Iterated multiple points I: Equations and pathologies

This is the first of a two‐part work on Kleiman's iterated multiple point spaces. Here we show general properties of these spaces and solve the problem of finding their equations in the case of maps of any corank between complex manifolds. We also describe pathologies regarding dimension and lack of symmetry.


INTRODUCTION
The multiple point spaces of maps ∶ → are a key tool in areas such as enumerative geometry [11-13, 15, 31, 32], the study of Thom polynomials [10,27], the vanishing (co)homology of disentanglements [8,9,21,30] and the study of finite determinacy of map-germs [1,2,16,18,25,28,29] but, despite their relevance, these spaces are not well understood. While it is clear that the multiple point spaces must contain the strict multiple points (that is, -tuples ( 1 , … , ) such that ( ) = ( ) and ≠ , for all ≠ ), there is no consensus about the best way to include diagonal points in order to get a reasonable structure.
There are several approaches to the definition of multiple point spaces, some based on deformations [25], on the Hilbert scheme [13] or on Fitting ideals [23]. The approaches on deformations and Fitting ideals require the map to be finite, thus require dim ⩽ dim . Here we study the more general approach of Kleiman's iterated multiple point spaces [11], defined for any separated morphism of schemes ∶ → , regardless of the dimensions of and and the finiteness of .
The double point space of is the residual space of the fibered product × along the diagonal Δ (see 1.1 for the definition of residual space). Composition of the structure map with the first projection gives a map 2 → . Higher order multiple point spaces are defined iteratively: The triple point space 3 is the double point space of 2 → , and comes with a map 3 → 2 . The quadruple point space 4 is the double point space of 3 → 2 , and so on.
If and are smooth and has only corank one singularities (that is, if is curvilinear in Kleiman's terminology), then coincides with Mond's multiple point space , the subspace of given by the vanishing of the iterated divided differences (see [20]). Since the number of equations is ( − 1) , one deduces that is a local complete intersection in , whenever it has the correct dimension − ( − 1) , with = dim and = dim (see Definition 5.5). A remarkable theorem of Marar and Mond [16] states that a corank one map is stable if and only if all are smooth of the correct dimension, and that a corank one map germ is finitely determined if and only if the spaces whose correct dimension is positive are isolated complete intersection singularities of the correct dimension and the spaces whose correct dimension is negative are confined to the origin.
The main difficulty with Kleiman's construction is to find explicit equations for in the presence of singularities of corank ⩾ 2. In this paper, we propose an alternative description of which solves this problem for maps between smooth spaces and , allowing singularities of any corank. For any fixed , the spaces of the maps ∶ → can be embedded as closed subspaces of a universal multiple point space = ( ). By definition, is the multiple point space of the constant map → * (observe that the definition of requires using multiple point spaces of a non-finite maps). If is smooth of dimension , then is smooth of dimension ; indeed, it is the blowup of −1 × −2 −1 along Δ −1 . Our main result, Theorem 3.1, states that where ∶ → −1 × −2 −1 is the blowup map, is the exceptional divisor and ∶ stands for the zero locus of the quotient ∶ of the defining ideal sheaves of two subspaces and . The right-hand side of the previous equality is taken in [22] as the definition of . This expression is easier to compute, but it hides the iteration principle, which is essential in many instances, and the equality requires for and to be smooth.
From this result, we derive explicit local equations of inside , which are natural generalizations of the iterated divided differences in a convenient atlas of . As in the corank one case, is locally defined by ( − 1) equations in , hence it is a local complete intersection whenever it has the correct dimension. We remark that for = 2, our description of coincides with Ronga's double point space [33].
Lastly, Section 6 is devoted to pathologies exhibited by when is a finite map with singularities of corank ⩾ 2 and is big enough. The first pathology is that always fails to have the correct dimension, even if is stable. In particular, it must have components of different dimensions, since it is dimensionally correct along the corank one points. Second, the image of by does not coincide, even set-theoretically, with the multiple point space in the target given by Fitting ideals, as defined by Mond and Pellikaan in [23]. The last pathology is the lack of symmetry of with respect to permutations of coordinates. Even for triple points, the natural action of 3 on 3 cannot be lifted to 3 . This anomaly comes from the choice of the projection in the iteration process. It appears that fixing these pathologies would require a totally different approach, which seems out of our reach at the present moment. For better exposition, some technical proofs are given in the Appendix.
In a forthcoming paper [26], the local properties of and their relation to stability and finite determinacy of maps will be studied. On one hand, we will show that 3 is smooth when is stable and that it provides a desingularization of the multiple point space 3 , which is always singular when has singularities of corank ⩾ 2. The analogous result for 2 was proven by Ronga in [33]. On the other hand, smoothness fails from quadruple points onwards for generically one-to-one maps of corank ⩾ 2. These pathologies will be used to give a simple criterion for finite determinacy within a wide range of dimensions, analogous to the Marar-Mond criterion for the corank one case [16].
Based on Sections 4 and 5, we have implemented a library in Singular [3] to compute the generalized divided differences of any polynomial map ∶ ℂ → ℂ , that is, local equations of in the charts of the smooth space . The library IteratedMultPoint.lib is freely available (see [24]) and its usage is illustrated in Example 5. 12. We think that the library will be a useful tool for anyone interested in working with examples.
The technical obstacles of higher corank have forced authors to restrict their work to singularities of corank one. However, the attention paid to higher corank singularities has been growing over the years. They are finding a place in works about finite determinacy [2,17,18,21,28,29], enumerative geometry [5] as well as some other topics [6]. We hope that this work will clarify some aspects of the higher corank case.
Some of the results of this paper appear in a simplified form in the recent monograph about singularities of mappings [22].

