Rev Mat Complut (2016) 29:439–454 DOI 10.1007/s13163-015-0187-5 Geometry and equisingularity of finitely determined map germs from C n to C 3, n > 2 A. J. Miranda1 · V. H. Jorge Pérez2 · E. C. Rizziolli3 · M. J. Saia2 Received: 8 September 2014 / Accepted: 29 December 2015 / Published online: 11 January 2016 © Universidad Complutense de Madrid 2016 Abstract In this article we describe the geometry and the Whitney equisingularity of finitely determined map germs f :(Cn, 0) → (C3, 0) with n ≥ 3. In the study of the geometry, we first investigate the critical locus �( f ) of the germ, which is in the source. Then the discriminant �( f ), the image of the critical locus by the germ f , is studied. Last, but not least we investigate the set X ( f ), which is the inverse image by f of the discriminant. If the critical locus is not empty, the set X ( f ) is an hypersurface in the source that has nonisolated singularity at the origin. Concerning the Whitney equisingularity of families, we use some of the properties of the strata to prove that the Whitney equisingularity of an unfolding F is equivalent to the con- stancy of the Lê numbers of the hypersurfaces �( f ) and X ( f ). From this study we describe some relationship among the invariants needed to describe theWhitney equi- singularity of families in these dimensions, we reduce the number of invariants needed This work was partially supported by CAPES, CNPq, FAPESP and FAPEMIG. B E. C. Rizziolli eliris@rc.unesp.br A. J. Miranda aldicio@famat.ufu.br V. H. Jorge Pérez vhjperez@icmc.usp.br M. J. Saia mjsaia@icmc.usp.br 1 Faculdade de Matemática, Universidade Federal de Uberlândia, Uberlândia, MG, Brazil 2 Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, SP, Brazil 3 Departamento de Matemática, Instituto de Geociências e Ciências Exatas, Universidade Estadual Paulista Júlio Mesquita Filho, Rio Claro, SP, Brazil 123 http://crossmark.crossref.org/dialog/?doi=10.1007/s13163-015-0187-5&domain=pdf 440 A. J. Miranda et al. to a total of 2n + 2, which improves substantially the number required by Gaffney’s theorem. Keywords Geometry of map germs · Whitney equisingularity · Numerical invariants · Lâ numbers Mathematics Subject Classification Primary 32S15 · 14B05 1 Introduction The study of the geometry of the singularities of map germs is one of the main questions in singularity theory, a key tool to better understanding of the geom- etry is the description of all strata which appear in the critical locus �( f ), in the discriminant �( f ), and in the hypersurface X ( f ). Moreover, in these sets we can study the numerical invariants that control triviality conditions in families of map germs. In this article, first we investigate the geometry of finitely determined map germs f : (Cn, 0) → (C3, 0) with n ≥ 3, in the second section we give an explicit description of all strata in these dimensions and, with the aid of a computer system, we show in an explicity way how to compute them in several examples. Concerning the Whitney equisingularity, Gaffney describes in [1] the follow- ing problem: “Given a 1-parameter family of map germs F : (C × C n, (0, 0)) → (C×C p, (0)), find analytic invariants whose constancy in the family implies the fam- ily is Whitney equisingular.” He shows that for the class of finitely determined map germs of discrete stable type, the Whitney equisingularity of such a family is guaran- teed by the invariance of the zero stable types and the polar multiplicities associated to all stable types. A natural question is to find a minimal set of invariants that guarantee the Whitney equisingularity of the family. Gaffney and Vohra in [2] studied map germs from n- space to the plane, with n ≥ 3. In this case they showed that the top dimensional stratum of the inverse image of the discriminant carries all information necessary to determine if the family of map germs is Whitney equisingular, in fact they showed that Lê numbers of this stratum control all other invariants needed for the Whitney equisingularity. In [3] the finitely determined map germs inO(n, 3)with n ≥ 3 are investigated and 2n+ 2 invariants are obtained under the restriction that the map germ has corank one, it is also shown in [3] that the Lê numbers of X ( f ) control theWhitney equisingularity of this set. Here we continue the study of such map germs, the main difference on one hand, we show that the Whitney equisingularity of X ( f ) also implies the Whitney equisingularity of the strata in�( f ), and on the other hand,we use the Lê numbers also for the discriminant �( f ), instead of the polar multiplicities as done in [3], moreover we show that the corank one condition is not needed. We use the Surfex program [4], to produce the pictures in this work. 123 Geometry and equisingularity of finitely determined map germs… 441 2 Geometry of finitely determined map germs f ∈ O(n, 3) 2.1 Stable types of finitely determined map germs We follow Gaffney in [1] and denote by O(n, p) the