A formula to calculate the invariant J of a quasi-homogeneous map germ
AA formula to calculate the invariant J of aquasi-homogeneous map germ O.N. Silva ∗ Abstract
In this work, we consider a quasi-homogeneous, corank 1, finitely determined map germ f from ( C ,
0) to ( C , m ( f ( D ( f )) and J , where m ( f ( D ( f )) denotes the multiplicity of the image of the doublepoint curve D ( f ) of f and J denotes the number of tacnodes that appears in a stabilization of the transversal slicecurve of f ( C ). We present formulas to calculate m ( f ( D ( f )) and J in terms of the weights and degrees of f . In this work, we consider a quasi-homogeneous, corank 1, finitely determined map germ f from ( C ,
0) to ( C , f ( x, y ) = ( x, ˜ p ( x, y ) , ˜ q ( x, y )), forsome quasi-homogeneous function germs ˜ p, ˜ q ∈ m , where m is the maximal ideal of the local ring of holomorphicfunction germs in two variables O (see Lemma 2.11).In [7], Nu˜no-Ballesteros and Marar studied the transversal slice of a corank 1 map germ f : ( C , → ( C ,
0) (see[7, Section 3]). They show in some sense that if a set of generic conditions are satisfied, then the transverse slice curvecontains information on the geometry of f . They also introduced the invariants C , T and J , which are defined as thenumber of cusps, triple points and tacnodes that appears in a stabilization of the transversal slice of f , respectively.They showed in [7, Prop. 3.10] that the numbers C and T are the same as the numbers of cross-caps and triple pointsthat appear in a stabilization of f , which are usually denoted by C ( f ) and T ( f ), respectively, and were defined byMond in [11]. On the other hand, the invariant J is related to both, the delta invariant of the transverse slice curveof f ( C ) and N ( f ) (see [7, page 1388]), where N ( f ) is the Mond’s invariant defined in [11].In [12], Mond presented formulas to calculate the invariants C ( f ), T ( f ) and µ ( D ( f )) of a quasi-homogeneousfinitely determined map germ of any corank in terms of the weights and degrees of f , where µ ( D ( f )) denotes theMilnor number of the double point curve D ( f ) of f (see Section 2.1 for the definition of D ( f ) and Th. 2.13). So, anatural question is: Question 1:
Let f : ( C , → ( C ,
0) be a quasi-homogeneous, corank 1, finitely determined map germ. Can theinvariant J be calculated in terms of the weights and the degrees of f ?It follows by Propositions 3.5 and 3.10 and Corollary 4.4 of [7] that J = 12 (cid:18) µ ( D ( f )) − C ( f ) − (cid:19) − T ( f ) + m ( f ( D ( f )), ∗ O.N. Silva: Instituto de Matem´aticas, Universidad Nacional Aut´onoma de M´exico (UNAM), Cuernavaca, M´exico.e-mail: [email protected] a r X i v : . [ m a t h . C V ] J a n here m ( f ( D ( f )) denotes the multiplicity of the image of the double point curve D ( f ). So, using Mond’s formulas for C ( f ), T ( f ) and µ ( f ( D ( f )) (Th. 2.13) we conclude that Question 1 has a positive answer if and only if there is a for-mula to calculate the invariant m ( f ( D ( f )) in terms of the weights and the degrees of f . So, another natural question is: Question 2:
Let f be as in Question 1. Can the invariant m ( f ( D ( f )) be calculated in terms of the weights anddegrees of f ?In [13, Proposition 6.2], Ruas and the author provided answers to both questions above in the case where f ishomogeneous. In this work, using a normal form for f (Lemma 2.11), we present a positive answer for both questionswithout any restriction on the weights and degrees of f . More precisely, we present in Theorem 3.2 formulas tocalculate both invariants, m ( f ( D ( f )) and J , in terms of the weights and the degrees of f . We finish this work bycalculating the invariants m ( f ( D ( f )) and J in some examples to illustrate our formulas. Throughout this paper, given a finite map f : C → C , ( x, y ) and ( X, Y, Z ) are used to denote systems of coordinatesin C (source) and C (target), respectively. Also, C { x , · · · , x n } (cid:39) O n denotes the local ring of convergent powerseries in n variables. The letters U, V and W are used to denote open neighborhoods of 0 in C , C and C , respectively.We also use the standard notation of singularity theory as the reader can find in Wall’s survey paper [18]. In this section, we deal only with of corank 1 maps from C to C . For the general definition of double pointspaces, see for instance [5, Section 1], [6, Section 2] and [11, Section 3].Consider a finite and holomorphic map f : U → C , where U is an open neighbourhood of 0 in C . The doublepoint