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Abstract

Let $\varphi :X\to S$ be a morphism between smooth complex analytic spaces and let $f=0$ define a free divisor on $S$. We prove that if the deformation space $T^1_{X/S}$ of $\varphi $ is a Cohen–Macaulay $\mathcal {O}_X$-module of codimension 2, and all of the logarithmic vector fields for $f=0$ lift via $\varphi $, then $f\circ \varphi =0$ 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 $\varphi :\mathbb {C}^{n+1}\to \mathbb {C}^n$ with critical set of codimension 2 has a $T^1_{X/S}$ with the desired properties. Finally, if $X$ is a representation of a reductive complex algebraic group $G$ and $\varphi $ is the algebraic quotient $X\to S=X\!{/\!/} G$ with $X\!{/\!/} G$ smooth, then we describe sufficient conditions for $T^1_{X/S}$ to be Cohen–Macaulay of codimension 2. In one such case, a free divisor on $\mathbb {C}^{n+1}$ lifts under the operation of ‘castling’ to a free divisor on $\mathbb {C}^{n(n+1)}$, partially generalizing work of Granger–Mond–Schulze on linear free divisors. We give several other examples of such representations.
LIFTING FREE DIVISORS
RAGNAR-OLAF BUCHWEITZ AND BRIAN PIKE
Abstract. Let ϕ:XSbe 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 fϕ= 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 XS=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=f1(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 ϕ1(D)≡ {fϕ = 0} ⊆ Xagain a free divisor?
We give sufficient conditions for ϕ1(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
ϕ1(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 ηDerS(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 ϕ1(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 ηDerS(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:XS=
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:XSis 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 mthe maximal ideal in Oof 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:XSis 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 ϕ1(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:XSis 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 :ϕ1
S
=OXOS1
S1
Xthat in turn sends
1OSds to d() 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
(sjϕ)
∂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 =δT0
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
∂sjT0
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 2.102.12.
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=
ϕ1(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
ST1
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 DT0
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:XSbe a morphism
from an analytic germ Xto S. If Xis Cohen–Macaulay and the inverse image
ϕ1(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//ϕDerS(log f)ϕσf)
//T0
S(OX)⊕ OXα//OX//Oϕ1(Σ) //0,
where α=ϕ[(Jac(f), f) = (Jac(f)ϕ, f ϕ).
If furthermore 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 OXOS 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 i0, we have Hi(C) = TorOS
i(OX,OΣ), and so each homology module is
supported on ϕ1(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 k1. Thus, if C0=OX, C1, . . . are the free modules
in C, then depth(J, Ci)> i 1 for all i1. This fact, and the earlier observation
that J·Hi(C) = 0 for i1, are enough to ensure that Hi(C) = 0 for all i1 (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 to
define a free divisor in X.
3.4. Theorem. Let ϕ:XSbe a morphism of smooth germs and let f
OSdefine a free divisor V(f)Swith singular locus ΣV(f). Assume that
Image(ϕ)*V(f), i.e., fϕ is not zero. Let gbe a reduction of 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
ST1
X/S vanishes on DerS(log f), that is,
δKS σf= 0, and
(c) the inverse image ϕ1(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 ϕDerS(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⊕ OXT0
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
ϕDerS(log f)
ϕ˜σf
ω//T1
X/S
==
T0
S(OX)⊕ OX
α
δ+0
88
0//DerX(log fϕ)˜σ
//
77
T0
X⊕ OX
β
77
γ////OX//
OX/I //
xx
0
T0
X/S
OO77
Oϕ1(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=δϕ[σf). Hence if DDerS(log f)T0
Sand 1D∈ OXOSDerS(log f)
is the pulled-back vector field in ϕDerS(log f), then ω(1 D) = δϕ[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)//ϕDerS(log f)//0
and
0//T1
X/S //OX/I //Oϕ1(Σ) //0.
Since DerS(log f) is a free OS–module by assumption, and thus ϕDerS(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:XSbe a morphism of smooth germs and let f∈ OS
define a free divisor V(f)Swith singular locus ΣV(f). If both
(b) the Kodaira–Spencer map δK S :T0
ST1
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 ϕ1(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 ϕ1(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 //Oϕ1(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 Oϕ1(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:XSis 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 fϕ= 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
ST1
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=C3S=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=CnS=C(and hence fϕfor 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 ϕJϕ, 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 ϕJϕ. Hence, the hypotheses of
Theorem 3.5 are satisfied exactly when ϕJϕ, 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:XSbe a morphism of smooth germs with module
L= ker(δKS )T0
Sof liftable vector fields. Let f∈ OSdefine a free divisor
with singular locus ΣV(f). Let (w1, . . . , wm)be a set of nonnegative integral
weights for the coordinates (s1, . . . , sm)on S. Let E=Pm
i=1 wisi
∂siT0
Sbe the
corresponding Euler vector field, so that T0
S(ϕ[)(E) = Pm
i=1 wi(siϕ)
∂siT0
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 fϕdefines a free divisor.
