned by ϕ(x1, x2, x3)=(x2

1+x3

2, x2

2+x1x3).

The critical locus V(x1, x2x3) has codimension 2, and the discriminant is the plane

curve deﬁned by ∆ = s2

1−s3

2. A Macaulay2 [GS] computation shows that the

liftable vector ﬁelds are exactly DerS(−log ∆). By Proposition 6.1 and Theorem

3.5, we conclude that ϕ−1(∆) is a free divisor deﬁned by

∆ϕ=x1(−3x4

2x3−3x1x2

2x2

3−x2

1x3

3+ 2x1x3

2+x3

1).

A generating set of DerX(−log ∆ϕ) consists of lifts of a generating set of DerS(−log ∆),

and the vertical vector ﬁeld −3x1x2

2∂

∂x1+ 2x2

1∂

∂x2−(4x1x2−3x2

2x3)∂

∂x3.

6.3. Example. Let ϕ:C4→C3be deﬁned by ϕ(x1, x2, x3, x4)=(x1x3, x2

2−

x3

3, x2x4). The critical locus C(ϕ) has codimension 2, and so by Proposition 6.1

the module T1

X/S is Cohen-Macaulay of codimension 2. Although the module of

all liftable vector ﬁelds is not free, thus not associated to a free divisor, each si∂

∂si,

i= 1,2,3, is liftable. Hence, any free divisor in C3containing the normal crossings

divisor s1s2s3= 0 will lift via ϕto a free divisor in C4.

7. Coregular and Cofree Group Actions

For a reductive linear algebraic group Gacting on X, we now consider the

algebraic quotient ϕ:X→S=X//G.

Castling. Our initial example is related to the classical castling of prehomogeneous

vector spaces.

7.1.Let G= SL(n, C) act on the aﬃne space V=Mn,n+1 of n×(n+ 1) matrices

over Cby left multiplication. Use coordinates {xij : 1 ≤i≤n, 1≤j≤n+ 1}for

V, and let ∆ibe (−1)itimes the n×nminor obtained by deleting the ith column

of the generic matrix (xij ). The quotient space V//G is then again smooth and

the corresponding invariant ring R=C[V]Gis the polynomial ring on the n×n

minors {∆i}i=1,...,n+1 (e.g., [VP94,§9.3,9.4]). In particular, dim R=n+ 1, and

the quotient map ϕ:V→V//G is smooth outside the null cone ϕ−1(0) that in

turn is the determinantal variety deﬁned by the vanishing of the maximal minors

of the generic matrix, thus, Cohen–Macaulay of codimension 2.

16 RAGNAR-OLAF BUCHWEITZ AND BRIAN PIKE

7.2. Theorem. Let f∈ OSdeﬁne a free divisor in S=Cn+1 that is not suspended,

equivalently [GS06],DerS(−log f)⊆mST0

S. Then f(∆1, ..., ∆n+1)deﬁnes a free

divisor on Cn(n+1).

Proof. Let X=V∼

=Cn(n+1), S =V//G ∼

=Cn+1 and let ϕ:X→Sbe the natural

morphism, smooth oﬀ the codimension 2 null cone ϕ−1(0).

That the Kodaira–Spencer map restricted to the logarithmic vector ﬁelds along

fvanishes is due to our assumption that DerS(−log f)⊆mST0

Sand the fact that

we can exhibit lifts of a generating set of mST0

S. Indeed, a computation shows that

for 1 ≤p, q ≤n+ 1 with p6=qand any 1 ≤r≤n,

Jac(ϕ) −

n

X

i=1

xiq

∂

∂xip != ∆p

∂

∂sq

=T0

S(ϕ[)sp

∂

∂sq

Jac(ϕ)

n+1

X

j=1

xrj

∂

∂xrj

−

n

X

i=1

xiq

∂

∂xiq

= ∆q

∂

∂sq

=T0

S(ϕ[)sq

∂

∂sq.