space of f , denoted by D ( f ), is defined (as a set) by D ( f ) := { ( x, y ) ∈ U : f − ( f ( x, y )) (cid:54) = { ( x, y ) }} ∪ Σ( f ),where Σ( f ) is the ramification set of f . We also consider the lifting of the D ( f ) in U × U , denoted by D ( f ), given bythe pairs (( x, y ) , ( x (cid:48) , y (cid:48) )) such that either f ( x, y ) = f ( x (cid:48) , y (cid:48) ) with ( x, y ) (cid:54) = ( x (cid:48) , y (cid:48) ) or ( x, y ) = ( x (cid:48) , y (cid:48) ) with ( x, y ) ∈ Σ( f ).We need to choose convenient analytic structures for the double point space D ( f ) and the lifting of the doublepoint space D ( f ). As we said in Introduction, when f has corank 1, local coordinates can be chosen so that thesemap germs can be written in the form f ( x, y ) = ( x, ˜ p ( x, y ) , ˜ q ( x, y )), for some function germs ˜ p, ˜ q ∈ m , where m isthe maximal ideal of O . In this case, we define the lifting of the double point space D ( f ), (as a complex space) by D ( f ) = V (cid:32) x − x (cid:48) , ˜ p ( x, y ) − ˜ p ( x, y (cid:48) ) y − y (cid:48) , ˜ q ( x, y ) − ˜ q ( x, y (cid:48) ) y − y (cid:48) (cid:33) where ( x, y, x (cid:48) , y (cid:48) ) are coordinates of C × C .Once the lifting D ( f ) ⊂ U × U is defined as a complex analytic space, we now consider its image D ( f ) (also as acomplex analytic space) on U by the projection π : U × U → U D ( f ) by f , denoted by f ( D ( f )), which will also be consider with thestructure given by Fitting ideals.We remark that given a finite morphism of complex spaces h : X → Y the push-forward h ∗ O X is a coherentsheaf of O Y − modules (see [3, Chapter 1]) and to it we can (as in [10, Section 1]) associate the Fitting ideal sheaves F k ( h ∗ O X ). Notice that the support of F ( h ∗ O X ) is just the image h ( X ). Analogously, if h : ( X, x ) → ( Y, y ) is afinite map germ then we denote by F k ( h ∗ O X ) the k th Fitting ideal of O X,x as O Y,y − module. In this way, we havethe following definition. Definition 2.1
Let f : U → V be a finite mapping, where U and V are open neighbourhoods of in C and C ,respectively.(a) Let π | D ( f ) : D ( f ) ⊂ U × U → U be the restriction to D ( f ) of the projection π . The double point space of f isthe complex space D ( f ) = V ( F ( π ∗ O D ( f ) )) .Set theoretically we have the equality D ( f ) = π ( D ( f )) .(b) The double point space of f in the target is the complex space f ( D ( f )) = V ( F ( f ∗ O )) . Notice that the underlyingset of f ( D ( f )) is the image of D ( f ) by f .(c) Given a finite map germ f : ( C , → ( C , , the germ of the double point space of f is the germ of complexspace D ( f ) = V ( F ( π ∗ O D ( f ) )) . The germ of the double point space of f in the target is the germ of the complex space f ( D ( f )) = V ( F ( f ∗ O )) . Remark 2.2 If f : U ⊂ C → V ⊂ C is finite and generically -to- , then D ( f ) is Cohen-Macaulay and hasdimension (see [6, Prop. 2.1 ] ). Hence, D ( f ) , D ( f ) and f ( D ( f )) are complex analytic curves. In this case, withoutany confusion, we also call these complex spaces by the “lifting of the double point curve”, the “double point curve”and the “image of the double point curve”, respectively. C ( f ) and T ( f ) Definition 2.3 (a) Two map germs f, g : ( C , → ( C , are A -equivalent, denoted by g ∼ A f , if there exist mapgerms of diffeomorphisms η : ( C , → ( C , and ξ : ( C , → ( C , , such that g = ξ ◦ f ◦ η .(b) A map germ f : ( C , → ( C , is finitely determined (or A -finitely determined) if there exists a positive integer k such that for any g with k -jets satisfying j k g (0) = j k f (0) we have g ∼ A f . Consider a finite map germ f : ( C , → ( C , f is finitelydetermined if and only if there is a finite representative f : U → V , where U ⊂ C , V ⊂ C are open neighbourhoodsof the origin, such that f − (0) = { } and the restriction f : U \ { } → V \ { } is stable.This means that the only singularities of f on U \ { } are cross-caps (or Whitney umbrellas), transverse doubleand triple points. By shrinking U if necessary, we can assume that there are no cross-caps nor triple points in U .Then, since we are in the nice dimensions of Mather ([8, p. 208]), we can take a stabilization of f , F : U × D → C , F ( z, s ) = ( f s ( z ) , s ),3here D is a neighbourhood of 0 in C . Definition 2.4