LIFTING FREE DIVISORS 11
Proof. Let tbe a coordinate on C, and let ϕ= (ϕ1, . . . , ϕm). Define θ:Y=
X×CSby 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(fϕ)×Cin Y. It follows that fϕdefines 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) =
ewit
∂sishows that deleting the exponential coefficients in (5) gives an isomorphic
cokernel. Thus, T1
Y/S
=coker Jac(θ) is isomorphic to NOXOY, 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
∂siT0
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
∂xjT0
X,
so that aiϕ=Pn
j=1 bj∂ϕi
∂xjfor i= 1, . . . , m. Let ξ0T0
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·(aiϕ)
∂si
=
m
X
i=1
aiew1t·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=C3S=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,
fϕ= (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=C3S=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(s1s3s2
2), by Corollary 4.2 each such fϕdefines 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×CpSbe 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) = {ηDerX: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:XSby 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/(IOCmOX)
=
(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
∂ziDerCmis 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 hJhthen 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=Mn1,n, and let π:MNbe the projection that
deletes the last row. Differentiate ρ: GL(n1,C)×GL(n, C)GL(N) defined
by ρ(A, B)(X) = AXB1to obtain a finite-dimensional Lie algebra gof linear
vector fields on N. Let DDerNbe 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)< n1} ⊂ 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:NCdeletes column iand takes the determinant. Since Pn
i=1(1)n+igixni =
det on M, by Proposition 5.1, (fπ)·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 (fπ)·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
,
(fπ)·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:CpCr,pr, if C(g) has the expected dimension r1, then coker(Jac(g))
is a Cohen–Macaulay OCp–module of dimension r1.
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:C3C2be 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
1s3
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 ϕ1(∆) is a free divisor defined by
ϕ=x1(3x4
2x33x1x2
2x2
3x2
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(4x1x23x2
2x3)
∂x3.
6.3. Example. Let ϕ:C4C3be 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:XS=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 in, 1jn+ 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:VV//G is smooth outside the null cone ϕ1(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:XSbe the natural
morphism, smooth off the codimension 2 null cone ϕ1(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 rn,
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⊗OXT0
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 ). 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 n21 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:M2,3M2,3
// 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(s1s3s2
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,nm, 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,nm
defined by replacing a m×mminor ∆Ion Mn,m with the (nm)×(nm) minor
0
Ion Mn,nmformed 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,nmwhich 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 ∆1415(∆14 25 1524 )(∆3445 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:XSis 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:VV//G is flat, equivalently (see
[VP94, 8.1]), ϕ:VV//G is coregular and equidimensional in that all fibres
have the same dimension;
(c) coreduced , if the null cone ϕ1(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
fjC[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:VV//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 XSof 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:XSis
smooth outside a set of codimension 2in X, then
(i) The natural OX–homomorphism g⊗ OXT0
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⊗ OXT0
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)2depth(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⊗ OXT0
X/S is an isomorphism, then either
N=Pd
ν=1 δνand dim(T1
X/S ) = N2, or
LIFTING FREE DIVISORS 19
N6=Pd
ν=1 δνand dim(T1
X/S ) = N1.
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(Nd)
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
tδνNt1+Nd!
=NPd
ν=1 δν+ (1 t)p(t, t1)
(1 t)N1
for some Laurent polynomial p(t, t1)Z[t, t1]. In particular, HT1
X/S has a pole
at t= 1 of order N1 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:XX//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:XSis equivariant with respect to the action of H,
then all vector fields on Sobtained by differentiating ρSlift across ϕ.
Proof. Differentiating gives representations Xand Sof h, the Lie algebra of
H, as Lie algebras of vector fields on X, respectively, S. Since ϕis equivariant, for
each Yh,X(Y) is ϕ-related to S(Y), and hence S(Y) lifts to X(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)A1,
where adj(A) is the adjugate of A. (If Mn,n+1 'VW, with dim(V) = n,
dim(W) = n+ 1, and M1,n+1 'CW, 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 Sproduces a generating set of
20 RAGNAR-OLAF BUCHWEITZ AND BRIAN PIKE
DerS(log{0}), any ηDerS(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
∂xiT0
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]GC[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
(sjϕ)
∂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:XS=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 gI. It
follows that JI.
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)
=VVof 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
gC[V]θ1//VVC[V]//VC[V](1)
=T0
X.
LIFTING FREE DIVISORS 21
Split (10) into short exact sequences and take OX–duals to get the exact sequence
0//Nψ//1
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 ET0
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 bker(ρ). By the exactness of (11), there exists an
nNsuch 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:XS=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=CxCy, be the standard rep-
resentation of G= SL(2,C). Differentiating this representation gives the vector
fields
(12) (e) = x∂y, dρ(f) = y∂x, dρ(h) = x∂xy∂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=xniyifor i= 0, . . . , n. Differentiating this G–representation shows
that e,f, and hact on each xniyiby the corresponding differential operator in
(12). Let ϕ:XX//G =S.
For 1 n4, 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
1z36z0z1z2z3+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 ηDerS(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
24z1z3+z0z4and f2=z3
22z1z2z3+z0z2
3+z2
1z4z0z2z4.
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
127s2
2)). Since all reduced plane curve singularities are free divisors,
by Theorem 3.5 any reduced plane curve containing s3
127s2
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
1s2
2)). By Proposition
7.12 and Theorem 3.5, any reduced plane curve singularity which contains 64s3
1s2
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
1x2i
2xj
1x2j
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
110s3
1s2
2+ 4s4
1s3+
27s4
218s1s2
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
1x3i
2xj, where 0 i3, 1 j2. 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
1s2)(2s3
13s2). 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
2g6)(2g3
23g6) 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+h1tn1+· · · +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+sntn1+· · · +s1