set of origin preserving germs of holomorphic mappings from C n to C p, Oe(n, p) denotes the set of germs at the origin but not necessarily origin preserving. For a germ f ∈ Oe(n, p), call m = min{n, p} and denote the ideal generated by the set of m × m minors of the derivative of f by J ( f ), the critical set of f , denoted �( f ), consists of the set of points x ∈ C n such that the differential is not an epimorphism. The discriminant �( f ) of f is the image of �( f ) by f . When n ≥ p there appear naturally in the inverse image of �( f ) other points than the points in the critical set, moreover these points form a set in C n with the same codimension than �( f ) inCp. In this case we denote by X ( f ) the hypersurface( f −1(�( f )) − �( f ) ) with reduced structure. As usual,wedenote byR theMather’s groupof local diffeomorphismson the source (Cn, 0), by L, the Mather’s group of local diffeomorphisms on the target (Cp, 0), A denotes the group the product L × R and K denotes the group of contact. A given map-germ f ∈ O(n, p) is said to be k-G-determined (G = A or K) if whenever g ∈ O(n, p) and j k f (0) = j kg(0), then g isG-equivalent to f ; f is finitely G-determined if it is k-G-determined for some k < ∞. In order to lighten notation, and in accordance with standard practice, whenever we say “finitely determined (without specifying whether A or K)”, we shall mean “finitely A-determined”. Map germs which are finitely K-determined are called map germs of finite sin- gularity type [5, p. 201]. Using the symmetry between the theory of A-equivalence and the theory of K-equivalence, we know that, in particular if f ∈ O(n, p) with n ≥ p is finitelyA-determined, then f has finite singularity type. In this situation, the restriction f |�( f ) : �( f ) → � f is a finite map [any point in the target has at most a finite number of pre-images], and hence dim�( f ) = dim� f . We say f is a stable germ if every “nearby germ is A-equivalent to f ”. A stable type is an A-equivalence class of stable germs. A stable type Q appears in a versal unfolding F of f , if for any representative F = (id, fu(x)) of F , there exists a point (u, y) ∈ C s ×C p such that the germ fu : (Cn, S) → (Cp, y) is a stable germ of type Q, S = f −1(y) ∩ �( fu). The points (u, y) and (u, x) with x ∈ S are called points of stable type Q in the target and in the source, respectively. A finitely determined germ f has discrete stable type if there exists a versal unfolding of f in which only a finite number of stable types occur. In particular, if the numbers (n, p) are in Mather’s “nice dimensions” [which is true for (n, 3), n ≥ 3, our focus here] or on the boundary thereof, then every finitely determined germ f ∈ O(n, p) has discrete stable type. The Mather–Gaffney geometric criterion of finite determinacy states that: A germ f is finitely determined if, and only if, there exist open neighborhoods U of 0 in the source and V of 0 in the target such that f −1(0) ∩ U ∩ �( f ) = 0 and for each y ∈ V , y �= 0, the germ f : (Cn,S) → (Cp, y) is a stable germ, where S = f −1(y) ∩U ∩ �( f ). See [1, Proposition 1.3], or [6]. 123 442 A. J. Miranda et al. 2.2 Geometry of the critical set and the discriminant The critical set �( f ) of f is described by its Thom–Boardman stratification, which we recover next. For any Boardman symbol i = (i1, . . . , ir ) with 1 ≤ r ≤ 3, denote by �i ( f ) the set of points in �( f ) of type i . The sets �i ( f ) are obtained as the zero sets of the iterated Jacobian ideals, notion due to Morin in [7]. Let f : (Cn, 0) → (Cp, 0) be an analytic map-germ and I ⊂ On an ideal generated by the system g1, . . . , gr . For each � ∈ {1, . . . , n}we define the jacobian extension of rank � for the pair ( f, I ), by ��( f, I ) := I + I�(d( f1, . . . , f p, g1, . . . , gr )), where I�(d( f1, . . . , f p, g1, . . . , gr )) denotes the ideal generated by minors of size � × � from the jacobian matrix of the f1, . . . , f p, g1, . . . , gr and (d( f1, . . . , f p, g1, . . . , gr )). If i = (i1, . . . , ik) is a Boardman number, we inductively define the Iterated Jaco- bian Ideal for i , Ji ( f ) in the followingmanner: Ji ( f )= { �n−i1+1( f, {0}) if k=1 �n−ik+1( f, Ji1,...,ik ( f )) if k>1. Therefore for all n ≥ 3 in any finitely determinedmap germ from (Cn, 0) to (C3, 0), �( f ) = �n−2( f ) is the zero set of the Jacobian ideal Jn−2( f ), �n−2,1( f ) is the zero set of the first iterated Jacobian ideal Jn−2,1( f ) and �n−2,1,1( f ) is the zero set of the ideal Jn−2,1,1( f ). Lemma 2.1 For any finitely determined map germ from (Cn, 0) to (C3, 0) with ≥ 3, (i) �( f ), the critical set of f, is either empty or 2-dimensional; (ii) The set �n−2,1( f ) is either empty or 1-dimensional; (iii) The set �n−2,1,1( f ) is either empty or 0-dimensional. Proof These results follow by the Mather–Gaffney geometric criterion of finite deter- minacy of the germ f , this implies that f has isolated instability at the origin, therefore the sets above