(8)

(In each case, Jac(ϕ) applied to the sum over igives a sum where the coeﬃcient

of ∂

∂skis of the form Pn

i=1 xiq ∂∆k

∂xip , which simpliﬁes to ±∆p,±∆k, or 0, depending

on p, q, k. Applying Jac(ϕ) to the sum over jgives Pn+1

k=1 ∆k∂

∂sk, as each minor

is linear in row r. Or, see 7.17.) This shows that condition (b) of Theorem 3.5 is

satisﬁed.

It suﬃces to establish condition (d). This will follow from the dual Zariski–Jacobi

sequence, once we show that the OX–module T0

X/S of vertical vector ﬁelds along

the map ϕis free. However, the Lie algebra slnacts through derivations on OX,

deﬁning a OX–linear map sln⊗OX→T0

X/S . This map is an isomorphism outside

the null cone, as the smooth ﬁbres there are regular orbits for the SL(n, C)–action.

Now both source and target of the exhibited map are reﬂexive OX–modules and

the map is an isomorphism outside the null cone of codimension 2, whence it must

be an isomorphism everywhere.

7.3. Remark. Two types of vector ﬁelds on Mn,n+1 generate Der(−log fϕ). The

ﬁrst are lifts of a generating set of Der(−log f), which may be found using (8).

The second are the linear vector ﬁelds arising from the SL(n, C) action on Mn,n+1;

these generate the module T0

X/S of vertical vector ﬁelds. Note that this is a minimal

generating set, and that if the generators of Der(−log f) are linear vector ﬁelds then

Der(−log fϕ) is also generated by linear vector ﬁelds.

7.4. Example. The normal crossings divisor in S=Cn+1 is the linear free divisor

deﬁned by f=s1· · · sn+1 = 0. By Theorem 7.2, this pulls back to the linear free

divisor fϕ = ∆1· · · ∆n+1 = 0, previously seen in [BM06, 7.4]. A generating set of

Der(−log fϕ) consists of the n2−1 vector ﬁelds arising from the SL(n, C) action on

Mn,n+1, and lifts (as in (8)) of the n+ 1 generators nsi∂

∂sion+1

i=1 of DerS(−log f).

7.5. Example. Let f= 0 be a reduced deﬁning equation of a free surface in C3

which is not suspended. Such free surfaces exist in abundance, see, for example,

[Dam02,Sek09]. Pulling back fvia ϕ:M2,3→M2,3

// SL(2,C)∼

=C3produces the

free divisor

f(−(x12x23 −x13 x22),(x11 x23 −x13x21 ),−(x11x22 −x12 x21)) = 0

LIFTING FREE DIVISORS 17

in M2,3. For instance, f=s1(s1s3−s2

2) pulls back to the linear free divisor

(x12x23 −x13 x22)·(−x12 x23x11 x22 +x2

12x23 x21 +x13x2

22x11 −x13 x22x12 x21

+x2

11x2

23 −2x11x23 x13x21 +x2

13x2

21)=0.

7.6.The classical castling construction relates a representation ρof a group G

on Mn,m,m<n, to a representation ρ0of some G0on Mn,n−m, and vice versa.

Then ρhas a Zariski open orbit if and only if ρ0has a Zariski open orbit, and the

hypersurface component of the complement of each is deﬁned by a homogeneous

polynomial (H, respectively, H0) in the respective generic maximal minors (§2.3 of

[GMS11]). There is a bijection between the maximal minors of Mn,m and Mn,n−m

deﬁned by replacing a m×mminor ∆Ion Mn,m with the (n−m)×(n−m) minor

∆0

Ion Mn,n−mformed by using the complementary set of rows and an appropriate

sign. As polynomials in the minors, via this correspondence Hand H0are the same

up to multiplication by a unit.

