J. R. Borges-Zampiva, B. Oréfice-Okamoto, G. Peñafort Sanchis,J.N. TomazellaDepartamento de Matemática, Universidade Federal de São Carlos, Caixa Postal 676, 13560-905, São Carlos, SP, BRAZILjrbzampiva@estudante.ufscar.brDepartament de Matemàtiques,Universitat de València, Campus de Burjassot, 46100 BurjassotSPAIN.guillermo.penafort@uv.esDepartamento de Matemática, Universidade Federal de São Carlos, Caixa Postal 676,13560-905, São Carlos, SP, BRAZILbrunaorefice@ufscar.brDepartamento de Matemática, Universidade Federal de São Carlos, Caixa Postal 676,13560-905, São Carlos, SP, BRAZILjntomazella@ufscar.br
Abstract.
A reflection mapping is a singular holomorphic mapping obtained by restricting the quotient mapping of a complex reflection group. We study the analytic structure of double point spaces of reflection mappings. In the case where the image is a hypersurface, we obtain explicit equations for the double point space and for the image as well. In the case of surfaces in , this gives a very efficient method to compute the Milnor number and delta invariant of the double point curve.
Key words and phrases:
Reflection groups, singular mappings, multiple points
2000 Mathematics Subject Classification:
Primary 32S25; Secondary 58K40, 32S50
The first author has been partially supported by CAPES. The second author has been suported by Grant PGC2018-094889-B-100 funded by MCIN/AEI/ 10.13039/501100011033 and by “ERDF A way of making Europe”. The third has been supported by FAPESP-Grant 2022/15458-1. The third and fourth author have been partially supported by FAPESP Grant 2019/07316-0.
1. Introduction
In the theory of singular mappings, there are few known examples which are degenerate and also have desirable properties. The problem, rather than an actual lack of mappings of this kind, seems to be the difficulty of the calculations.Reflection mappings were introduced in the prequel of this work [17], as a means to produce degenerate mappings which are easy to understand. This stablished the existence of the examples we were looking for in certain dimensions. In this paper we go one step further and show how to compute the double point spaces—and, in the hypersurface case, the image as well— for reflection mappings. In short, a reflection mapping is a holomorphic singular mapping obtained by restricting the quotient map of a reflection group to a submanifold of the vector space where the group acts (see Section 2 for details). The idea is that the group action, together with the way sits in , must encode the geometry of the mapping.
The reflection mapping class contains the first families of quasi-hom*ogeneous finitely determined map-germs having unbounded multiplicity, for arbitrary and or [17, Theorems 9.5 and 9.6] (see Example 2.5 for low dimensional examples). These mappings where later shown by Ruas and Silva to be counterexamples to a long standing conjecture of Ruas [21]. Brasselet, Thuy and Ruas used them to show the density of the finitely determined mappings among certain spaces of quasi-hom*ogeneous mappings [2]. Silva noticed that these reflection mappings show that the topological type of a generic transverse slice, in the case of -finite quasihom*ogeneous mappings, is not determined by their weights and degrees [22]. Rodrigues Hernandes and Ruas have shown that every finitely determined monomial mapping , with , is a reflection mapping [19]. In contrast, for , the only finitely determined reflection mappings are folding maps [17, Theorem 8.5]. Reflection mappings were also shown to satisfy an extended version of Lê’s conjecture, a rather misterious problem relating injectivity and corank [17, Proposition 4.3].
Apart from the theory of singular mappings, reflection mappings have proven to be relevant in differential geometry. The original idea is due to Bruce and Wilkinson [3, 4], who noticed that the singularities produced by folding a surface with respect to a plane (in our terminology these are -reflection mappings) reveal interesting extrinsic information of the surface. This idea has since been extended to the other spaces [1, 7, 15] as well as to different reflection groups [18]. The reflection mappings used by Bruce and Wilkinson are known as folding mappings and have been widely studied (apart from the previous references, see [6]). They belong to a subclass called reflected graphs [17, Definition 13], which we describe in Example 2.4. This class includes other previously studied classes of mappings, such as double folds [8, 16] and -folding mappings [18]. Double folds provided the first examples of finitely determined quasi-hom*ogeneous corank two map-germs. Inspecting the classification of all simple map germs [12] reveals that they are either folding mappings or -folding mappings.
Now we can discuss the content of the present work. Singular mappings are understood via the study of their multiple point spaces, the most fundamental of which are the double point spaces. Moreover, given the coordinate functions of a map-germ , one would like to know the equation of its image. As it turns out, both the image and the double points spaces become degenerate very fast as the complexity of the coordinate functions increases. We show how to study these objects for reflection mappings, in a much simpler way than in the case of arbitrary singular mappings. In the hypersurface case, the image of reflection mappings is described as well.
For reflection mappings, there are three decompositions of double point spaces indexed by the reflection group, namely
These decompositions appeared in [17, Sections 6 and 7] already, but just at the set theoretical level. In contrast, here the analytic structure of the branches , and is described explicitly. These analytic structures are a fundamental part of the theory of singular mappings, which requires , and to be complex spaces, sometimes with a non-reduced structure indicating higher degeneracy.
For arbitrary mappings with , the double point space and the image are hypersurfaces (or more precisely, complex spaces defined locally by principal ideals, as they are not always reduced), but computing their equations is usually very hard. One of the main achievements of this work are the explicit formulas
for reflection mappings. As a bonus, the formula for gives a much faster way to compute the Milnor number and delta invariant of for germs of reflection mappings .