have the expected dimension. � Lemma 2.2 For any finitely determined map germ from (Cn, 0) to (C3, 0) with ≥3, the sets f −1(0), �n−2( f ), �n−2,1( f ) and �n−2,1,1( f ) have isolated singularity. Proof Since f is finitely determined, it follows from the Geometric criterion of Mather-Gaffney (see [6]), that there exist neighborhoods U and V of 0 in C n such that f −1(0) ∩ U ∩ �( f ) = 0 and for each y ∈ V , y �= 0, x �= 0 with x ∈ f −1(�) and x ∈ �( f ), the germ f : (Cn, x) → (C3, y) is stable, hence it is written as f (x1, x2, . . . , xn) = (x1, x2, g(x1, x2, x3) + ∑n j=4 x 2 j ) and the germ (x1, x2, g(x1, x2, x3)) is a stable germ from C 3 to C 3, therefore the result follows from [8, Section 3]. We remember that such germ f is called a suspension of the germ (x1, x2, g(x1, x2, x3)). � To describe the discriminant�( f ) = f (�n−2( f )), we remember that for all n ≥ 3, the stable singularities which appear in the target are precisely the same type than map germs fromC 3 toC3, namely the ordinary double point curve, the cuspidal edge curve, the ordinary triple points, swallowtails, the transversal crossings of a cuspidal edge with a plane and the regular part, denoted A1( f ). According to Arnold’s notation we denote these sets by A1,1( f ), A2( f ), A1,1,1( f ), A3( f ) and A2,1( f ) respectively, and 123 Geometry and equisingularity of finitely determined map germs… 443 call A( f ) the set A( f ) = A1,1,1( f ) ∪ A2,1( f ) ∪ A3( f ) ∪ A2( f ) ∪ A1,1( f ), hence �( f ) = A1( f ) ∪ A( f ). The curves A1,1( f ) and A2( f ) are the 1-stable singularities of f and the sets A1,1,1( f ), A3( f ) and A2,1( f ) are the 0-stable singularities of f . Definition 2.3 We call source double point curve, denoted by D( f ) the curve given by the intersection f −1(A1,1( f )) with �n−2( f ). To obtain this curve we consider its associated ideal I (D( f ) as the reduction of the corresponding ideals, or I (D( f )) := √〈I ( f −1(A1,1( f ))), Jn−2( f )〉. We denote by g : (C3, 0) → (C, 0), the defining equation of the discriminant with reduced structure or g−1(0) = �( f ). Example 2.4 Let f (x, y, z, w) = (x, y, z6 + xz + yz2 + w2). This germ from C 4 to C 3 is a suspension of the germ f0(x, y, z) = (x, y, z6 + xz + yz2) and in this case it is easier to compute the stables types which appear in �( f ) and in �( f ) from the suspension. First we compute the critical set �( f0) = �1( f0) and the curve �1,1( f0) using the iterated Jacobian ideals J1( f0) = I (�( f0)) := 〈x + 2 yz + 6 z5〉 and J1,1( f0) = I (�1,1( f0)) := 〈x − 24z5, y + 15z4〉. For the germ f we have the iterated Jacobian ideals J2( f ) = I (�( f )) := 〈x + 2 yz + 6 z5, w〉 and J2,1( f ) = I (�2,1( f )) := 〈x − 24z5, y + 15z4, w〉. To obtain the source double point curve of the germ f0 we apply the method described in [8]which uses theVandermondematrix associated to the set D2( f0) ⊂ C 4 consisting of the points (x, y, z1, z2) with z1 �= z2 in C 4 such that f0(x, y, z1) = f0(x, y, z2). The associated ideal I (D2( f0)) is: I (D2( f0)) := 〈x + 12z21z 3 2 + 12z31z 2 2 + 6z1z 4 2 + 6z41z2, y − 3z41 − 3z42 − 12z1z 3 2 − 12z2z 3 1 − 15z21z 2 2, 4z 3 1 + 6z21z2 + 6z22z1 + 4z32〉. The projection of D2( f0) to C3 gives the double point set D( f0) and I (D( f0)) := 〈x + 2yz + 6z5, 16y3 + 153y2z4 + 540yz8 + 675z12〉. For the germ f we can not apply the results shown in [8] since the results there are for map germs from C n to Cn , then in general we compute D( f ) using the definition of its associated ideal I (D( f )), however, as this germ is a suspension of f0 we know that I (D( f )) := 〈I (D( f0)), w〉 (Fig. 1). Fig. 1 The real part of the set �( f0) and the real parts of the curves �1,1( f0) and D( f0) 123 444 A. J. Miranda et al. Fig. 2 The real part of the discriminant �( f ) = �( f0) in the target Fixing (X,Y, Z) as the target variables, we obtain the defining equation of the discriminant �( f ): g(X,Y, Z) = 46656Z5 + 43200Z2Y 2X2 + 13824Y 3Z3 + 1024ZY 6 + 22500ZY X4 + 256Y 5X2 + 3125X6. The cuspidal curve A2( f ) satisfies the equations 25X2 + 96Y Z = 0 and 135Z2 + 4Y 3 = 0 and the ideal of the double point curve in the target is I (A1,1( f )) := 〈25X3 + 36XY Z , 675X2Z + 432Y Z2 + 64Y 4, 135X Z2 − 16XY 3, 27Z3 + 5X2Y 2 + 4Y 3Z〉. or I (A1,1( f )) := 〈27Z2 + 4Y 3, X〉 ∩ 〈−135Z2 + 16Y 3, 25X2 + 36Y Z〉. To obtain these equations in the target we use the Fitting ideals of the discriminant, concept presented byMond and Pellikaan in [9, sections 1. and 2.], for the computation of the Fitting ideals we apply the results in [10,11] (Fig. 2). The Milnor numbers μ(�( f )), μ(�2,1( f )) and μ(D( f )) are shown in the table below. Germ f μ(�( f )) = μ(�( f0)) μ(�2,1( f )) = μ(�1,1( f0)) μ(D( f )) = μ(D( f0)) (x, y, z6 + xz + yz2 + w2) 0 0 22 Next we show the numbers of 0-stable singularities and also the Milnor numbers μ(A2( f )) and μ(A1,1( f )). Germ f �A3( f ) �A1,1,1( f ) �A2,1( f ) μ(A2( f )) μ(A1,1( f )) (x, y, z6 + xz + yz2 + w2) 3 1 6 8 17 We remember that the numbers �A3( f ), �A2,1( f ) and �A1,1,1( f )mean the number of corresponding points in a stable perturbation of the germ f . In general, the space curves A1,1( f ) and A2( f ) are not a complete intersection and the Milnor number 123 Geometry and equisingularity of finitely determined map germs… 445 refers to the Buchweitz–Greuel definition of the Milnor number for such curves, see [12]. To compute these numbers we use the software Singular [13]. 