Castling sends linear free divisors to linear free divisors by Proposition 2.10(4)

of [GMS11]. For arbitrary free divisors, our Theorem 7.2 addresses the n=m+ 1

situation (in one direction), and it is reasonable to ask whether it holds more

generally for arbitrary (n, m). One diﬃculty is that there is generally no morphism

between Mn,m and Mn,n−mwhich sends ∆Ito ∆0

I, or vice-versa, and hence it is

unclear how to lift vector ﬁelds, or even what this means. In the classical situation,

an underlying representation θof a group Hon a n-dimensional space is used in

the construction of both ρand ρ0, and so gives a correspondence between the vector

ﬁelds generated by the action of θon the two spaces.

The general situation remains mysterious:

7.7. Example. For (n, m) = (5,2), let ∆ij denote the minor on M5,2obtained by

using only rows iand j. A calculation using the software Macaulay2 or Singular

shows that ∆14∆15(∆14 ∆25 −∆15∆24 )(∆34∆45 −∆2

35) = 0 deﬁnes a (non-linear)

free divisor on M5,2. Another computation shows that the corresponding divisor on

M5,3is not free. It is unclear what additional hypotheses are necessary to generalize

Theorem 7.2.

Group actions. We now generalize the ideas behind Theorem 7.2 to the case when

ϕ:X→Sis given by the quotient of Xunder a group action. We work now in the

algebraic category of schemes of ﬁnite type over C. Recall the following deﬁnitions.

7.8. Deﬁnition. If Gis any reductive complex algebraic group, then a ﬁnite di-

mensional linear representation Vis

(a) coregular if the quotient space V//G is smooth;

(b) cofree, if further the natural projection ϕ:V→V//G is ﬂat, equivalently (see

[VP94, 8.1]), ϕ:V→V//G is coregular and equidimensional in that all ﬁbres

have the same dimension;

(c) coreduced , if the null cone ϕ−1(0) is reduced.

In algebraic terms, with C[V] the ring of polynomial functions, coregularity

means that the ring of invariants R=C[V]Gis again a polynomial ring, while

cofreeness means that further C[V] is free as an R–module (e.g., [VP94,§8.1]3).

3The reference there for the algebraic result needed to justify this interpretation of cofreeness

is incorrect, and should be Bourbaki’s Groupes et Alg`ebres de Lie, Chap. V, §5, Lemma 1.

18 RAGNAR-OLAF BUCHWEITZ AND BRIAN PIKE

If R=C[f1, ..., fd] is the polynomial ring over the indicated invariant functions

fj∈C[V], then in the cofree case these functions form a regular sequence in C[V].

7.9. Remark. A famous conjecture by Popov suggests that equidimensionality of

(the ﬁbres of) the projection ϕ:V→V//G already implies coregularity and then

automatically cofreeness for Gconnected semi-simple.

There are many examples of cofree representations, and even more that are coreg-

ular. We just mention Kempf’s basic result that a representation is automatically

cofree whenever dim V//G 62; see [VP94, Thm.8.6] or [Kem80]. For further lists

of such representations see [Sch79,Lit89,Weh93].

7.10. Remark. In the case of Theorem 7.2 above, the action of SL(n, C) on Mn,n+1

is coregular, but not cofree.

7.11.To apply our main theorems to the quotient X→Sof a coregular represen-

tation, T0

X/S must be free. There is a straightforward suﬃcient criterion for the

stronger condition that T1

X/S is Cohen–Macaulay of codimension 2.

7.12. Proposition. Let X=Vbe a coregular representation of the reductive com-

plex algebraic group Gwith Lie algebra gand quotient S=V//G. If the generic

stabilizer of Gon Xis of dimension 0and the natural morphism ϕ:X→Sis

smooth outside a set of codimension 2in X, then

(i) The natural OX–homomorphism g⊗ OX→T0

X/S is an isomorphism;

(ii) T1

X/S is a Cohen–Macaulay OX–module of codimension 2.