Terminology
We use preimages and intersections as defined for any category that has fibered products (such as the arrow categories introduced before Definition 2.17). This is done by replacing the notion of subset by that of monomorphism. An intersection 1 ∩ 2 inside  is the fibered product of two monomorphisms ↪ , and the preimage of ↪ by a morphism ∶ → is the fibered product × . From now on, an arrow ↪ between two complex spaces stands for a locally closed embedding of complex spaces (as opposed to just a monomorphism) and we will call it simply an embedding. This means that ↪ factors as the composition of an open and a closed embedding. In particular, any embedding ( , ♯ ) is a monomorphism in the category of complex spaces and ♯ ∶  → *  is an epimorphism of sheaves.
An ' * ' symbol on a diagram stands for the complex manifold consisting of a single point. Given two closed complex subspaces = ( ) and = ( ) of a complex space , we write ∶ = ( ∶ ) ⊆ .
Given a holomorphic map ∶ → between manifolds, the corank of at ∈ is This work is written in the category of complex spaces, but it can be adapted to the category of schemes with separated morphisms. Also, some results cited here are stated for schemes in their original sources.

Residual spaces
Here we introduce some properties satisfied by residual spaces, and their relation to blowups. These two constructions are similar, the difference being that the symmetric algebra plays for residual spaces the role played by the Rees algebra for blowups. Proofs which we could not find in the literature are contained in the Appendix. Our definition of residual (complex) space is just an adaptation of Kleiman's definition of residual scheme. We refer to [11,Section 2] for more details about residual schemes. For the definition and basic properties of the symmetric algebra we refer to [4].
Definition 1.1. Given a closed complex subspace = ( ) ⊆ , the residual space of along is the relative homogeneous spectrum where ( ) stands for the symmetric algebra of , regarded as an  -module. The residual space comes equipped with a canonical proper morphism Res → .
The residual space Res can be described locally (on the base ) as follows: Given a point ∈ , since is a coherent sheaf, we can choose an open neighborhood of where we have a presentation for some × -matrix = ( ) with entries in  ( ). This induces a graded presentation wherê= (̂1, … ,̂) and thêare the linear formŝ It follows that −1 ( ) is isomorphic to the closed complex subspace of × ℙ −1 given by the vanishing of̂, with = 1, … , . Moreover, the restriction ∶ −1 ( ) → can be identified with the projection onto the first component. It is obvious from this description that is an isomorphism on ⧵ . Another relevant property is that Res = Bl when is regularly embedded in (which also follows from Micali [19]). By definition, is regularly embedded in if is locally generated by a regular sequence g 1 , … , g . In this case, thêcorrespond to the trivial relations between the g . We have = and by replacing the subindex by ( , ), with 1 ⩽ < ⩽ , This agrees with the local description of the blowup Bl along a regularly embedded subspace . The local description also makes clear that, in general, computing a residual space amounts to computing the matrix of syzygies. By contrast, the blowup is a more difficult object to deal with. Proof. See Proof 2 in the Appendix. □ The previous property is key to this work, and it does not hold if one replaces residual spaces by blowups. For example, taking  = ℂ 2 , 1 = ( ), 2 = ( − 2 ) and  = ( , ), one obtains Bl ( 1 ∩ 2 ) = ∅, but Bl 1 ∩ Bl  2 ≠ ∅. This is the main reason why, in order for multiple point spaces to satisfy many desirable properties, they are defined by means of residual spaces rather than blowups.
Residual schemes and blowups are nevertheless very much related. The next result, due to Micali [19] (see also [ Going back to the previous example, we obtain Res  ( ) = Bl  ( ). In order not to contradict Proposition 1.3, the spaces Bl  ( 1 ∩ 2 ) and Res  ( 1 ∩ 2 ) must disagree, which is allowed by the fact that  = ( , ) is not regularly embedded in the complex space 1 ∩ 2 = ( 2 , ).
Remark 1.5. Taking into account that −1 ( ) is a Cartier divisor, the previous property can be restated as follows: If is regularly embedded in , then Observe that −1 ( ′ ) is the total transform of ′ in Bl but, in general, Res ′ is not the strict transform. In fact, we see in Section 6 that Res ′ may have greater dimension than ′ .
We remark that some authors like Fulton [7, Definition 9.2.1] consider only the case where is a Cartier divisor in and use the right-hand side of the above equality as a definition of residual scheme. Lemma 1.6. Let ↪  and let  be a closed regularly embedded subspace of . If  ∩ is regularly embedded in , then for the blowup maps  ∶ Bl   →  and ∶ Bl ∩  → .