2.3 The geometry of f−1(�) For map germs from C n to Cp with n ≥ p there appear naturally in the inverse image of �( f ) other points than the points in the critical set, moreover these points form a set in Cn with the same codimension than �( f ) in Cp. When p = 3 in f −1(�( f )) there appear the inverse images by f of the sets A1,1( f ), A2( f ), A1,1,1( f ), A3( f ), A2,1( f )which are also codimension preserving, or in other words, f −1(A1,1( f )) and f −1(A2( f )) are (n − 2)-dimensional, since A1,1( f ) and A2( f ) are curves inC3, f −1(A1,1,1( f )), f −1(A3( f )) and f −1(A2,1( f )) are (n−3)- dimensional, since A1,1,1( f ), A3( f ) and A2,1( f ) are isolated points inC3.We remark also that f −1(0) is (n − 3)-dimensional. In general the critical sets of f −1(A( f )), f −1(A1,1( f )), f −1(A2( f )), f −1(A1,1,1( f )), f −1(A3( f )) and f −1(A2,1( f )) possible have nonisolated singu- larity at the origin. Remark 2.5 When n = p = 3 the hypersurface f −1(�) in C n has two components, the surfaces X ( f ) := ( f −1(�) − �( f )) and�( f ). The intersection of X ( f )∩�( f ) is formed by the curves �1,1( f ) and D( f ) together with the origin. In �1,1( f ) this intersection is tangential while in D( f ) this intersection is transversal. On the other hand, for n > 3 the critical set �( f ) is two dimensional while X ( f ) is (n − 1)-dimensional, in this case �( f ) ⊂ X ( f ). In fact, in these dimensions one has X ( f ) = f −1(�). Now we study the geometry of f in the source from the point of view of the defining equation of the discriminant, denoted g : (C3, 0) → (C, 0). We denote by h : (Cn, 0) → C the composition h := g ◦ f . Lemma 2.6 1. If �( f ) �= ∅, then �( f ) ∩ f −1(0) = {0}. 2. g is a submersion at each point of �( f )\A( f ). 3. �( f ) ⊆ �(h) ⊆ �( f ) ∪ f −1(A( f )). 4. When n > 3, X := X ( f ) = V(h), and I(X) = (h). Proof (1) If �( f ) is nonempty, it is 2-dimensional, and it is clear that both �( f ) and f −1(0) contain the point {0} in the source. The desired result now holds from the finite determinacy of the germ f as this implies that f |�( f ) is a finite map. (2) For all points z in �( f )\A( f ) sufficiently close to 0, we know that �( f ) is smooth at z; since g defines�( f )with reduced structure, we conclude that Dg(z) �= 0. (3) We first show that �( f ) ⊆ �(g ◦ f ). It clear that this inclusion holds if �( f ) = ∅; consider the case of �( f ) �= ∅, then for any z ∈ �( f ), with z �= 0 and z close to zero, we know that z is in the set �n−2( f ) (fold) or in the set �n−2,1( f ) (cusp), and in both cases we see that all minors 3 × 3 of the Jacobian matrix at z are equal to 0, this also holds for the point z = 0 since it is part of �( f ), then it follows that locally the inclusion �( f ) ⊆ �(g ◦ f ) holds since the gradient matrix of g ◦ f is also zero at these points. 123 446 A. J. Miranda et al. Fig. 3 The real part of the hypersurface f −1 0 (�( f0)) in C3 In regard the second inclusion �(g ◦ f ) ⊆ �( f ) ∪ f −1(A( f )), consider a point z ∈ �(g ◦ f )\�( f ); it suffices to show that z belongs to f −1(A( f )). At such point z, we see that f is a submersion, since z /∈ �( f ) and as D(g ◦ f )(z) = 0 it follows from the chain rule that we then must have Dg( f (z)) = 0. Since we also know that g is a submersion at each point of � f \A( f ), we conclude that z ∈ f −1(A( f )). (4) This proof is done by using Serre’s criterion R0 and S1, see [14] or [15]. Since h defines a hypersurface in C n it is Cohen–Macaulay and we have S1. Condition R0 means that the hypersurface defined by h is smooth in codimension ≥ 1, but this follows from that fact that�(h) = �(g ◦ f ) has dimension≤ n−2 by item 2. Hence, h is reduced by Serre’s criterion. � Remark 2.7 For n > 3 we see from the item 4 of this lemma that one of the following inequalities �( f ) ⊆ �(g ◦ f ) ⊆ �( f ) ∪ f −1(A) of the item 3 is always an equality. But as the discriminant set is given by the union �( f ) = A1( f ) ∪ A( f ) the equality �( f ) = �(g ◦ f ) occurs if, and only if, A( f ) is empty, or in other words, f is a fold and this case is simple. Therefore we consider from now on that f is not a fold and that the equality �(g ◦ f ) = �( f ) ∪ f −1(A( f )) always holds. We remember also that for all n ≥ 3, f −1(A2)∩ f −1(A1,1) = f −1(0) as the germ f is finitely determined. Example 2.8 Let f (x, y, z, w) = (x, y, z6 + xz + yz2 + w2). First we obtain the defining equation of the hypersurfaces f −1 0 (�( f0)) in C 3 and f −1(�( f )) in C 4 for both germs, f0 and its suspension f . Then