Proof. ϕis smooth outside of a set of codimension 2 in Xand T1

X/S is supported

on the critical locus of ϕ, so codim(supp T1

X/S )≥2, or dim(T1

X/S )≤dim(X)−2.

Since the generic stabilizer of Gon Xis of dimension zero, thus, a ﬁnite group,

the OX–homomorphism ρ:g⊗ OX→T0

X/S is an inclusion. On the set in X

where ϕis smooth, ρis also locally surjective. As g⊗ OXis free and T0

X/S is a

second syzygy module (by the dual Zariski–Jacobi sequence), ρis a homomorphism

between reﬂexive modules which is an isomorphism oﬀ a set of codimension ≥2,

and hence ρis an isomorphism. This proves (i).

By (i) and the dual Zariski–Jacobi sequence, projdimOXT1

X/S ≤2. By the

Auslander–Buchsbaum formula and the usual relation between depth and dimen-

sion,

dim(X)−2≤depth(T1

X/S )≤dim(T1

X/S ).

As dim(T1

X/S )≤dim(X)−2, T1

X/S is Cohen–Macaulay of codimension 2.

There are coregular representations for which T0

X/S of the quotient is free, but

T1

X/S is not Cohen–Macaulay of codimension 2 (e.g., Example 8.10). Our next result

gives some insight into these cases, and also gives a necessary numerical condition

for Proposition 7.12 to apply.

7.13. Proposition. Let X=Vbe a coregular representation of the reductive com-

plex algebraic group Gwith Lie algebra gand quotient S=V//G. Let N= dim(X),

d= dim(S), and let δ1, . . . , δd>1be the degrees of the generating invariants. If

the natural OX–homomorphism g⊗ OX→T0

X/S is an isomorphism, then either

•N=Pd

ν=1 δνand dim(T1

X/S ) = N−2, or

LIFTING FREE DIVISORS 19

•N6=Pd

ν=1 δνand dim(T1

X/S ) = N−1.

Proof. If T0

X/S is free and generated by the group action, then the dual Zariski–

Jacobi sequence provides a graded free resolution of the graded OX–module T1

X/S

of the form

(9)

0//O⊕(N−d)

X//⊕N

ν=1OX(1) //⊕d

ν=1OX(δν)//T1

X/S //0.

First, (9) implies projdimOX(T1

X/S )≤2, and then the Auslander–Buchsbaum for-

mula shows dim(T1

X/S )≥dim(X)−2. Also by (9), the Hilbert–Poincar´e series of

T1

X/S satisﬁes

HT1

X/S (t) = 1

(1 −t)N d

X

ν=1

t−δν−Nt−1+N−d!

=N−Pd

ν=1 δν+ (1 −t)p(t, t−1)

(1 −t)N−1

for some Laurent polynomial p(t, t−1)∈Z[t, t−1]. In particular, HT1

X/S has a pole

at t= 1 of order N−1 if and only if N6=Pd

ν=1 δν. Finally, the order of this pole

equals dim(T1

X/S ).

7.14. Remark. If we inspect the tables of cofree irreducible representations of sim-

ple groups in [VP94], we check readily that when the generic stabilizer is ﬁnite, the

equation dim(X) = Pd

ν=1 δνis satisﬁed. However, the tables of cofree irreducible

representations of semisimple groups in [Lit89] show that this is not automatic;

for instance (in the notation there), the representation ω5+ω0

1of B5+A1has

dim(X) = 64 and (δi) = (2,4,6,8,8,12), which falls 64 −40 = 24 short.

7.15.To prove that vector ﬁelds lift across ϕ:X→X//G, the following technique

is sometimes useful.

7.16. Lemma. Let X=Vbe a coregular representation of the algebraic group G

with quotient S=V//G. Let ρXand ρSbe representations of an algebraic group H

on X, respectively, S. If ϕ:X→Sis equivariant with respect to the action of H,

then all vector ﬁelds on Sobtained by diﬀerentiating ρSlift across ϕ.