Proof. See Proof 3 in the Appendix. By construction, the multiple point spaces satisfy the iteration principle: Proposition 2.2. For any map between complex spaces, and any integers , ′ ⩾ 0 and , ′ ⩾ 1, such that + = ′ + ′ , the multiple points satisfy Away from the preimage of the diagonal, the double point space 2 is isomorphic to × , and measures the failure of injectivity of . What goes on over the diagonal is a more delicate matter, specially if we allow to be singular. ) .
Proof. By the iteration principle, it suffices to show the result for = 2. Since is a submersion, Δ ⊆ × ⊆ × are submanifolds and, by Proposition 1.4, 2 = Res Δ ( × ) = Bl Δ ( × ). One checks easily that 2 → is submersive. □ The reader may be surprised that we care about multiple points of submersions. After all, multiple points have always been regarded as a tool for the study of finite and generically one-to-one maps. As we will see, submersions do play a role in the study of multiple points of general maps between manifolds.
Before being able to compute for possibly non-submersive maps, we need to figure out certain relations between spaces = ( ) and ′ = ( ′ ), in terms of relations satisfied by maps ∶ → and ′ ∶ ′ → ′ . This is the starting point: Proof. By the iteration principle, it suffices to show the first step of the extension, because every square in the extended diagram is of the form ( ). Notice that the embedding × ↪ ′ × ′ , ′ has Δ as the preimage of Δ ′ . Moreover, the construction of this embedding is functorial in the same sense of Item 2, and gives an isomorphism if the hypotheses of Item 3 are met. Applying Proposition 1.2, we obtain the embedding 2 ↪ ′ 2 , also in a functorial way. Therefore, the resulting embedding also satisfies Items 2 and 3. □ The simplest diagram ( ) gives an interesting outcome: For a fixed space , the multiple point spaces of all maps → can be embedded into unique universal spaces. where → * is the constant map. In order to distinguish the universal spaces of different complex spaces, we write ( ) = .
Proposition 2.7. The universal multiple point spaces satisfy the following: (1) For every map ∶ → , there is a canonical embedding ↪ .
(2) For every embedding ↪ ′ , there is a canonical embedding The construction of is functorial in the sense that the embeddings in Item 2 commute with compositions. The embeddings of Items 1 and 2 commute with the morphisms → −1 , → −1 and ′ → ′ −1 , and they are compatible with the structure maps to the fibered products.
Proof. Extend, respectively, the following two diagrams:

Proof.
Letting = ( ), the claim follows inductively from the isomorphisms Proof. This is a particular case of Proposition 2.4, because → * is a submersion. □ By abuse of notation, all these blowup maps are written as Now consider a map ∶ ℂ → , with a manifold. Once embedded in 2 (ℂ ), the double point space 2 of has a nice geometric interpretation, going back to work of Ronga [33]. Proposition 2.11. As a set, the space 2 of a map ∶ ℂ → between manifolds consists of the following points: Observe that functoriality plays an impoortant role in the previous constructions. In a diagram of the form ( ), the spaces , and ′ are canonically embedded in ′ ; but the functoriality of the involved constructions gives us more: Extending the horizontal arrows in the diagram we obtain that the compositions ↪ ↪ ′ and ↪ ′ ↪ ′ are equal. By the universal property of the intersection of and ′ in ′ , we obtain the following result: Proof. By the iteration principle Proposition 2.2, it suffices to prove the theorem in the case = 2. Moreover, we know that × ′ = × , so we may assume = ′ and consider the commutative diagram: The embedding ↪ ′ factors as the composition of an open and a closed embedding. By Proposition 1.2 we can suppose that ↪ ′ is either open or closed. But the case that ↪ ′ is open is obvious from the definition, so we only need to consider the case that ↪ ′ is closed. Now we apply Proposition 1.
we have which gives 2 = ′ 2 ∩ 2 , the intersection being made in ′ 2 . □ Something that illustrates the usefulness of previous result is that it allows us to extend a result about mappings between complex manifolds to a result about mappings between complex spaces: Proposition 2.14. Given two maps ∶ → , = 1, 2, let Inside , we have Proof. In the case where , 1 and 2 are complex manifolds, the statement follows immediately from Theorem 5.2. This theorem gives the generators of in explicitly, and it is clear that the generators of ( ) are obtained by putting together those of ( 1 ) and ( 2 ) (see Remark 5.3). The proof will thus be complete only after Theorem 5.2 is proved; here we show how to reduce the general case to that of maps between complex manifolds.
Given a point ∈ , to prove the statement it suffices to find an open neighborhood of in , such that Letting  =  1 ×  2 and = ( 1 , 2 ), Theorem 5.2 implies that ( ) = ( 1 ) ∩ ( 2 ), as we mentioned before. Take = ( ), which is embedded in ( ) naturally, and can also be considered as an open subset of = ( ) containing . Now consider the diagrams and the analogous diagrams for 1 and 2 . Applying Theorem 2.13, we obtain the desired equality: The following result of Kleiman (see [11,Proposition 2.4]) can be recovered from our previous results.