2. Notation and preliminaries
In this section we summarize the prerequisites and set the notation for reflection groups and reflection mappings. For a more detailed account and proper citations, we refer to [17].To avoid repetition, we fix the meaning of the symbols and notations summarized in this section, which will not be reintroduced.
Reflection groups and the orbit mapping
Throughout the text, stands for a (complex) reflection group acting on a -vector space . We write .
The orbit of a subset (or better, the union of the orbits of the points in ) is denoted by .We adopt the convention that the action of on a function in isHence, a zeroset is transformed by into the set .The stabilizer of is
and the pointwise stabilizer of is
is the maximal subgroup of acting on . The pointwise stabilizer is a normal subgroup of and the quotient acts faithfully on .
The union of the reflecting hyperplanes of all reflections in is called the hyperplane arrangement and is written as.The reflecting hyperplanes induce a partition of into subsets, called facets, consisting of those points contained in exactly the same hyperplanes. The set of facets is denoted by and called the complex of .Clearly, facets are open subsets of linear subspaces, hence have the same tangent everywhere, which may be identified with the closure . Similarly, we often write instead of , for any .
The celebrated Shephard-Todd-Chevalley Theorem characterizes reflection groups as the only subgroups of the group of unitary transformations of for which the quotient mapping can be realized as a polynomial mapping
whose coordinate functions are hom*ogeneous polynomials (they are a set of generators of the ring of -invariant polynomials). More geometrically, the map identifies an orbit to a point, that is, for any , we have that .
Reflection groups act faithfully, hence is generically -to-one, and it is well known that the ramification locus is precisely the hyperplane arrangement. More precissely, given a point , contained in a facet , we have that .
By the universal property of quotient mappings, the mapping is well defined up to -equivalence (that is, up to changes of coordinates in the target). As it will become clear, replacing by an -equivalent map does not change the -class of the associated reflection mappings. Since we study reflection mappings up to -equivalence, we may pretend to be unique, and abusively call it the orbit map of .
Example 2.1 (Products of cyclic groups).
The product is a reflection group acting on by
An element is a reflection if and only if exactly one is nonzero. The orbit map is defined by
Example 2.2 (Dihedral groups, ).
These are the groups of symmetries of regular polygon of sides. We will use the group , consisting of the identity, four reflections
and three rotations
The orbit map of is the mapping , given by
Example 2.3 (The group of symmetries of the tetrahedron).
Consider a regular tetrahedron centered at the origin of , for example the one with vertices
The group of unitary automorphisms of the tetrahedron is a reflection group, isomorphic as an abstract group to the permutation group of its four vertices. This group is generated by the permutations , which in matrix form are
The group contains reflections, which are precisely the permutations . The orbit map is
Reflection mappings
Take an embedding of an -dimensional complex manifold into . The image of is an -dimensional complex submanifold
A reflection mapping is a map obtained as the composition of the orbit map and the embedding , that is,
It is often convenient to replace by the inclusion , obtaining a reflection mapping which, abusively, is also denoted
The choice between these two equivalent settings will be clear from the context.To finish fixing our notation, locally at any point, is defined in by a collection of regular equations, which we write as
(technically, it would be better to express our results in terms of the ideal sheaf of holomorphic functions vanishing on . For the sake of clarity, we have chosen to ignore this issue, which can be fixed by a standard glueing process).
Example 2.4 (Reflected graphs).
Take a reflection group acting on , and a mapping . We may regard as a reflection group acting on , by extending the action trivially on , and take to be the graph of (or equivalently, take the graph embedding ) . The resulting reflection mapping is called a reflected graph and has the form
A typical and much studied example of reflected graphs are folding maps
Observe that, after a target change of coordinates, can be taken to be of the form . Similarly, double folds are -reflected graphs of the form and -folding mappings are -reflected graphs (See the Introduction for references for folding maps, double folds and -folding mappings).
An interesting example is the -reflection graph , defined by the graph of the function . It parametrizes the -singularity , but it does so in a generically -to-one way, as explained in Example 2.9 (observe that being normal prevents it from being parametrized in a generically one-to-one way).
Also, taking the group (Example 2.2) and the functions and , we obtain the reflection graphs
The last two examples are depicted in Figure 1.
![Double points and image of reflection maps (1) Double points and image of reflection maps (1)](https://i0.wp.com/arxiv.org/html/2312.06792v2/x1.png)
Example 2.5 (Reflection mappings with unbounded multiplicity).
The map-germs
with pairwise coprime positive integers, belong to a family of map-germs , introduced in [17, Theorems 9.5 and 9.6]), which, to this day, are the only known family of -finite map-germs whose coordinate functions have unbounded order. These are the germs studied by Brasselet, Ruas, Silva and Thuy mentioned in the Introduction.
Unfoldings and -unfoldings
In the theory of singularities of mappings, the notion of deformation of a space-germ is replaced by that of unfolding of a map-germ. An unfolding of a map-germ is a map-germ
of the form , satisfying . Fixed a small representative of and a value of the parameter , the map is called a perturbation of .
A reflection mapping may be perturbed into a non reflection mapping, for example, by perturbing in a way unrelated to the action of the group . At the same time, perturbing while keeping intact gives a family of reflection mappings containing the original reflection mapping defined by . If we want to study unfoldings of reflection mappings without leaving the reflection mapping seting, these are the deformations we want to consider. In the following lines we formalize this construction.
Let be a reflection mapping, and consider a complex submanifold , such that defines a trivial deformation of over an open subset containing the origin. We may extend the action of to , trivially on the coordinates, so that the corresponding orbit mapping is .