Proof. Diﬀerentiating gives representations dρXand dρSof h, the Lie algebra of

H, as Lie algebras of vector ﬁelds on X, respectively, S. Since ϕis equivariant, for

each Y∈h,dρX(Y) is ϕ-related to dρS(Y), and hence dρS(Y) lifts to dρX(Y).

7.17. Example. This argument may be used in the castling situation of Theorem

7.2. There, GL(n+ 1,C) has representations ρXand ρSon X=Mn,n+1 and

S=M1,n+1 deﬁned by

ρX(A)(X) = XATρS(A)(Y) = Yadj(A) = Ydet(A)A−1,

where adj(A) is the adjugate of A. (If Mn,n+1 'V⊗W, with dim(V) = n,

dim(W) = n+ 1, and M1,n+1 'C⊗W∗, then ρXis a representation on W

and ρSis the contragredient representation of ρX.) A calculation shows that ϕ

is equivariant with respect to ρXand ρS. Since dρSproduces a generating set of

20 RAGNAR-OLAF BUCHWEITZ AND BRIAN PIKE

DerS(−log{0}), any η∈DerS(−log{0}) will lift. (Note that the lifts in (8) have

been simpliﬁed.)

7.18.We now investigate a method for determining the generating invariants of

lowest degree. First we observe that if Jac(ϕ) was known, then it would be easy to

determine a generating set of invariants.

7.19. Proposition. Let X=Vbe a coregular representation of G, and let ϕ:X→

S=V//G with N= dim(X)and d= dim(S). If E=PN

i=1 xi∂

∂xi∈T0

Xis the

Euler vector ﬁeld and we write

Jac(ϕ)(E) =

d

X

j=1

˜

fj

∂

∂sj

∈T0

S(OX),

then the coeﬃcient functions ˜

fjform a generating set of the invariants in C[V]G⊆

C[V].

Proof. Observe that ϕ[:OS=C[s1, . . . , sd]→C[f1, . . . , fd] = C[V]G⊆C[V] is the

canonical inclusion, that is, ϕ[(sj) = fj. Since each fjis homogeneous, we have

Jac(ϕ)(E) =

d

X

j=1 N

X

i=1

xi

∂(sj◦ϕ)

∂xi!∂

∂sj

=

d

X

j=1

deg(fj)fj

∂

∂sj

.

Now each deg(fj)>0, and we are in characteristic zero, so that the functions

˜

fj= deg(fj)fjalso form a generating set of invariants.

7.20.We now describe a way to compute the OX–ideal generated by the invariants,

even without knowledge of the invariants. From this ideal we may recover the

invariants of lowest degree.

7.21. Proposition. Let X=Vbe a coregular representation of Gwith ﬁnite generic

stabilizer, and let ϕ:X→S=X//G, with N= dim(X). Let f1, . . . , fdbe

generating invariants, and let J= (f1, . . . , fd)OX. Let K⊆(OX)Nbe the OX–

module of b= (bi)such that PN

i=1 biai= 0 for any linear vector ﬁeld PN

i=1 ai∂

∂xi

on Xarising from the action of the Lie algebra gof G. Let Ibe the OX–ideal

consisting of PN

i=1 bixi, where (bi)∈K. If T1

X/S is a Cohen–Macaulay OX–module

of codimension 2, then J=I.

Proof. For a homogeneous invariant g∈ OX, ( ∂g

∂xi)∈K, and hence g∈I. It

follows that J⊆I.

Since dim(T1

X/S ) is the dimension of the critical locus, ϕis smooth oﬀ a set

of codimension 2. By Proposition 7.12,T0

X/S is generated by the action θ:g→

gl(V)∼

=V⊗V∨of g. As OX=C[V], the ﬁrst map ρin the dual Zariski–Jacobi

sequence

0//g⊗ OX

ρ//T0

X

Jac(ϕ)

//T0

S(OX)//T1

X/S //0(10)

is given by the composition

g⊗C[V]θ⊗1//V⊗V∨⊗C[V]//V⊗C[V](1) ∼

=T0

X.