we have f −1 0 (�( f0)) := (7424 y3xz3 + 13200 y2x2z2 + 10000 x3yz + 5616 z16y + 9072 z12y2 + 6672 z8y3 + 9156 x3z5 + 10908 x2z10 +6048 xz15 +24156 x2z6y+21312 xz7y2 +19872 xz11y+1296 z20 + 2176 y4z4 + 3125 x4 + 256 y5)(x + 2 yz + 6 z5)2. Here f −1 0 (�( f0)) splits in two components �( f0) and X ( f0), as �( f0) := x + 2 yz + 6 z5, we obtain X ( f0) := 7424 y3xz3 + 13200 y2x2z2 + 10000 x3yz + 5616 z16y + 9072 z12y2 + 6672 z8y3 + 9156 x3z5 + 10908 x2z10 + 6048 xz15 + 24156 x2z6y+21312 xz7y2+19872 xz11y+1296 z20+2176 y4z4+3125 x4+256 y5. We remark that since n = 3 one has �( f0) ∩ X ( f0) = D( f0) ∪ �1,1( f0) (Fig. 3). On the other hand, X ( f ) := f −1(�( f )) = 1024 y7z2 + 1024 y6w2 + 2799360 z15xyw2 + 2799360 z9xyw4 + 2799360 z11xy2w2 + 2799360 z10x2yw2 +1399680 x2z4yw4 + 1486080 x2z6y2w2 + 933120 x3z5yw2 + 1399680 xz5y2w4 +1016064 xz7y3w2 + 933120 xz3yw6 + 86400 y2x3zw2 + 127872 y3x2z2w2 + 41472 y3xzw4 + 82944 y4xz3w2 + 933120 z19xw2 + 1399680 z14yw4 + 933120 z20yw2+1399680 z17xy2+1399680 z16y2w2+ 1399680 z14x2w2 + 933120 z11x3y + 1399680 z13xw4 +1399680 z16x2y+933120 z21xy+1399680 z8x2w4 +933120 123 Geometry and equisingularity of finitely determined map germs… 447 z9x3w2 + 1442880 z12x2y2+974592 z13xy3+933120 z7xw6+1399680 z10y2w4+ 974592 z12y3w2 +933120 z8yw6+255780 x4z6y+466560 x3z3w4+233280 x4z4w2 + 552960 x3z7y2+594432 x2z8y3+466560 x2z2w6+316224 xz9y4+233280 xzw8 +508032 y3z6w4+316224 y4z8w2+466560 y2z4w6+233280 yz2w8 +65700 y2x4z2 +100224 y3x3z3+84672 y4x2z4+43200 y2x2w4 +41472 y4z2w4+41472 y5z4w2+ 41472 y5xz5+1024 y6xz+22500 x5yz+22500 x4yw2+256 x2y5 + 233280 z25x+ 233280 z26y + 233280 z24w2 + 466560 z20x2 + 466560 z22y2 + 466560 z18w4 + 466560 z15x3 + 480384 z18y3 + 466560 z12w6 + 233280 z10x4 + 274752 z14y4 + 233280 z6w8 + 46656 x5z5+88128 y5z10+14848 y6z6+13824 y3w6+46656w10+ 46656 z30 + 3125 x6. In X ( f ) there exists the singular surface obtained as the inverse image of the double point curve with associated ideal I ( f −1(A1,1( f ))) := 〈x, 4y3 + 27w4 + 54yz2w2 + 27y2z4 + 54z6w2 + 54yz8 + 27z12〉 ∩ 〈−3125x4 + 768y5, 25x2 + 36xyz + 36yw2 + 36y2z2 + 36yz6, 64y4 + 375x3z + 375x2w2 + 375x2yz2 + 375x2z6,−32y3 − 105x2z2 + 540xzw2 + 270w4 − 270y2z4 + 540xz7 + 540z6w2 + 270z12〉. There exists also the singular surface obtained as the inverse image of the cuspidal curve and its associated ideal is I ( f −1(A2( f ))) := 〈25x2 + 96xyz + 96yw2 + 96y2z2 + 96yz6, 4y3 + 135x2z2 + 270xzw2 + 135w4 + 270xyz3 + 270yz2w2 + 135y2z4 + 270xz7 + 270z6w2 + 270yz8 + 135z12〉. If we consider w = 0 in the ideals above one has the corresponding ideals of these curves for the germ f0 : (C3, 0) → (C3, 0): I ( f −1 0 (A1,1( f0))) := 〈25x3+36x2yz+36xy2z2+36xyz6, 200x2y2+416xy3z− 160y4z2−2955x3z3−4455x2yz4−1080xy2z5−920y3z6−4455x2z8−1080y2z10+ 1080xz13 + 1080yz14 + 1080z18, 64xy3 − 165x3z2 − 540x2yz3 − 1080x2z7 − 540xyz8−540xz12, 64y4+375x3z+675x2yz2+432xy2z3+432y3z4+675x2z6+ 432xyz7 + 864y2z8 + 432yz12〉. We remark that this ideal is not radical and √ I ( f −1 0 (A1,1( f0))) := I (D( f0)) ∩ HD( f0), with HD( f0) := 〈75625x2 + 93800xyz + 69696y2z2 + 75696xz5 + 97200yz6 + 33696z10, 18875xy + 7360y2z + 14487xz4 + 16656yz5 + 8352z9, × 20y2 − 7xz3 + 19yz4 + 3z8〉. or HD( f0) := 〈125y2 + 138yz4 + 45z8, 25x + 11yz + 15z5〉 ∩ 〈4y + 3z4, x〉. The branch corresponding to the ideal HD( f0) belongs to X ( f0). 123 448 A. J. Miranda et al. The inverse image of the curve A2( f0) by f0 has corresponding ideal: I ( f −1 0 (A2( f0)) ) := 〈25x2 + 96xyz + 96y2z2 + 96yz6, 128y3 + 3195x2z2 + 4320xyz3 + 8640xz7 + 4320yz8 + 4320z12〉. With radical ideal: √ I ( f −1 0 (A2( f0))) := I (�1,1( f0)) ∩ H1,1( f0), where H1,1( f0) := 〈2000y3 + 3528y2z4 + 2565yz8 + 675z12,−20y2 + 42xz3 + 9yz4 + 27z8, 300xy+352y2z−75xz4 +360yz5, 375x2 +240xyz+32y2z2 +300xz5〉. The branch corresponding to the ideal H1,1( f0) belongs to X ( f0). The Buchweitz–Greuel Milnor numbers of the curves √ ( f −1 0 (A2( f0))),√ (g−1(A1,1( f0))), H1,1( f0), HD( f0) and HD( f0) in the set X ( f0) of the germ f0 are shown below. Germ f0 μ √ ( f −1 0 (A2( f0))) μ √ (g−1(A1,1( f0))) μ(H1,1( f0)) μ(HD( f0)) ( x, y, z6 + xz + yz2 ) 31 67 16 16 Example 2.9 Let f : C4 → C 3 : f (x, y, u, v) = (x, y, yu + xv + u3 + v3). This germ from C 4 to C 3 is not a suspension, then we need to compute all its stable types directly. The critical set inC4 is two dimensional with Jacobian ideal J2( f ) = I (�2( f )) := 〈y + 3u2, x + 3v2〉. The ideal for the source critical curve is: J2,1( f ) = I (�2,1( f )) := 〈x + 3v2, y + 3u2, uv〉 and μ(J2,1( f )) = 1. Now compute the defining equation of the discriminant �( f ): g(X,Y, Z) := 〈729Z4 + 216X3Z2 + 216Y 3Z2 + 16X6 − 32X3Y 3 + 16Y 6〉. The ideals of the singular curves of the discriminant are: I (A2( f )) := 〈XY, 27Z2+ 4X3+4Y 3〉withμ(A2( f )) = 7 and I (A1,1( f ))) := 〈Z , X3−Y 3〉withμ(A1,1( f )) = 4. In the source C4 we obtain the ideal of the hypersurface X ( f ) = f −1(�( f )): I (X ( f )) := 〈16x6 − 32x3y3 + 16y6 + 216x3y2u2 + 216y5u2 + 432x4yuv + 432xy4uv + 