Definition 2.6.
In the above setting, the reflection mapping
is called a -unfolding of . If is a germ of reflection mapping, a -unfolding of is the germ at of a -unfolding of a representative.
Observe that the triviality condition on the deformation ensures that and are isomorphic. In particular, every -unfolding of a germ is, up to -equivalence, an unfolding of .
Example 2.7 (A family of tetrahedral reflection mappings).
Let be the group of symmetries of a tetrahedron, as in Example 2.3. Considerthe family of reflection mappings
Equivalently, we may parametrize by and think of as mappings of the form
As it turns out, is one of the reflecting hyperplanes of . Since acts transitively on its reflecting hyperplanes, we have that , thus
which means that the image of is precisely the discriminant of the orbit map .
The degree of a reflection mapping
Before turning our attention to the image and double points space, we show how the degree of a reflection map is encoded by the stabilizers of in a very simple way. This is essential to our study of the image in the hypersurface case, but it applies to all dimensions.
In this work, when we talk about degree of a mapping , we mean the number of preimages of a generic point in . For this to make sense, there must be an open dense subset of on which this number of preimages is contant. This happens if is irreducible and is proper. The degree for finite map-germs is defined by taking a proper representative. For mappings where fails to be irreducible, a different degree is associated to on each irreducible component of .
Proposition 2.8.
Let be a reflection mapping such that is proper and is irreducible. The degree of is .
Proof.
The degree of is the number of preimages of a generic point in , that is, the number of points in , for in a certain open dense subset . Observe that, given and , the condition may hold even if does not fix the set , that is, even if . But, by definition of , this must happen only on a proper closed subset of . Hence, there is an open dense subset such that the orbit of by the action of is . Since acts faithfully on , there is an open dense subset , such that the orbit of a point consists of points. Since , the claim follows.∎
Example 2.9.
Consider the mapping of Example 2.4. One sees that the subgroup of preserving is
while no non trivial element of preserves pointwise. Therefore, the mapping is generically -to-one. The mapping of Example 2.7 has
Therefore, parametrizes the discriminant of in a two-to-one way.
Corollary 2.10.
Let be a reflection mapping such that is proper. Then, is generically one-to-one if and only if, for all , .
Proof.
We may assume to be irreducible, for neither the generically one-to-one property nor the condition will be affected if we consider the irreducible components of separately. The condition is equivalent to the statement that any must satisfy either or , that is, that any must be contained in .∎
In order to prove our formulas for , we need the next result. Since the proof is slightly involved and the statement is quite clear, we postpone the proof until Appendix A.
Lemma 2.11 (Generically One-To-One Unfolding).
Any multi-germ of reflection mapping admits a one-parameter -unfolding, given by , such that , for all .In particular, this unfolding is generically one-to-one.
Fitting ideals
It is well known that the image of a finite holomorphic mapping is an analytic set (indeed it is enough for the mapping to be proper). However, given an unfolding , the ideal of the image of need not be the same as the result of computing the ideal of the image of and replacing . This is a problem for the study of singular mappings, where deformations are regarded as an essential part of the theory. Luckily enough, there is a solution consisting on declaring the image of a finite mapping to be, rather than just a set, a complex space
where stands for the th Fitting ideal sheaf of the pushforward module. This sometimes gives a non-reduced analytic structure, but this is the price we pay in order for the analytic structure to behave well under deformations. For map-germs, one uses the th Fitting ideal, written . For information about Fitting ideals, we refer to [11]. We only include here the results we need.
Proposition 2.12.
If is a finite mapping and then .
Proposition 2.13.
Let be a finite mapping defined on a reduced -dimensional Cohen-Macaulay space, and assume the irreducible decomposition of to be . Let be a generator of the ideal of in , and let be the degree of restricted to . Then is a principal ideal, generated by
The first result follows from [11, Lemma 1.2], the second is [11, Proposition 3.2]. The next one is a slight modification, tailored to our needs:
Proposition 2.14.
Let be a finite map-germ defined on an -dimensional Cohen-Macaulay space. Let be -dimensional Cohen-Macaulay complex spaces, forming a set theoretical decomposition ,where and have no common components if .Assume each to be isomorphic to on an open dense subset of . Then,
Proof.
Let be the disjoint union of the spaces and take the obvious mapping . Since being Cohen-Macaulay is a local property satisfied by each of the , the space is Cohen-Macaulay. Since
it follows from the construction of Fitting ideals that . Since and are the same map on an open dense subset of their targets, it follows thatthe stalks of and are the same on that open dense subset. But and are principal ideals, hence they must agree.∎
3. The image of a reflection mapping
In this section we show explicit formulas for the image of a reflection map . For any finite map-germ between complex manifolds, the ideal is principal (this holds, more generally, whenever is an -dimensional Cohen Macaulay space, and it follows from [11, Section 2.2]). Hence, letting be a generator of , the image of is
Putting together Proposition 2.13 and Corollary 2.10, one obtains what follows:
Proposition 3.1.
For any reflection mapping , the space is reduced if and only if is generically one-to-one, if and only if , for all , .
Theorem 3.2.
For any reflection mapping , is the zero locus of
where is a section of . Equivalently, .
Proof.
It is immediate that vanishes precisely at , hence we must check that is holomorphic and that the analytic structure of is that of . This may be verified locally in the target and, for simplicity, we will do it just at the origin.