LIFTING FREE DIVISORS 21

Split (10) into short exact sequences and take OX–duals to get the exact sequence

0//N∗ψ//Ω1

X

ρ∗

//g∗⊗ OX,(11)

where N= Image(Jac(ϕ)), ψis the dual of Jac(ϕ), and Ω1

X∼

=(T0

X)∗= (Ω1

X)∗∗ as

the smoothness of Ximplies the reﬂexivity of Ω1

X. By this identiﬁcation, the Euler

derivation E∈T0

Xgives a map ˜

E: Ω1

X→ OXdeﬁned by ˜

E(Paidxi) = Paixi.

Observe that under the obvious identiﬁcation of (OX)Nwith Ω1

X,K∼

=ker(ρ∗),

and I=˜

E(ker(ρ∗)). Let b∈ker(ρ∗). By the exactness of (11), there exists an

n∈N∗such that b=ψ(n). Then by the form of ψand the homogeneity of

f1, . . . , fd, we have

˜

E(b)=x1· · · xn

b1

.

.

.

bn

=x1· · · xn

∂f1

∂x1· · · ∂fd

∂x1

.

.

.....

.

.

∂f1

∂xn· · · ∂fd

∂xn

n1

.

.

.

nd

∈J.

8. Examples of group actions

8.1.We now apply the results of §7to a number of coregular and cofree group ac-

tions. For each ϕ:X→S=X//G, we determine when T1

X/S is Cohen-Macaulay of

codimension 2, and determine the liftable vector ﬁelds. These examples come from

classiﬁcations that provide the number and degrees of the generating invariants.

To check our hypotheses for ϕ, however, it is necessary to choose speciﬁc gener-

ating invariants. For many of the examples below, we have used Macaulay2 [GS]

to ﬁnd all invariants of the given degrees4, make a choice of generating invariants

to ﬁnd an explicit form for ϕ, compute the dimension of the critical locus of ϕ, and

ﬁnd the module of liftable vector ﬁelds. A diﬀerent choice of generating invariants

gives a diﬀerent presentation of C[X]Gas a polynomial ring, a new ϕ0(equal to

ϕcomposed with a diﬀeomorphism in S), and a diﬀerent module of liftable vector

ﬁelds.

Note also that there are many other examples in, e.g., [Lit89].

Special linear group.

8.2. Example. Let ρ: SL(2,C)→GL(V), V=Cx⊕Cy, be the standard rep-

resentation of G= SL(2,C). Diﬀerentiating this representation gives the vector

ﬁelds

(12) dρ(e) = x∂y, dρ(f) = y∂x, dρ(h) = x∂x−y∂y

on V, where sl2=C{e, f, h}.

Consider the nth symmetric power X=Symn(V) of ρ, where Symn(V) has the

C-basis zi=xn−iyifor i= 0, . . . , n. Diﬀerentiating this G–representation shows

that e,f, and hact on each xn−iyiby the corresponding diﬀerential operator in

(12). Let ϕ:X→X//G =S.

For 1 ≤n≤4, the resulting representation appears in the list of cofree repre-

sentations of [Lit89], along with the dimension gof the generic isotropy subgroup,

4The vector space of degree dinvariants of a linear representation of a connected group is just

the space of degree dpolynomials annihilated by the linear vector ﬁelds corresponding to the Lie

algebra action.

22 RAGNAR-OLAF BUCHWEITZ AND BRIAN PIKE

and the number (= dim(S)) and degrees of the generating invariants. For n= 1,2,

Proposition 7.12 does not apply because g= 1.