Proposition 2.15.
For any map ∶ × → × of the form ( , ) = ( , ( )), the following hold: Proof. The first item follows by putting Propositions 2.8 and 2.14 together. For the second item, apply Theorem 2.13 to the diagram Theorem 2.13 has less obvious applications than the previous ones. As an example, we sketch the computation of double points of reflection maps, introduced in [29]. Example 2.16. Let be a reflection group acting on ℂ . The orbit map (or quotient map) of is a polynomial map ℂ ⟶ ℂ , taking a -orbit to a point, that is, −1 ( ( )) = , for all ∈ ℂ . A reflection map is a map ∶ → ℂ forming a commutative diagram for some embedding ℎ.
As it turns out, some basic theory of reflection groups suffices to describe the double point space of the orbit map, which is a union of smooth components indexed by . To be precise, the double points are the reduced space where each g ⊆ 2 (ℂ ) is obtained by blowing up the graph of the map g ∶ ℂ → ℂ along are -finitely determined if the integers , , , , , are pairwise coprime and , and are odd.
As already noted, functoriality is key for our results. We finish this section by introducing a new layer of formalism, where the multiple point spaces are replaced by multiple point functors, and then giving reinterpretations of our results from this point of view.
The arrow category () of a category  has as objects the arrows of , and as morphisms → ′ the commutative diagrams of the form which we simply call squares. The identity square and the composition of squares are defined in the obvious way. Now let  be the category of complex spaces. We write ′ () and ′′ (), respectively, for the wide subcategories of () (that is, subcategories including all objects but only some of the morphisms) whose squares are, respectively, of the forms Thanks to Proposition 2.5, we may disguise Kleiman's multiple point spaces as functors.
Definition 2.17. The double point functor is taking a map → to the map 2 → . At the level of morphisms, 2 is given by In accordance with the iteration principle, the iterated multiple point functors are defined as These functors give maps ∈ Hom( , −1 ), for any map ∶ → . To be consistent with the convention that 1 = and 0 = , the functor 1 is set to be the identity ′ () → ′ (). Now that we have turned multiple point spaces into functors, we shall revisit the results obtained so far. From the fact that complex spaces have fibered products, it follows easily that (), ′ () and ′′ () have them as well. Indeed, the fibered product 1 × 2 of two arrows ∶ → , each equipped with a square to ∶  → , is the canonical arrow equipped with the two squares forming the top and left faces of the commutative cubical diagram satisfying the usual universal property of fibered products. We call such a diagram a cartesian cube. Now Theorem 2.13 and Proposition 2.14 can be rephrased as the fact that, for cartesian cubes of the forms the multiple point functors commute with fibered products. In other words, given a diagram of any of the two forms above, we have that However, at this point we do not have a proof or a counterexample to the fact that commutes with general fibered products.