216x5v2 + 216x2y3v2 + 432x3yu4 + 1161y4u4 + 432x4u3v + 3348xy3u3v + 4374x2y2u2v2 + 3348x3yuv3 + 432y4uv3 + 1161x4v4 + 432xy3v4 + 216x3u6 + 3132y3u6 + 8748xy2u5v + 8748x2yu4v2 + 3348x3u3v3 + 3348y3u3v3 + 8748xy2u2v4 + 8748x2yuv5 + 3132x3v6 + 216y3v6 + 4374y2u8 + 8748xyu7v + 4374x2u6v2 + 8748y2u5v3 + 17496yu4v4 + 8748x2u3v5 + 4374y2u2v6 + 8748xyuv7 123 Geometry and equisingularity of finitely determined map germs… 449 + 4374x2v8 + 2916yu10 + 2916xu9v + 8748yu7v3 + 8748xu6v4 + 8748yu4v6 + 8748xu3v7 + 2916yuv9 + 2916xv10 + 729u12 + 2916u9v3 + 4374u6v6 + 2916u3v9 + 729v12〉. We also have the corresponding ideals to the (n − 2)-dimensional surfaces f −1(A2( f )) and f −1(A1,1( f )). I ( f −1(A2( f ))) := 〈y, 4x3 + 27x2v2 + 54xu3v + 54xv4 + 27u6 + 54u3v3 + 27v2〉 ∩ 〈x, 4y3 + 27y2u2 + 54yv3u + 54yu4 + 27u6 + 54u3v3 + 27v2〉 and I ( f −1(A1,1( f ))) := 〈x3 − y3, yu + xv + u3 + v3〉. We remark that these two ideals are radical, complete intersections and the origin is not isolated singularity. The source double point curve D( f ) is obtained from the definition of its associated ideal I (D( f ). I (D( f )) = 〈 f −1(A1,1( f )), J2( f )〉 = 〈x + 3v2, y + 3u2, u3 + v3〉. This ideal is radical and the curve is an isolated complete intersection singularity, or ICIS for short, and μ(D( f )) = 4. We can check that this curve is already the source double point curve as follows. From the generators of I (D( f )) write x = −3v2, y = −3u2, u3 = −v3, then as u = −v, one has f|D( f )(x, y, u, v) = (−3v2,−3u2,−2u3 − 2v3) = (−3v2,−3u2, 0). Now, calling X, Y and Z for the variables in the target one has : X = −3u2, Y = −3v3 and Z = 0. Therefore since u3 = −v3, u6 = v6 and X3 = Y 3, or X3 − Y 3 = 0, therefore as Z = 0 one has I ( f (D( f ))) = I (A1,1( f )) = 〈X3 − Y 3, Z〉. 3 Whitney equisingularity of map germs Let F : (C × C n, (0, 0)) → (C × C p, (0, 0)), F = (t, f (t, x)), be a 1-parameter unfolding of a finitely determined germ f , such that f (t,−) preserves the origin for all t . Let T := C × {0}. F is a good unfolding of f if there exist neighbourhoods U , W of the origin in C×C n and C×C p respectively such that F−1(W ) = U , F maps U∩�(F)−T toW−T and if (t0, y0) ∈ W−T , then thegerm ft0 : (Cn, S) → (Cp, y0) is stable, where S = F−1(t0, y0) ∩ �(F). A good unfolding is excellent if all the 0-stable invariants are constant in the unfold- ing and f is of discrete type. An unfolding F of f isWhitney equisingular along the parameter space T if there exists a regular stratification of the source and the target, with T a stratum of the source and the target and these stratifications are Whitney equisingular along T , i.e. satisfy the Whitney conditions a and b and Thom’s AF condition. One of the questions of main interest is to show when an excellent unfolding F : (C × C n, (0, 0)) → (C × C p, (0, 0)) of a finitely determined germ f ∈ O(n, p) is Whitney equisingular. 123 450 A. J. Miranda et al. Using the polar invariants, i.e, the polar multiplicities of the polar varieties of the stable types Gaffney showed in the proof of the Theorem 7.1 of [1] the following. Theorem 3.1 [1, Theorem 7.1] Suppose that F : (C×C n, (0, 0)) → (C×C p, (0, 0)) is an excellent unfolding of a finitely determined germ f ∈ O(n, p). Also suppose that the polar invariants of all stable types defined in: 1. the discriminant �( ft ) = ft (�( ft )), 2. the singular set �( ft ), 3. X ( ft ) := ( f −1 t (�( ft )) − �( ft )), are constant at the origin for all t . Then the unfolding is Whitney equisingular. As a consequence of a convenient version of Thom’s isotopy lemma [1, Theo- rem 6.1], the Theorem 7.1 of Gaffney shows the topological triviality of the family. The theorem above remains valid if we replace the term “an excellent unfolding” in the hypothesis by “a 1-parameter unfolding” which, when stratified by stable types and by the parameter axis T , has only the parameter axis T as 1-dimensional stratum at the origin [2]. Here we use the geometry of the germ to control the polar invariants needed to ensure the Whitney equisingularity of a family of map germs inO(n, 3) when n > 3. Now we fix F : (C × C n, (0, 0)) → (C × C 3, (0, 0)) with n > 3 be a good 1-parameter unfolding of a finitely determined map germ f ∈ O(n, 3). For each fixed t we call ft (x) = F(t, x), then according to the Theorem 3.1 of Gaffney we need to control all polar invariants in the target and also in the source of the map germs ft . Call Xt := X ( ft ) = ( f −1 t (�( ft )) − �( ft )), X1( ft ) := ( f −1 t (A1,1( ft )) − D( ft )) and X2( ft ) := ( f −1 t (A2( ft )) − �n−2,1( ft )). We remember that if gt is the defining equation of the discriminant �( ft ) and ht := (gt ◦ ft ) : (Cn, 0) → (C, 0), then Xt = X ( ft ) = V(ht ). Consequently, if we suppose that F : (C × C n, (0, 0)) → (C × C 3, (0, 0)), n > 3 is an excellent unfolding of a finitely determined germ f ∈ O(n, 3), n > 3, we need of the constancy of 4n + 10 polar invariants (at the origin for all t) to guarantee the unfolding