When n= 3, then Xis the space of so-called “binary cubics”, and g= 0,

dim(S) = 1. As a sole generating invariant one can take

f1=−3z2

1z2

2+ 4z0z3

2+ 4z3

1z3−6z0z1z2z3+z2

0z2

3.

Since it is readily checked that ϕ= (f1) is smooth oﬀ a set of codimension 2, it

follows from Proposition 7.12 that T1

X/S is Cohen-Macaulay of codimension 2. We

compute that any η∈DerS(−log s1) will lift, so Theorem 3.5 implies that f1itself

will deﬁne a free divisor. Note that a linear change of coordinates takes f1to the

example [GMS11, 2.11(2)].

When n= 4, the case of “binary quartics”, then g= 0, dim(S) = 2, and the

generating invariants are

f1= 3z2

2−4z1z3+z0z4and f2=z3

2−2z1z2z3+z0z2

3+z2

1z4−z0z2z4.

The map ϕ= (f1, f2) is smooth oﬀ a set of codimension 2, so by Proposition 7.12,

the module T1

X/S is Cohen-Macaulay of codimension 2. The liftable vector ﬁelds are

DerS(−log(s3

1−27s2

2)). Since all reduced plane curve singularities are free divisors,

by Theorem 3.5 any reduced plane curve containing s3

1−27s2

2as a component lifts

through this group action to a free divisor in Sym4(V)∼

=C5.

8.3. Example. Let Vbe the standard representation of G= SL(3,C). Then

X=Sym3(V)∼

=C10, the space of “ternary cubics”, has ﬁnite generic isotropy

subgroup, S=X//G has dimension 2, and the invariants gS,gThave degree 4 and

6 (e.g., [Stu08, 4.4.7, 4.5.3]). Then ϕ= (gS, gT) is smooth oﬀ a set of codimension

2, and the liftable vector ﬁelds are exactly DerS(−log(64s3

1−s2

2)). By Proposition

7.12 and Theorem 3.5, any reduced plane curve singularity which contains 64s3

1−s2

2

as a component lifts via ϕto a free divisor in X.

8.4. Example. Let Vbe the standard representation of SL(2,C), and let X=

Sym2(V)⊗Sym2(V), a representation of G= SL(2,C)×SL(2,C). Use the basis

yij =xi

1x2−i

2⊗xj

1x2−j

2, 0 ≤i, j ≤2, for X. By [Lit89], this cofree representation

has ﬁnite generic isotropy subgroup, S=X//G of dimension 3, with invariants g2,

g3, and g4, deg(gi) = i. We compute generating invariants as

g2=2y2

11 −2y12y10 +y20 y02 −2y21y01 +y22 y00,

g3=y20y11 y02 −y21y10 y02 −y20y12 y01 +y22y10 y01 +y21y12 y00 −y22y11 y00,

g4=−4y4

11 + 8y12y2

11y10 −4y2

12y2

10 + 2y20y12 y10y02 −4y21 y11y10 y02 + 2y22y2

10y02

−1

2y2

20y2

02 −4y20y12 y11y01 + 8y21 y2

11y01 −4y22 y11y10 y01 + 2y21y20 y02y01

−4y2

21y2

01 + 2y22y20 y2

01 + 2y20y2

12y00 −4y21 y12y11 y00 + 2y22y12 y10y00

+ 2y2

21y02 y00 −3y22y20 y02y00 + 2y22 y21y01 y00 −1

2y2

22y2

00.

For ϕ= (g2, g3, g4), ϕis smooth oﬀ a set of codimension 2 and the liftable vector

ﬁelds are Der(−log ∆) for the free divisor deﬁned by ∆ = s6

1−10s3

1s2

2+ 4s4

1s3+

27s4

2−18s1s2

2s3+5s2

1s2

3+2s3

3.By Proposition 7.12 and Theorem 3.5, any free divisor

in C3containing ∆ as a component lifts to a free divisor in X∼

=C9. Note that ∆

is equivalent to the classical swallowtail.