A FORMULA FOR INSIDE
We give an explicit expression for , embedded in , in the case where is a complex manifold. This will lead us to the formulas for from Section 5. For any ⩾ 2, we may assume inductively that we have embedded −1 in −1 and that we have a map −2 → −2 (for the initial case of = 2, we take the identity map → and → * ). This induces an embedding Recall that we write ∶ → −1 × −2 −1 for the blowup maps, and for their exceptional divisors.
Theorem 3.1. For any map ∶ → between manifolds and any ⩾ 2, Step 1: We show the property analogous to Theorem 2.13 for . For any commutative diagram of maps between manifolds we have = ′ ∩ . In other words, = ℎ −1 ( ′ ), for the embedding ↪ ′ canonically induced by ↪ ′ .
The right to left inclusion holds in general, so we are left with the inclusion ( is a complex subspace of ′ , and, applying ′−1 , it follows that ′ ∩ is a complex subspace of ′−1 ( ′ ) or, equivalently, that ⊆ . Moreover, the ideal is principal, because ′ is a the exceptional divisor in ′ , so locally we may write = ⟨ ⟩. Now let ∈ ( + ) ∶ ( + ) = ( + ) ∶ . We have = + , with ∈ and ∈ . Since ⊆ , then = , for some . Since is smooth and is not contained in the exceptional divisor, we have ∶ = , and hence ∈ . We have ( − ) = ∈ , so − ∈ ∶ , and therefore ∈ ( ∶ ) + .
Step 2: We reduce the proof to showing = for submersions. For any map → between manifolds, consider the commutative diagram where Γ is the graph embedding and the projection on the second factor. The embedding Γ induces embeddings ( ) ↪ ( × ), and is obviously a submersion. Assuming that = for submersions, from Theorem 2.13 and the previous step we obtain the equality Step 3: We show = for submersions. By induction, we may assume −1 = −1 and −2 = −2 . We know from Proposition 2.4 that all maps → −1 are submersions between smooth spaces. This in turn implies the smoothness of Since submanifolds are regularly embedded, from Lemma 1.6 we obtain Remark 3.2. Theorem 3.1 was already known for = 2, that is, for double points of maps between smooth spaces, see, for example, [7, Section 9.3]. However, unlike other results found here, this one does not admit an easy reduction to the case = 2. Observe that the source and target of higher maps → −1 may not be smooth, and that we do not know a priori that = −1 ( −1 × −2 −1 )∶ satisfies any short of iteration principle. In general, arguments about the algebraic structure of higher are delicate, and it is apparent that our proof relies heavily on the machinery from previous sections.

COORDINATES FOR THE UNIVERSAL SPACES
We give coordinates for ( ) in the case where is a complex manifold. First, we justify that it suffices to give coordinates for the spaces (ℂ ) Let be a partition 1 + ⋯ + = of . We say that a point ∈ is of type if the components of its projection in consist of different points ( ) ∈ , each repeated times (in a possibly disordered way).

Proposition 4.2. Let ∶ → be a map between complex manifolds. Around a point
Proof. See Proof 4 in the Appendix. □ Remark 4.3. Let ∈ be a point of type as above, for a manifold . Then is locally isomorphic at to for some disjoint coordinate open subsets , which we may regard them as subsets ⊂ ℂ . Since both and ℂ are smooth of the same dimension, it follows from Propositions 2.9 and 2.7 that ( ) is an open submanifold of (ℂ ). Hence, coordinates for ( ) are obtained by restriction of the coordinates for (ℂ ).

A pyramid of maps
We need one last ingredient, the triangular diagram below, before giving coordinates for the spaces . We write the -fold fibered product of a map → as Now we are ready to give coordinates for = (ℂ ). For technical reasons, we need to describe the spaces ( ∕ −1 ) in the diagram ( ) and the maps between them. To gain intuition, we start with the universal double and triple points for ℂ 2 . This will also fix the notation for double and triple points of maps corank ⩽ 2, as in Examples 5.4 and 5.12.

4.2
Atlas for the universal double point space (ℂ ) Let = ℂ 2 . As stated in Example 2.10, the universal double point space of is The space 2 is covered by the open subsets 1 , 2 , where These open subsets are isomorphic to ℂ 4 , respectively, via the maps
Recall how the coverings for the case of = ℂ 2 were indexed: If we are only interested in double points, that is, if = 2, then two open subsets indexed by = 1, 2 are enough. If we are interested in triple points, that is, = 3, then we covered the spaces 2 and ( 2 ∕ℂ 2 ) 2 by three open subsets indexed by = 1, 2, 3. The space 3 was then covered by six open subsets , with multi-indices ( , ), with = 1, 2, 3 and = 1, 2.
From now on, we fix a positive integer , satisfying + ⩽ + 1. The unusual placement of coordinates in ( ) is designed to match the diagram ( ) of Remark 4.5. We refer to the columns of ( ) decreasingly from + − 2 to 0, so that the th column is the one containing ( ) . which drops the top row of ( ) (in the case of = 2, the index ( 1 , … , −2 ) is meant to be 0 ∈  1 ).
To simplify notation, once a multi-index is fixed, we write = .
The remaining ( , ) , marked with '•', are determined by the '•' entries, by means of the relations
Since we are mainly interested in the spaces = ( ∕ −1 ) 1 , we may take = . We write  =  and = 1 , as well as

EQUATIONS FOR THE MULTIPLE POINT SPACES
Here we give an explicit set of local equations for ( ) in the coordinates of the affine open subsets described above. where ( −1, ) stands for the function ( −1) + ( ( ) ). We omit the multi-indices if there is no risk of confusion.
Proof. We proceed by induction on . By Theorem 3.1, we have: is the projection onto the first factor, and hence The induction hypothesis states that −1 ( −1 ) is defined in the coordinate system ( , ) of by the ideal sheaf generated by the coordinate functions of [ , (1) ], … , [ , (1) , … , ( −2) ].
Since ( −1) is not a zero divisor in  ∕ , this implies that the defining ideal sheaf of as a complex subspace of −1 ( −1 ) is generated in ( , ) by the classes of the coordinate functions of [ , (1) , … , ( −1) ]. □ Remark 5.3. It follows from Theorem 5.2 that, for a mapping between complex manifolds, the local generators of the ideal sheaf defining ( ) are obtained one by one from the coordinate functions of . This implies that Proposition 2.14 holds for mappings between manifolds (the generalization to mappings between complex spaces is given next to Proposition 2.14). Definition 5.5. We say that is dimensionally correct if dim = − ( − 1).