Before comparing both structures, we show that is holomorphic and study the space . The ring hom*omorphism may be identified with the inclusion , where stands for the subring of -invariant germs. This identifies the ideal in with . From the fact that is a principal ideal generated by a -invariant function, it follows that this ideal is precisely the ideal generated by in . In particular, is a principal ideal generated by a holomorphic function , such that . Since is surjective, the function is uniquely determined, not only among the holomorphic functions, but among all functions . Since the function satisfies the desired equality, we conclude and, in particular, is holomorphic. Moreover, we have identified the coordinate ring of with the quotient of by . Note that this ring is a subring of .
Now we show . Obviously, the explicit formula given for behaves well under -unfoldings in the same way does for any unfolding (see Proposition 2.12). Consequently, it suffices to show the isomorphism for an unfolding of . Hence, in view of the Generically One-to-one Unfolding Lemma 2.11, we may assume to satisfy . Since and are the same at the set theoretical level, it suffices for both spaces to be reduced for them to be equal. The space is reduced by Proposition 3.1. At the same time the condition is equivalent to the condition that is reduced, which forces to be reduced because, as we mentioned before, the stalk of is a subrings of stalks of .∎
Remark 3.3.
The same formula applies to compute images of singular hypersurfaces. To be precise,if is a singular hypersurface (instead of a complex manifold), then computes a generator of the ideal , which defines the image of by . Briefly speaking, the formula holds because it is correct on the dense open subset where is smooth, but this forces it to be correct everywhere. If is non reduced, one shows the claim by taking a -unfolding with reduced generic fiber.
Example 3.4.
Consider the group acting on . Then,
One does not need to worry about the definition of the complex square root, because any choice of a section of will give the same final result.
Take, for example the fold mappings of Example 2.4. The ideal of is generated by , hence the image is the zero locus of
For double folds (also Example 2.4) one has and, writing , one gets
Example 3.5 (Image of ).
Consider the -reflected graph of Example 2.4,
A generator of the ideal of is and a section of is, for example,
Multiplying the elements in the orbit of gives the function
Taking the composition with , the square roots vanish and we obtain the expression
The same method gives the equation of the image of
but it is too big to be written here. Both surfaces are depicted in Figure 1.
Remark 3.6.
Let be a -reflected graph, and let be the degrees of the coordinates functions of . If is hom*ogeneous of degree , then is obviously hom*ogeneous, with degrees . As a consequence of Theorem 3.2, the function defining is quasi-hom*ogeneous with weights and degree .
Computing the image via elimination of variables
Computing explicit expressions of sections of orbit mappings can be hard and, even if we manage to find them, the formula becomes too tedious to compute by hand when we take bigger reflection groups. If we try to implement the expression in some computer software, we face the fact that the coordinate functions of sections are non-polynomial, which is a problem when using commutative algebra software such as Singular. Here we introduce two alternative ways of computing the ideal defining the image, without resorting to sections of . As a trade-off, this methods use elimination of variables, which is less explicit, but it is a task computers are happy to do, or at least to try to.
In the case at hand, elimination of variables means what follows: Consider the ring of polynomials on the variables and (the same works for, say, the ring of germs of holomorphic functions at the origin). Contained in , there is the ideal
and the subring of polynomials in . Then, is an ideal in , said to be obtained by eliminating the variables in . Alternatively, consider the ring hom*omorphism
given by . Then, the elimination can be expressed as
Proposition 3.7.
For any reflection mapping , is the zero locus of
Proof.
The first equality was shown in the proof of Theorem 3.2. For the second equality, let
Since , it follows that is radical, hence is radical. This means that is the (radical) ideal of the image of . At the same time, we know by Proposition 2.8 that the degree of is . Putting both things together, the equality is a direct application of Proposition 2.13.∎
In practice, our experience shows that computing is faster than computing , probably due to the fact that is a -invariant function.
Example 3.8.
Consider the family of -reflection mappings of Example 2.7. Sections of the orbit mapping of have horrendous expressions, so computing by means of the expression of Theorem 3.2 is not convenient. In contrast, Singular computes in no time. The equation of the image of the unfolding is too long to write here but, for and , we obtain the following equations:
The hypersurces and are depicted in Figure 2. Recall that is the discriminant of , with non-reduced structure, and that the exponent in the equation reflects the fact that has degree two (see Example 2.9).
![Double points and image of reflection maps (2) Double points and image of reflection maps (2)](https://i0.wp.com/arxiv.org/html/2312.06792v2/x2.png)
Remark 3.9.
The calculations shown before have been carried out by means of a Singular library for reflection mappings, being developed by the authors of this paper. We are happy to send the latest version to anyone interested.
4. Decomposition of the double point spaces and .
Given a germ of singular mapping, studying only the equations of the image somehow means forgetting that these are parametrizable singularities, that is, that these singularities are obtained by glueing one disk by means of a holomorphic mapping (or several disks, in the multi-germ case). Instead, it usual to study singular mappings by means of their multiple-point spaces, which are spaces designed to encode how points in the source glue together to form the image of the map.
There are double point and triple point spaces, as well as higher multiplicity ones, but here we restrict ourselves to double points. Even then, there are several different double point spaces one can look at. Our original interest is to prove an explicit formula for the double point space in the case where is a reflection mapping given by a hypersurface (see Theorem 5.2. The definitions of and other double point spaces are given later on this section). However, as the logic structure of this and the following section reveal, this is best done by studying first the double point space . This in turn requires looking at a more abstract double point space , which enjoys a key functorial property (Proposition 4.2). The results before Section 5 do not require to be a hypersurface and are given for all codimensions.