8.5. Example. Let Vbe the standard representation of SL(2,C), and let X=

Sym3(V)⊗V, a representation of G= SL(2,C)×SL(2,C). On X, use the basis

LIFTING FREE DIVISORS 23

zij =xi

1x3−i

2⊗xj, where 0 ≤i≤3, 1 ≤j≤2. By [Lit89], this cofree representation

has ﬁnite generic isotropy subgroup, S=X//G of dimension 2, with invariants g2

and g6, deg(gi) = i. We compute the invariants as

g2=3z22z11 −3z21 z12 −z32z01 +z31 z02,

g6=27z3

22z3

11 −81z21z2

22z2

11z12 + 81z2

21z22 z11z2

12 −27z3

21z3

12 −27z32z2

22z2

11z01

+ 27z32z21 z22z11 z12z01 + 27z31 z2

22z11 z12z01 + 9z2

32z2

11z12 z01

−27z31z21 z22z2

12z01 −18z31 z32z11 z2

12z01 + 9z2

31z3

12z01 + 9z32 z21z2

22z2

01

−9z31z3

22z2

01 + 6z2

32z22 z11z2

01 −15z2

32z21 z12z2

01 + 9z31z32 z22z12 z2

01

−2

3z3

32z3

01 + 27z32z21 z22z2

11z02 −9z2

32z3

11z02 −27z32 z2

21z11 z12z02

−27z31z21 z22z11 z12z02 + 18z31 z32z2

11z12 z02 + 27z31z2

21z2

12z02 −9z2

31z11 z2

12z02

−18z32z2

21z22 z01z02 + 18z31 z21z2

22z01 z02 + 9z2

32z21 z11z01 z02

−21z31z32 z22z11 z01z02 + 21z31 z32z21 z12z01 z02 −9z2

31z22 z12z01 z02

+ 2z31z2

32z2

01z02 + 9z32 z3

21z2

02 −9z31z2

21z22 z2

02 −9z31z32 z21z11 z2

02

+ 15z2

31z22 z11z2

02 −6z2

31z21 z12z2

02 −2z2

31z32 z01z2

02 +2

3z3

31z3

02.

Then ϕ= (g2, g6) is smooth oﬀ a set of codimension 2, and the liftable vector ﬁelds

are Der(−log ∆), for the plane curve ∆ = (s3

1−s2)(2s3

1−3s2). By Proposition 7.12

and Theorem 3.5, any reduced plane curve containing ∆ among its components

lifts to a free divisor in X∼

=C8. In particular, (g3

2−g6)(2g3

2−3g6) deﬁnes a free

divisor.

Special orthogonal group.

8.6. Example. Let Vbe the standard representation of G= SO(n, C). Consider

the representation Sym2(V), which we identify with the action of Gon the space

X= Symmn(C) of n×nsymmetric matrices by A·M=AMAT. Since multiples

of the identity are ﬁxed by G,Xdecomposes as the direct sum of the trivial

1-dimensional representation (on C·Ifor the identity I) and a representation

(on the traceless matrices) which appears on the lists of [VP94] and [Lit89] of

irreducible representations. As a result, we know that the generic stabilizer is

ﬁnite, the generating invariants are g1, . . . , gn, with deg(gi) = i, and S=X//G has

dimension n.

Since Gacts by conjugation, it preserves the characteristic polynomial det(t·I−

M) = tn+h1tn−1+· · · +hnof M. When restricted to the subspace Dof diagonal

matrices, hi= (−1)iσi, where σiis the ith elementary symmetric polynomial in the

diagonal entries; it follows that each hk+1 /∈C[h1, . . . , hk], and hence gi=hiare

generating invariants for i= 1, . . . , n. Let ϕ= (gn, . . . , g1); under the identiﬁcation

of (s1, . . . , sn)∈Swith the monic degree npolynomial tn+sntn−1+· · · +s1