Corollary 5.6. Let → be a map between complex manifolds. The dimension of is at least − ( − 1) at any point. If is dimensionally correct, then it is locally a complete intersection in .
Proof. This follows from the fact that is locally defined by ( − 1) equations in □ We finish this section by explaining how the computations can be simplified for maps of lower corank, and giving an example. Recall that two maps ∶ → and ′ ∶ ′ → ′ between manifolds are called -equivalent if there exist two biholomorphisms ∶ → ′ and ∶ → ′ , such that ′ = • ′ • −1 . As a consequence of Theorem 2.13, we obtain the following: A map ∶ × → × of the form ( , ) = ( , ( )) is said to be an unfolding of the map 0 ∶ → , for each 0 ∈ . The manifold is called the parameter space. Recall that, by Proposition 2.15, the multiple point space ( ) is canonically embedded in × (ℂ ), and ( ) ∩ { = 0 } = ( 0 ). One checks easily that ( ) is computed as follows: Proposition 5.8. Let ∶ × ℂ → × ℂ be an unfolding of the form ( , ) = ( , ( )) and fix a covering collection of (ℙ −1 ) −1 . In each of the open subsets × , ∈  , the multiple point space ( ) is given by the vanishing of [ , (1) ], … , [ , (1) , … , ( −1) ]. Definition 5.9. In the setting above, we call [ , (1) , … , ( ) ] the relative divided differences of ( ). These expressions were introduced by Marar and Mond's in [16] as equations for their multiple point space ( ) of a corank one map. Consequently, we obtain the following result. For arbitrary corank, a space 2 ( ) ⊆ × was introduced by Mond in [20]. A general construction of multiple point spaces ( ) ⊆ was given by the authors in [25] (see also [22,Section 9.2]). Example 5.12. We are going to compute the spaces 2 and 3 of the map ∶ ℂ 3 → ℂ 4 given by ( , , ) ↦ ( , 2 + , 2 − , 3 + 3 + ).

F I G U R E 1 Maps unfolded by
One checks easily that has an isolated point of corank 2 at the origin. Furthermore, and is a one-parameter unfolding and, for ≠ 0, the mapping ( , , ) has three crosscaps and a triple point, collapsing to the origin as tends to zero. One may think of the mappings (0, , ) and ( , , ) as complex versions of the mappings depicted in Figure 1.
Since is an unfolding, Proposition 5.8 ensures that we may compute as a subspace of ℂ × (ℂ 2 ) by means of the relative divided differences. As already mentioned, the expressions from Example 5.4 compute double and triple points of maps of corank two, leaving as a parameter. In what follows, the notation for the atlas and divided differences is taken from there.
The first entry L [1] has the equations of 2 on the open subset 1 . The variables l(1),a(1) correspond to and . The space 2 ∩ 1 has dimension 2 and thus 2 is dimensionally correct and a complete intersection on on 1 . The projection 2 ∩ 1 → ℂ × (ℂ 2 ) 2 , with coordinates ( , ( , ), ( ′ , ′ )), is described by In order to cover 2 , the divided differences on 2 must be computed. We omit them, as they yield nothing new. The equations for 2 ∩ 2 on Singular are the content of the entry L [2]. Now we move to the computation of triple points 3 . As part of the process for triple points, we must compute double points in the extra open subset 3 . Again, this computation is uninteresting and omitted.
Somehow surprisingly, the images of 3 ∩ 11 and 3 ∩ 12 on ℂ × (ℂ 2 ) 3 are the same, despite coming from spaces of different dimensions (again, this can be checked with Singular). A moment of thought will convince the reader of the fact that this implies that 3 has an irreducible component contained in the exceptional divisor of 3 → 2 × ℂ 3 2 . This and other related pathologies are explained in the next section.
Remark 5.13. When using the library IteratedMultPoint.lib on an -parameter unfolding , it is convenient to introduce the parameters in the front of the list of polynomials defining . This way, the procedure ItMP(f,r); makes computations in ℂ × (ℂ − ), as indicated in Proposition 5.8. If, for example, we were to reorder the coordinate functions of the previous example as list f=x2+ty,y2-tx,x3+y3+xy,t, the equations will be given in (ℂ 3 ). The equations and the coverings would still be correct, but more complicated.
Remark 5.14. The computation of ItMP(f,r) involves choosing a covering collection for (ℙ −1 ) −1 , which the procedure does internally. A different collection will give the same space , but may result in very different covering and equations.