Decomposition of the double point space
Here we study Kleiman’s double points for reflection mappings. We skip many details, for which we refer to [13, 20].For this description of the double point space for mappings (not necessarily reflection mappings) let be an -dimensional complex manifold and assume that it admits a global coordinate system (This is for the sake of simplicity, a standard glueing process shows the results in this section to be valid in general). We write the blowup of the product of two copies of along the diagonal as
where is a shortcut to indicate the vanishing of all minors of the matrix
We write the exceptional divisor as .
Now let be a holomorphic mapping between manifolds (not necessarily a reflection mapping) of dimensions and , both admiting a global coordinate system. Think of and as a vectors with entries in , of sizes and , respectively. By Hilbert Nullstellensatz, there exist a matrix , with entries in , such that
Then, Kleiman’s double point space is
The proof that does not depend on is in [10, Proposition 3.1]. Away from the exceptional divisor, is just the fibered product
via the blowup map.On the exceptional divisor, keeps track of the kernel of the differential of , as follows:
Proposition 4.1.
[13, Corollary 5.6]The dimension of is at least . If has dimension , then it is locally a complete intersection.
The double point spaces enjoy the following key functorial property, very much related to the construction of reflection mappings. First, any embedding of complex manifolds, , induces an embedding .
Proposition 4.2.
[13, Theorem 2.13]Given a mapping between complex manifolds, the double point space of the restriction is
In particular, the double points of reflection mappings are slices of the double points of orbit mappings. Conveniently enough, we understand quite well. Fixed , let and choose linear mappings and , so that
The following description of is found (with different notation) in [17, Theorem 7.6]:
Proposition 4.3.
The double point space is a -dimensional reduced locally complete intersection, with irreducible decomposition
where is the blow-up of along , embedded in as
In view of Proposition 4.2, the space of reflection mappings must inherit a -indexed decomposition:
Definition 4.4.
For each , we let
As we just mentioned, this gives the set-theoretical decomposition
Observe that neither nor need to be reduced, so there is no clear way to upgrade the previous decomposition into a union of complex spaces, because (to the best of the authors’ knowledge) the meaning of a union means in the category of complex spaces is unclear. In any case, since the order of is finite, and are locally equal topological spaces at points not contained in any other . Moreover, since both spaces are reduced, and are locally isomorphic complex spaces at such points. This means that the comparison of analytic structures in the above equality is only problematic at points where different branches and meet. To be precise, we have observed what follows:
Lemma 4.5.
Let be a point contained in . If is not contained in , for any , then and are locally isomorphic at .
Now we want to refine our description of .Again, we may think of and as vectors with entries in . Then, there is a matrix , with entries in , such that
Also, consider the involution , given by . This mapping lifts to an involution taking onto itself.
Proposition 4.6.
With the above notations, the branch is the complex space
and it satisfies the following properties:
- (1)
The dimension of is at least . If has dimension , then it is locally a complete intersection.
- (2)
The blowup mapping takes isomorphically to the space
- (3)
The involution takes isomorphically to .
Proof.
The space in question is defined as . Consider the matrix , with entries in , satisfying . This allows to describe inside as
To see this, it suffices to check that the space on the right hand side of the equality is the strict transform of in . This in turn is true because this space has dimension , making it a complete intersection by counting equations, and it is isomorphic to away from , while the equations prevent it from having irreducible components contained in . Now, to finish the description of in the statement, it suffices to show that, for points , the conditions and are equivalent. For simplicity, we may identify , so that any vector is of the form , with and . Any point satisfies , which means that the equation has the form
Taking , the equation is equivalent to the one above.
Item (1) is an easy equation counting: is a -dimensional manifold, on which we impose equations. To see item (2), observe that is contained in if and only if , and that away from , the equations and are equivalent. For item (3), we already know that is an automorphism on and, since it has the form , it must take to . This must happen isomorphically, since otherwise would fail to be an automorphism.∎
Example 4.7.
Consider the map-germ
from Example 2.5. The equations of are .To illustrate the calculation of , take, for instance, the elements of the form , , with (see Example 2.1 for notation). The pointwise fixed spaces are given by the vanishing of . From the equations one concludes that is the matrix defined by the first columns of
Since, for , is injective, the equation forces . This, added to the condition for points in , forces and, since a point cannot have all its coordinates equal to zero, we conclude . is a line, with no component on the exceptional divisor, and it is a complete intersection. is a line, with an irreducible component on the exceptional divisor. is the union of a line and a 2-dimensional component lying on the exceptional divisor. This branch is not a complete intersection (indeed it is not even Cohen Macaulay).
Decomposition of the double point space
Now we study a different double point space. As a set, is just the projection of . As a complex space, this space was introduced in [10]. The results for which we give no proof can be found in [14]. Let be a holomorphic mapping between complex manifolds, as in the previous section. We define
where stands for ideal of the diagonal and is the ideal generated by the minors of (see [10, Proposition 3.1] for the independence of ). As it is the case with , the space has a nice analytic structure whenever it has the right dimension.
Proposition 4.8.
The dimension of is at least . If has dimension , then it is a Cohen-Macaulay space.
As we mentioned, set theoretically, is the projection of , that is,
Going back to the reflection mapping setting, by letting be the image of in , we obtain a set-theoretical decomposition
and clearly We may improve this description, by giving an adequate analytic structure:
Definition 4.9.
Let and .If , then let
where are the minors of . If , then let
The proof that this definition is independent of the matrix is, mutatis mutandis, the one of [10, Proposition 3.1].
Proposition 4.10.