is Whitney equisingular. To control all polar invariants needed to apply the Theorem 3.1 of Gaffney we show some equations involving all these polar invariants and invariants associated to the set Xt . First we recover basic definitions about the Lê numbers and relative polar multi- plicities necessary to better understand the results shown here. To study the set X ( f ), as it has possibly non isolated singularity at zero, we need to understand the Lê cycles and relative polar multiplicities associated to it. For this we recover these concepts, given by Massey in [16], for any analytic function h : (U, 0) → (C, 0) with U an open subset of CN+1 containing the origin. We assume that the reader is familiar with the notion of coherent sheaves, gap sheaves, schemes and cycles. In order to fix the notation, for a sheaf α and an analytic subsetW in some affine space, we denote by α/W the corresponding gap sheaf and by V (α)/W the scheme associated with the sheaf α/W , where V (α) denotes the analytic space defined by the vanishing ofα.We shall at times enclose cycles in square brackets, [ · ]. 123 Geometry and equisingularity of finitely determined map germs… 451 For the definitions in this section let h : (U, 0) → (C, 0) be an analytic function, U an open subset of CN+1 containing the origin and z = (z0, z1, . . . , zN ) a linear choice of coordinates in C N+1. We fix the dimension of the singular set �(h) of h is s with 0 ≤ s ≤ N − 1. Definition 3.2 For 0 ≤ k ≤ N , the kth (relative) polar variety, �k h,z , of h with respect to z is the scheme V ( ∂h ∂zk , . . . , ∂h ∂zN ) /�(h). The kth (relative) polar multiplicity , denoted mk(h, z), is the multiplicity of �k h,z On the level of ideals, �k h,z consists of the components of V ( ∂h ∂zk , . . . , ∂h ∂zN ) which are not contained in �(h). We denote by [�k h,z] the cycle associated with this scheme. In particular we note that �0 h,z is empty and we call �N+1 h,z = U . Definition 3.3 For 0 ≤ k ≤ N , the kth Lê cycle, k h,z , of h with respect to z is the cycle [ �k+1 h,z ∩ V ( ∂h ∂zk ) ] − [�k h,z]. In general we shall denote this cycle simply k h,z , and not [ k h,z], because unlike the polar varieties which are defined as schemes and we have to consider the associated cycle, this definition is given in terms of cycles. If the intersection of k h,z with the cycle of V (z0 − p0, . . . , zk−1 − pk−1) is purely 0-dimensional at a point p = (p0, p1, . . . , pN ), i.e., either p is an isolated point of the intersection or p is not in the intersection, it is possible to define the Lê numbers as follows: Definition 3.4 For 0 ≤ k ≤ N , the kth Lê number, λkh,z(p), of h with respect to z at p, is defined as the intersection number ( k h,z · V (z0 − p0, . . . , zk−1 − pk−1))p. Definition 3.5 For any linear generic subspace Lk ⊆ (CN+1, 0), with k = 0, . . . , N+ 1 the reduced Euler characteristic of the Milnor fiber of the h | Lk at 0, denoted χ(k), is defined by χ(k) := mk(h) + λN+1−k(h) − λN+1−(k−1)(h) + λN+1−(k−2)(h) + · · · + λN+1−s(h) with s = dim V (J (h)) and mk(h) denotes the kth (relative) polar multiplicity at 0. The Euler characteristic, denoted just χ(∗), is the following sequence: χ(∗) := (χ(N+1), . . . , χ(2)). The number χ(k) does not depend on the choice of the Lk . Next we show a condition for the parameter axis T = C × {0} in C × C 3 to be stratum of a Whitney stratification of the discriminant �(F). 123 452 A. J. Miranda et al. Theorem 3.6 [17, p.32] The pair (�(F), T ) is Whitney equisingular if, and only if, the sequence (m1(gt ),m2(gt ), χ∗ gt ) is independent of t . This theorem ensures the Whitney equisingularity of the pair (�(F), T ) using invariants defined in the target, since each germ gt : (C3, 0) → (C, 0) gives the defining equation of the discriminant �( ft ) in C3. Now we denote X (F) := (F−1(�(F)) − �(F)) and show how to control the Whitney equisingularity of the pair (X (F), T ) in the source. Theorem 3.7 [3, Theorem 6.3] Suppose that the stratification by the stable types of F has only the parameter space T = C × {0} in C × C n as a locus of instability and the singular set of X (F) is Cohen Macaulay. Then the pair (X (F), T ) is Whitney equisingular if, and only if, the sequence (m1(ht ), . . . ,mn−1(ht ), χ∗ ht ) is independent of t . From now on we consider that F : (C×C n, (0, 0)) → (C×C 3, (0, 0))with n > 3 is a good 1-parameter unfolding of a finitely determined map germ f ∈ O(n, 3). We also suppose that the stratification by the stable types of F has only the parameter space T = C× {0} in C×C n as a locus of instability and the singular set of X (F) is Cohen Macaulay. Then we have the following Proposition 3.8 The unfolding F is Whitney equisingular if, and only if, the sequences: (m1(gt ),m2(gt ), χ ∗ gt ) and (m1(ht ), . . . ,mn−1(ht ), χ ∗ ht ) are independent of t. Proof In this setup we obtain from the Remark 2.7 that the singular set of X (F) is equal to �(F) ∪ F−1(A) and from the Theorem 3.7 we obtain that the