PATHOLOGIES
As Kleiman observes in [11], the idea that is the double point space of −1 → −2 is just a definition, with a clear interpretation only for strict multiple points. As it turns out, in the presence of points of corank ⩾ 2 the iteration principle may yield too many points, and may do so in a nonsymmetrical way. These pathologies come as no surprise; the excess of dimension and the fact that and the target multiple points disagree are somehow easy set-theoretical considerations, while the lack of symmetry was already pointed out by Ran in [31, Section 1]. Our explicit description of just allows us to be more precise about them, something that will be crucial for results in the sequel of this work [26].

Excess of dimension
Assume that a map ∶ → between manifolds has corank ⩾ 2 at ∈ . Following the description of double points in Proposition 2.11, we may take any two different points (1) , (2) ∈ ℙ(ker ) to produce two double points ( , , (1) ) and ( , , (2) ). Since the map 2 → drops the ( ) , these two points form a point in 2 × 1 2 away from Δ 2 . This point is the image of a point in 3 under the map 3 → 2 × 1 2 , which locally is an isomorphism. Since this preimage is not contained in the exceptional divisor, it is contained in 3 = −1 ( 2 × 1 2 ) ∶ −1 (Δ 2 ). Summarizing, any pair of different points , ′ ∈ ℙ(ker ) produces a triple point, with no further conditions on , ′ . The argument carries on to higher multiple spaces; the following result counts exactly how many points are obtained this way. Proposition 6.1. Let be a map between manifolds, let be a point where the corank of is ⩾ 2 and let ⩾ 2. The preimage of ( , … , ) ∈ by the map → has dimension ( − 1)( − 1).

and target multiple points
Kleiman, Lipman and Ulrich [14] studied relations between the multiple point spaces given by iteration and by the Fitting ideals. For any finite map ∶ → , the subspace ( ) ⊆ is given by the vanishing of the ( − 1)st Fitting ideal of *  . The image of is 1 ( ), the double points of in are 2 ( ), and so on. Pulling back these points, we obtain the multiple point spaces ( ) = −1 ( ( )) ⊆ . Here, and are not assumed to be smooth, and a suitable extended definition of corank is used. Write +1 ⟶ for the usual maps, and ⟶ the maps obtained by composition.
For finite maps ∶ → of corank one, with dim = dim + 1, they show that and that all are finite maps of corank one. In this case, the maps are also finite and have corank one, and one obtains the set-theoretical equality This means that the projection ( ) of the iterated multiple points is the same as the inverse image −1 ( ( )) of the target multiple points.
The first problem that one encounters in the case of corank ⩾ 2 is that the maps , and hence the are not finite anymore. Therefore their pushforward modules are not finitely presented modules, their Fitting ideals are not defined and it is not clear how one should define the algebraic structure of the projections ( ). But the problem is worse, as ( ) and −1 ( ( )) do not agree even at the set-theoretical level. To see this, just observe that ( ) is empty, for every bigger than the multiplicity of , while is never empty in the presence of points of corank ⩾ 2, as a consequence of Proposition 6.1. It is worth mentioning that the spaces ( ) have their own disadvantages compared to ( ). The first is that they are only defined for finite mappings. The second is that they are hard to compute, since the calculation of Fitting ideals requires computing a presentation matrix of the pushforward module *  , which is a very hard task when the corank of is high. In contrast, and despite their pathologies, the spaces have advantage of being given by the explicit equations of Theorem 5.2.