For every , the complex space defined above is, as a set, the image of on , and it satisfies the following properties:
- (1)
The dimension of is at least . If , then is Cohen-Macaulay. If furthermore is a reflection or , then is locally a complete intersection.
- (2)
Away from , we have an equality of complex spaces
- (3)
Let , assume that and that , for all . Then, and are locally isomorphic at .
- (4)
The involution takes isomorphically to .
Proof.
We know that the image of in is the set of pairs , such that . Letting , the matrix has size . Therefore, must have non-trivial kernel automatically if . Whenever , the conditions and are equivalent. This shows to be the image of , as a set.
Now we prove item (1). If then is defined by the equations on the -dimensional manifold , hence it is a complete intersection if it has dimension . If is a reflection, then consists of a single entry and, then, we may compute just by dividing the entries of by , that is,
Now is generated by the entries of . In this case, modulo the entries of , the entries of are generated by the entries of , and can be discarded as generators of the ideal of . Just as in the case of , now is defined by equations and it is a complete intersection if it has dimension . If , the fact that is a Cohen-Macaulay space whenever it has dimension is a direct application of [5, Thm 2.7, Lemma 2.3].
Item (2) is obvious if . In the case of we need to show that, locally on a point , the minors of are in the ideal generated by the entries of and . Let be the square submatrix obtained by picking the rows of . Let be the vector with entries . Since , there must be some such that . Let be the matrix obtained by substitution of the -th column of by . By Cramer’s Rule, we obtain . The claim follows from the fact that is clearly in the ideal generated by the entries of and .
Observe that item (2) can be restated as the fact that and are isomorphic. At the same time, and are isomorphic (this is true because both spaces are isomorphic to . This easy to see from the equations of and from the definition of given in [13]). From this point of view, item (3) is the same as item (2) from Proposition 4.6. Finally, item (4) is proven in the same way as item (3) from Proposition 4.6.∎
Example 4.11.
Consider the map-germ
from Example 4.7. With the notations that we used there, we obtain because, in both cases, has maximal rank. On the other hand, for , , therefore
This spaces must be lines, because their equations are linear and we know a priori that is -finite [17, Lemma 9.10 and Proposition 9.8], which forces the double point space to be a reduced curve.
Remark 4.12.
In the case of , every branch having dimension is locally a complete intersection. This is remarkable, since it is well known that there are germs whose double point space has dimension but is not complete intersection. Proposition 4.10 shows that, for reflection mappings , we can split into the branches , and these branches are complete intersections.
Decomposition of the double point space
If is a finite holomorphic mapping between complex manifolds (not necessarily a reflection mapping), then the mapping , given by , is finite. The source double point space is defined as the image of , that is,
As a set, we have that
Now come back to the reflection mapping setting. Since we have the set theoretical decomposition , then the set must be the union of the sets .
Definition 4.13.
For any , we define .
We obtain the set theoretical decomposition
Since the equations are among the equations defining , the spaces and isomorphic. Hence the spaces may be described equivalently as follows: Let . If , then
If , then
As the spaces and are isomorphic, Proposition 4.10 can be recast as a result about .
Proposition 4.14.
For every , the complex space satisfies the following properties:
- (1)
The dimension of is at least . If , then is Cohen-Macaulay. If furthermore is a reflection or , then is locally a complete intersection.
- (2)
As complex spaces, .
- (3)
Let such that , for all . Then, and are locally isomorphic at .
- (4)
is isomorphic to via .
If there is no risk of confusion, it is common to write instead of . Similarly, we may sometimes write for . This notation appears, for example, in Example 5.3.
5. A formula for in the hypersurface case
Let be a finite mapping between complex manifolds (not necessarily a reflection mapping). The 0th Fitting ideal of the projection is then a principal ideal (if , this is true because is Cohen-Macaulay, see the first paragraph of Section 3. In the case of , one simply gets the zero ideal). We usually write for a generator of this ideal, so that
If is a reflection mapping, then we have that
where, according to the description given below Definition 4.13, the functions are
Observe that, even though vanishes on , we have chosen to keep the term in the expressions. This allows for the divisibility by to hold on the ambient space , making computations easier.
Before giving an explicit formula for the double point space , we need the following technical lemma, whose proof is in Appendix A.
Lemma 5.1 (Unfolding With Good Double Points).
Every multi-germ of reflection mapping admits a -unfolding , such that, for all ,
Theorem 5.2.
As a complex space, the double point space is the zero locus of
Proof.
The expression behaves well under -unfoldings obviously, and it is well known that behaves well under general unfoldings (see for example [14]). Therefore, we may assume to satisfy the conditions of Lemma 5.1 and to be generically one-to-one, by the generically one-to-one unfolding Lemma 2.11. The generically one-to-one condition gives , which forces to be Cohen-Macaulay, by Proposition 4.8. From the decomposition and Item (1) of Proposition 4.10, it follows that the branches are Cohen-Macaulay spaces of dimension as well. We claim that this, added to the fact that satisfies the conditions of Lemma 5.1, implies that and have no irreducible components in common, for all . This is true because points in are of the form , hence satisfy , but .Now it follows from Item (3) of Proposition 4.10 that every branch has a dense open subset on which and are locally isomorphic.Since and have no common irreducible components and the isomorphisms give , the result follows directly from Proposition 2.14.∎
Example 5.3.