independence of the sequence (m1(ht ), . . . ,mn−1(ht ), χ ∗ ht ) also guarantees the Whitney equisingularity along the parameter space T of all other strata of �(F) and F−1(A). Therefore the result appears as a consequence of the Theorems 3.6 and 3.7. � We also can show the following: Proposition 3.9 Let F : (C × C n, (0, 0)) → (C × C 3, (0, 0)) with n > 3 be a good 1-parameter unfolding of a finitely determined map germ f ∈ O(n, 3). Suppose that the stratification by the stable types of F has only the parameter space T = C × {0} in C × C n as a locus of instability and the singular set of X (F) is Cohen Macaulay. Then the unfolding F is Whitney equisingular if, and only if, the sequences: (m1(gt ),m2(gt ), λ 0(gt ), λ 1(gt )) and (m1(ht ), . . . ,mn−1(ht ), λ 0(ht ), . . . , λ n−2(ht )) are independent of t. 123 Geometry and equisingularity of finitely determined map germs… 453 Proof From [18, p.726] we can get the following equivalences: (i) (m1(gt ),m2(gt ), χ∗ gt ) is independent of t if, and only if, (m1(gt ),m2(gt ), λ 0(gt ), λ 1(gt )) is independent of t . (ii) (m1(ht ), . . . ,mn−1(ht ), χ∗ ht ) is independent of t if, and only if, (m1(ht ), . . . ,mn−1(ht ), λ 0(ht ), . . . , λ n−2(ht )) is independent of t and the result follows. � Remark 3.10 From the Proposition 3.9 we obtain that there are needed 2n + 2 invari- ants, for instance when n = 4, we only need 10 invariants, while 26 were required from the Theorem 3.1. References 1. Gaffney, T.: Polar multiplicities and equisingularity of map germs. Topology 32(1), 185–223 (1993) 2. Gaffney, T., Vohra, R.: A numerical characterization of equisingularity for map germs from n-space, (n ≥ 3), to the plane. J. Dyn. Syst. Geom. Theor. 2, 43–55 (2004) 3. Jorge Pérez, V.H., Rizziolli, E.C., Saia, M.J.: Whitney equisingularity, Euler obstruction and invariants of map germs from C n . In: Real and Complex Singularities, Trends in Mathematics, pp. 263–287. Birkhäuser, Basel (2007) 4. Holzer, S., Labs, O.: surfex 0.90, University of Mainz and University of Saarbrücken (2008). http:// www.surfex.AlgebraicSurface.net 5. Wall, C.T.C.: Lectures onC∞-stability and Classification. In: Lecture Notes in Mathematics, vol. 192, pp. 178–206. Springer, Berlin (1971) 6. Wall, C.T.C.: Finite determinacy of smooth map germs. Bull. Lond. Math. Soc. 13, 481–539 (1981) 7. Morin, B.: Calcul Jacobian. Ann. Sci. École Norm. Sup. 8, 1–98 (1975) 8. Jorge Pérez, V.H., Levcovitz, D., Saia, M.J.: Invariants, equisingularity and Euler obstruction of map germs. Journal fur die Reine und Angewandte Mathematik, Crelle’s Journal 587, 145–161 (2005) 9. Mond, D., Pellikaan, R.: Fitting ideals and multiple points of analytic mappings. In: Algebraic Geom- etry and Complex Analysis (Pátzcuaro, 1987), pp. 107–161. Lecture Notes in Mathematics, vol. 1414. Springer, Berlin (1989) 10. Jorge Pérez, V.H., Miranda, A.J., Saia, M.J.: Counting singularities via fitting ideals. Int. J. Math. 23(6), 1250062-1–1250062-18 (2012) 11. Hernandes, M.E., Miranda, A.J., Peñafort-Sanchis, G.: An algorithm to compute a presenta- tion of pushforward modules (2014) (Pre-print). https://sites.google.com/site/aldicio/publicacoes/ presentation-matrix-algorithm 12. Buchweitz, R.-O., Greuel, G.-M.: TheMilnor number and deformations of complex curve singularities. Invent. Math. 58(3), 241–281 (1980) 13. Greuel, G.-M., Pfister, G., Schönnemann, H.: Singular: A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern, free software under the GNU General Public Licence 1990–2007. http://www.singular.uni-kl.de 14. Grothendieck, A., Dieudonné, J.: Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie. Publications Mathématiques de l’IHÉS 24 (1965). doi:10.1007/bf02684322 15. Matsumura, H.: Commutative Algebra, 2nd edn. The Benjamim/Cummings Publishing Company, Inc., Reading (1980) 123 http://www.surfex.AlgebraicSurface.net http://www.surfex.AlgebraicSurface.net https://sites.google.com/site/aldicio/publicacoes/presentation-matrix-algorithm https://sites.google.com/site/aldicio/publicacoes/presentation-matrix-algorithm http://www.singular.uni-kl.de http://dx.doi.org/10.1007/bf02684322 454 A. J. Miranda et al. 16. Massey, D.: Lê cycles and hypersurface singularities. In: Lecture Notes in Mathematics, vol. 1615. Springer, Berlin (1995) 17. Gaffney, T., Massey, D.: Trends in equisingularity theory. In: Bruce, J.W., Mond. D. (eds.) Singularity Theory. London Mathematical Society Lecture Note Series 263, pp. 207–248. Cambridge University Press, Cambridge (1999) 18. Gaffney, T., Gassler, R.: Segre numbers and hypersurfaces singularities. J. Algebraic Geom. 08, 695– 736 (1999) 123 Geometry and equisingularity of finitely determined map germs from mathbbCn to mathbbC3, n >2 Abstract 1 Introduction 2 Geometry of finitely determined map germs f inmathcalO(n,3) 2.1 Stable types of finitely determined map germs 2.2 Geometry of the critical set and the discriminant 2.3 The geometry of f-1(Δ) 3 Whitney equisingularity of map germs References