Lack of symmetry
It is well known that the multiple point spaces ⊆ are invariant by the action of the symmetric group by permutation of the coordinates in . It would be reasonable to expect the spaces to have natural actions, lifting those on , and that these actions restrict to actions on compatible with those on . We show that, unfortunately, this is not the case for ⩾ 3. Before this, we show that, for a manifold , the action of 2 on 2 can be lifted to 2 , and indeed there is a unique way of doing it.
Let be a complex manifold and ∶ 2 → 2 be the map corresponding to the transposition (1 2). To lift the action of 2 on 2 , we must find an involutioñ∶ 2 → 2 , making the diagram commutative. For this we use the fact that 2 is the blowup of 2 along Δ . Let ℎ be the composite map 2 → 2 ⟶ 2 . Since the preimage of Δ by is again Δ , the preimage ℎ −1 (Δ ) is the exceptional divisor of 2 . The universal property of the blowup 2 → 2 applies to ℎ and gives a unique map̃satisfying the above commutativity. The fact that̃is an involution follows from the fact that̃can be identified with on 2 ⧵ , a dense subset of 2 . We observe that the liftĩ s defined in local coordinates in the obvious waỹ( 1 , 2 , ) = ( 2 , 1 , ). Now we show that, for a complex manifold of dimension at least 2, there is no action of 3 on 3 lifting the action on 3 . We assume for simplicity that = ℂ . Since the transposition (1 2) takes the set Δ 13 = {( , ′ , ) ∈ 3 } to Δ 23 = {( ′ , , ) ∈ 3 }, it suffices to show that the preimages in 3 of Δ 13 and Δ 23 are not isomorphic. We start by looking at the preimages of Δ 13 and Δ 23 in the space 2 × 2 . The space 2 × 2 consists of points At an open subset given by, say, ′ ≠ 0 ≠ ′′ , the preimage of Δ 12 is given by the equation ′ − = 0 and hence is a Cartier divisor. However, the preimage of Δ 23 equals ∪ Δ 2 , where the diagonal 2 is given by ′′ = ′ and ′ = , and is a codimension 2 component given by ′ = ′′ = 0, which is not contained in Δ 2 because ⩾ 2.
On one hand, it is clear that the preimage of Δ 13 is a divisor in 3 , because it is the preimage of the corresponding preimage in 2 × 2 , which is already a divisor. On the other hand, the space 3 is obtained by blowing up Δ 2 , and therefore the preimage of Δ 2 is also a divisor in 3 . However, since the structure map of the blowup is an isomorphism away from the exceptional divisor, and is not contained in Δ 2 , we conclude that the preimage of Δ 13 has a component of codimension 2 arising from . This shows that the preimages of Δ 13 and Δ 23 are not isomorphic.

APPENDIX: SOME PROOFS
Two basic lemmas will be used, which follow immediately from the definition of the symmetric algebra.
Lemma A.2. For any ring morphism ′ → and any -module , there is unique graded morphism of algebras ′ ( ) → ( ), which is ′ → in degree zero and id in degree one. For all ⩾ 1, the degree part ( ′ ( )) → ( ( )) is an epimorphism, and it is an isomorphism if ′ → is an epimorphism.
Proof 1 (Proposition 1.2). Given ∶ ↪ , let = −1 (). We have an epimorphism of   -modules which restricts to an epimorphism  ↠ * , where  and are the ideals which define  in  and in , respectively. By Lemma A.1, this morphism extends to an epimorphism By Lemma A.2, there is also an epimorphism   ( * ) ↠ *  . Composing these two epimorphisms we obtain   (  ) ↠ *  ( ), which induces the monomorphism Res ↪ Res  . Functoriality follows from functoriality of the elements involved.
Proof 2 (Proposition 1.3). Assume that = ( ) and  = ( ) for some coherent ideal sheaves 1 , 2 and on such that 2 ⊆ . We use the local description given after Definition 1.1 to show the required equality between the residual spaces. Let ∶ Res   →  be the structure map. We take a point ∈ and a small enough open neighborhood of in . Put =   ( ) for simplicity. Given any × -matrix = ( ) with entries in we denote bŷthe ideal in the polynomial ring [ 1 , … , ] generated by the linear formŝ= ∑ , with = 1, … , .
Proof 4 (Proposition 4.2). To use induction on we prove a slightly more detailed result. We write for ( ( ) ), as the ( ) are clear from the context. Let ∈ project to a point ∈ −1 , via the map → −1 . We claim that −1 ( ), around , is locally isomorphic to for a partition 1 , … , = − 1. But we also claim that one of the following statements holds: (1) is locally isomorphic to 1 × ⋯ × × 1 , and the map → −1 drops the last component. (2) is locally isomorphic to 1 × ⋯ × +1 × ⋯ × , the map → −1 being the restriction of There is nothing to prove for 1 and for 2 there are two cases: if ∈ 2 is not contained in the exceptional divisor of 2 , then locally 2 is isomorphic to × , that is, 2 ≅ 1 × 1 , which is case (1). On the exceptional divisor we are looking at 2 → 1 , which is case (2). Now assume that the statement holds up to ⩾ 2 and compute × −1 , around a point whose first projection is ∈ and its second projection is ′ ∈ . Consider the following cases: If is of the form (1) around both and ′ , then × −1 is isomorphic to We need to subdivide this case further: If = ′ , then the previous isomorphism takes Δ to 1 × ⋯ × × Δ 1 . In this case, the space +1 is locally isomorphic to The map +1 → is given by 2 → 1 on the last component and leaves the other components untouched. This is an instance of case (2 and we are in the case (2).
We leave to the reader to check the following cases: If is of the form (2) around and of the form (1) around ′ , then we are in case (1). When is of the form (2) around and ′ , we are in case (2).
Proof 5 (Lemma 4.4). We proceed by induction on , skipping the case of = 2, as it is analogous to the general case. Assume that the statement is true for − 1. We may use the induction hypothesis and usual commutativities for fibered products to obtain the equality of the following compositions:  .