Consider the -reflected graph
from Examples 2.4 and 3.5. With the notation of the Example 2.2, for the reflections , the generators of the ideals of are units, hence . The spaces and are lines, given by the vanishing of
This double point curves are depicted in Figure 3.Since , the branches and are glued together on a single branch of double points on the image of . In contrast, since , the branch is glued to itself to produce the branch on the image. This forces to be -symmetric (or, in other words, it forces to have as an automorphism). By contrast is not -symmetric, but takes isomorphically onto .
A more complex example is the mapping
also from Examples 2.4 and 3.5. This time the branches associated to reflections are nonempty, and they are given by the vanishing of
The curves are all regular, and each of them gets glued to itself to form a curve on the image of . Consequently, each acts as an automorphism on . The double point branches are depicted in Figure 4. The elements have associated functions
The space consists of two branches and . Since , the space is isomorphic to , and its two branches and (depicted as dashed lines) are identified with the branches of to form the two branches , with . Finally, the space consists of three branches , each being glued to itself to form the curve on the image of .
![Double points and image of reflection maps (3) Double points and image of reflection maps (3)](https://i0.wp.com/arxiv.org/html/2312.06792v2/x3.png)
![Double points and image of reflection maps (4) Double points and image of reflection maps (4)](https://i0.wp.com/arxiv.org/html/2312.06792v2/x4.png)
Remark 5.4.
It is well known that the jacobian (that is, the determinant of the differential matrix of ) equals the product of the equations of the reflecting hyperplanes of all reflections in . Consequently, the expression for can be rewritten as
Remark 5.5.
If is a reflected graph and is hom*ogeneous of degree , as in Remark 3.6, then is hom*ogeneous of degree , where is the number of reflections in .
The double point curve of a reflection mapping
In these dimensions the double point spaces are plane curves when they have the right dimension. Moreover, a germ is -finite if and only if is a reduced plane curve [9]. For these mappings, one is interested in computing the Milnor number and the delta invariant . This is much easier for reflection mappings than it is for arbitrary mappings, thanks to the following criterion, which follows easily by putting together Hironaka’s formula, the additivity of and the decomposition of Theorem 5.2.
Proposition 5.6.
A reflection mapping is -finite if and only if all the Milnor numbers and all the intersection numbers , with , are finite. In this case,
and
where is the intersection number of the branches and and is the number of elements for which (We follow the convention that ).
The data of the , , and whether a branch is empty can be stored easily in matrix form. Here is a convenient way to do it: First choose an ordering of the reflection group where . Let be the matrix with entries
Let be the size vector with entries
and define the vector analogously, but replacing every with . A value anywhere indicates that is non-reduced, hence that fails to be -finite. In the abscence of any , the mapping is -finite and the formulae in 5.6 turn into
Furthermore, the number of branches of is the sum of the number of branches of all the , which is equal to
Observe that the matrix is symmetric and that the vectors and must contain the same value at the positions corresponding to and , since these are isomorphic spaces by virtue of Item 4 of Theorem 4.
Example 5.7.
Consider the mapping of Example 2.4, whose double point branches were computed in Example 5.3. By ordering the group as (see Example 2.2), computing the Milnor numbers and the intersection number of all pairs of branches, one gets
From this, one computes and .
This same method is efficient when applied to bigger groups. For example, for the reflection mapping given by the reflection group of Example 2.3 and the embedding , the vectors and square matrix involved have size 23, but it is better to look at the information in this format than trying to understand the space formed by all branches together. For this particular example, one gets and .
Example 5.8.
If is a reflection graph with hom*ogeneous , as in Remarks 3.6 and 5.5, then, by ordering as identity first, then reflections and then non-reflections, one obtains
where is the number of reflections in .Therefore,
(observe that this also follows from Remark 5.5) and
For instance, the map from Example 2.5 has , and .
Appendix A Proofs of the unfolding lemmata 2.11 and 5.1
In this section we prove the Generically One-To-One Unfolding Lemma 2.11 and the Unfolding With Good Double Points Lemma 5.1, which are key in the proofs of the explicit equations of the image and double point spaces of reflection mappings in the hypersurface case. The statements of these lemmata are quite intuitive and, if there is anything surprising about them, it is that we could not prove them more easily. One has to keep in mind however that is any smooth analytic of codimension one germ, with no hypothesis on how bad its relation to is, and most usual transversality arguments do not apply in this setting.
For the most part, it is enough to apply an origin preserving rigid motion to in order to attain the conditions that we desire. However, certain elements of the group may impose problems that cannot be fixed in this way. We start by formalizing who the problematic elements are.
Lemma A.1.
Given a linear endomorphism , the following statements are equivalent:
- (1)
preserves all -dimensional vector subspaces of , for certain .
- (2)
preserves all vector subspaces of .
- (3)
The matrix representation of in some (or any) basis of is
for some .
Proof.
That the first statement implies the second follows easily after observing that, fixed , any vector subspace can be expressed by means of intersection and sums of -dimensional subspaces. The remaining implications are immediate.∎
Definition A.2.
Any linear endomorphism satisfying the conditions above is called a complex hom*othety (centered at the origin). Given a subspace , we say that acts as a complex hom*othety on if and is a complex hom*othety.
We will use the following basic result, whose proof is ommited.
Lemma A.3.
Let be a linear automorphism and let be a vector subspace. Let and be positive integers, with . Then, there is a non-empty Zariski open subset of the product of Grassmanians,
such that all satisfy the following conditions:
- (1)
, for all and all .
- (2)
, for all , if does not act as a complex hom*othety on .
Proof of the Generically One-To-One Unfolding Lemma 2.11
Let be a multi-germ of reflection mapping, given by a germ . Take the decomposition into mono-germs
We want to find a representative having a trivial deformation
such that , for all . Observe that, if the representatives are chosen adequately, the desired dimension drop is equivalent to the condition that, for all and all , we have that
To justify the existence of , we consider the consider the cases and separately.
In the case of , we have in our advantage the fact that no complex hom*othety fixes any of the points in . In particular, if is a complex hom*othety and the representatives are chosen small enough, then . Then, any small enough deformation satisfies , for all .
By Lemma A.3 and the Curve Selection Lemma, there exists a curve
such that, identifying as usual, we have that and
for all . Letting , there exists an analytic family of linear automorfisms , such that . Finally, the germs of biholomorphism
given by , define deformations of , such that , for all , and . We must check that, given and assuming that either or is not an hom*othety, the deformations satisfy for all . This is trivial if and, otherwise, it follows from the fact that , because then
Now consider the case where . Here there is the advantage that the orbit of is just . By the same reasoning as in the case of , there exists a family of linear automorphisms , such that, and, for all and every which is not a complex hom*othety,
Now let be the hom*otheties in , such that (we do not need to care about complex hom*otheties with , since they will satisfy , for any small enough deformation of ). We may assume , as our problem is trivial otherwise. Then, cannot be contained in any , because a complex hom*othety which is not the identity fixes the origin only. Consequently, we may find a point , such that for . Since , we may take a polynomial function on , such that, , and vanishes at the origin with order of vanishing at least two. Now take the equations of and define
For , the fact that vanishes at with order at least two implies that and that, for every which is not a complex hom*othety,
which clearly implies , as desired.
Finally, for and any of the complex hom*otheties above, it suffices to show that . On one hand, by construction, the conditions and are equivalent, and the second holds because is in and vanishes at . On the other hand, is equivalent to .Now observe that complex hom*otheties commute with linear transformations, and thus . Since , we conclude , and since , it follows that , as desired.
Proof of the Unfolding With Good Double Points Lemma 5.1
Recall that our goal is to show that any multi-germ of reflection mapping admits a -unfolding , such that, for all
Lemma A.4.
Any germ of codimension one submanifold admits a -unfolding, given by , such that, for every facet and for all , the intersection is smooth of dimension , and such that is transverse to the reflecting hyperplanes in .
Proof.
Taking the equation of and letting , the projection on the parameter is the Milnor fibration of inside . This implies the claim that is smooth of dimension . The transversality of to the reflecting hyperplanes, which have the form , is obvious from its equation.∎
Since a -unfolding of a -unfolding of is still a -unfolding of and we do not care about the number of parameters needed for the -unfolding in Lemma 5.1, the original may be replaced by the -unfolding given by the Generically One-To-One Unfolding Lemma. This may in turn be replaced by one satisfying the conditions of Lemma A.4 as well. In other words, in order to prove Lemma 5.1, we may assume our original multi-gem to satisfy the following conditions:
- (1)
intersects transversely all reflecting hyperplanes.
- (2)
intersects properly all facets of of dimension .
- (3)
, for all .
Now observe that, to show the existence of an unfolding satisfying , we do not need to deal with all pairs and at once. We may fix and , find an a good unfolding for them and move on to the next pair of elements. We may also assume to have at most two branches, one containing some point and, when , another one containing . If more branches were present, we would solve the problem taking consecutive unfoldings, one pair of branches at a time. Additionally, condition (2) allows us to assume to be a reflection, since otherwise the dimension of is already smaller than that of . By condition (1), the intersection is smooth of dimension . We are now reduced to showing the following result:
Lemma A.5.
Let and let be a reflection in . Let be a germ of -dimensional complex manifold satisfying the conditions (1), (2) and (3) above, and of one of the following forms:
- •
is a bigerm with .
- •
is a monogerm with .
Then, there exists a deformation of whose fibers, for , satisfy .
Proof.
In the bigerm case, we may proceed as in the proof of the Generically One-To-One Unfolding Lemma 2.11, but restricting everything to . To be precise, we start by observing that, by condition (1) and the fact that acts on the set of hyperplanes, the sets and are complex manifolds of dimension . Now using Lemma A.3 with and the Curve Selection Lemma, we produce a deformation of whose fibers and contain and , respectively, and satisfy
This implies , and the claim follows from the inclusion .
Now we deal with the monogerm case. Let be the union of all -dimensional components of . Transversality forces to be a complex manifold of dimension . Since the irreducible components of have dimension and , we conclude that is either empty or equal to . We may assume to be non-empty, because otherwise a trivial deformation already satisfies our claim. We distinguish two cases:
If , from the inclusion we obtain
This means that, for any representative of (also denoted by ), there exists , with , such that . Now take an equation for , and a polynomial , such that and , and define
For all , just as in the proof of the generically one-to-one unfolding Lemma 2.11, the fiber satisfies and , in particular, . This implies , hence , as desired.
Finally, consider the case where . This hypothesis requires to have dimension . Since is the closure of a facet of the complex of and intersects properly the facets of dimension , it follows that must be a reflecting hyperplane. Then, and are reflections with respect to this hyperplane and, since depends only on and , we may assume the reflection group to be .
Then, the hypothesis implies , which forces the germ to be non-immersive. Since , this implies that contains . In particular, we may choose coordinates on and , so that the expression of is
for some analytic function of the form
with . Since is the graph , it is isomorphic to its projection on and, under this isomorphism, becomes the subset and becomes the zero locus of
Clearly, we may perturb in a way that the perturbed function becomes not divisible by for , which forces , as desired.∎
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