We define T ( f ) := (cid:93) of triple points of f s and C ( f ) := (cid:93) of cross-caps of f s ,where s (cid:54) = 0 . It is well known that the numbers T ( f ) and C ( f ) are independent of the particular choice of the stabilization andthey are also analytic invariants of f (see for instance [9]).We remark that the space D ( f ) plays a fundamental role in the study of the finite determinacy. In [5, Theorem2.14], Marar and Mond presented necessary and sufficient conditions for a map germ f : ( C n , → ( C p ,
0) with corank1 to be finitely determined in terms of the dimensions of D ( f ) and other multiple points spaces. When ( n, p ) = (2 , Theorem 2.5 ([6,
Corollary 3.5 ]) Let f : ( C , → ( C , be a finite and generically -to- map germ. Then f isfinitely determined if and only if µ ( D ( f )) is finite (equivalently, D ( f ) is a reduced curve). D ( f ) When f : ( C , → ( C ,
0) is finitely determined, the restriction of a representative of f to D ( f ) is finite. In thiscase, f | D ( f ) is generically 2-to-1 (i.e; 2-to-1 except at 0). On the other hand, the restriction of f to an irreduciblecomponent D ( f ) i of D ( f ) can be generically 1-to-1 or 2-to-1. This motivates us to give the following definition whichis from [15, Def. 4.1] (see also [13, Def. 2.4] and [14, Sec. 3]). Definition 2.6
Let f : ( C , → ( C , be a finitely determined map germ. Let f : U → V be a representative, where U and V are neighbourhoods of in C and C , respectively. Consider an irreducible component D ( f ) j of D ( f ) .(a) If the restriction f | D ( f ) j : D ( f ) j → V is generically -to- , we say that D ( f ) j is an identification component of D ( f ) .In this case, there exists an irreducible component D ( f ) i of D ( f ) , with i (cid:54) = j , such that f ( D ( f ) j ) = f ( D ( f ) i ) .We say that D ( f ) i is the associated identification component to D ( f ) j or that the pair ( D ( f ) j , D ( f ) i ) is a pair ofidentification components of D ( f ) .(b) If the restriction f | D ( f ) j : D ( f ) j → V is generically -to- , we say that D ( f ) j is a fold component of D ( f ) .(c) We define the sets IC ( D ( f )) = { identification components of D ( f ) } and F C ( D ( f )) = { fold components of D ( f ) } .And we define the numbers r i ( D ( f )) := (cid:93)IC ( D ( f )) and r f ( D ( f )) := (cid:93)F C ( D ( f )) . Remark 2.7
Let f and g be finitely determined map germs from ( C , to ( C , . Suppose that g ∼ A f and write g = ξ ◦ f ◦ η as in Definiton Consider representatives f, g : U → V of f and g . Let D ( g ) i be an irreduciblecomponent of D ( g ) and consider its corresponding image by η , D ( f ) i := η ( D ( g ) i ) , which is an irreducible componentof D ( f ) . Note that , f | D ( f ) j : D ( f ) i → V is generically k -to- if and only if g | D ( g ) i : D ( g ) i → V is generically k -to- ,where k = 1 , . Hence r i ( D ( f )) = r i ( D ( g )) and r f ( D ( f )) = r f ( D ( g )).4he following example illustrates the two types of irreducible components of D ( f ) presented in Definition 2.6. Example 2.8
Let f ( x, y ) = ( x, y , xy − x y ) be the singularity C of Mond’s list ([11, p.378 ]). In this case, D ( f ) = V ( xy − x ) . Then D ( f ) has three irreducible components given by D ( f ) = V ( y − x ) , D ( f ) = V ( y + x ) and D ( f ) = V ( x ). Notice that ( D ( f ) , D ( f ) ) is a pair of identification components and D ( f ) is a fold component. Hence, wehave that r i ( D ( f )) = 2 and r f ( D ( f )) = 1 . We have also that f ( D ( f ) ) = V ( X, Z ) and f ( D ( f ) ) = f ( D ( f ) ) = V ( Y − X , Z ) (see Figure D ( f ) (real points) Remark 2.9
In the Example , we have made use of the software Surfer [17] . Definition 2.10
A polynomial p ( x , · · · , x n ) is quasi-homogeneous if there are positive integers w , · · · , w n , with nocommon factor and an integer d such that p ( k w x , · · · , k w n x x ) = k d p ( x , · · · , x n ) . The number w i is called the weightof the variable x i and d is called the weighted degree of p . We also write w ( p ) to denote the weighted degree of p . Inthis case, we say p is of type ( d ; w , · · · , w n ) . This definition extends to polynomial map germs f : ( C n , → ( C p ,
0) by just requiring each coordinate function f i to be quasi-homogeneous of type ( d i ; w , · · · , w n ). In particular, when f : ( C , → ( C ,
0) is quasi-homogeneous,we say that f is quasi-homogeneous of type ( d , d , d ; a, b ), where here we change the classical notation w , w of theweights of x and y by a, b , for simplicity.The following lemma describes a normal form for a class of quasi-homogeneous map germs from ( C ,
0) to ( C , Lemma 2.11 (Normal form lemma) Let g ( x, y ) = ( g ( x, y ) , g ( x, y ) , g ( x, y )) be a quasi-homogeneous, corank ,finitely determined map germ of type ( d , d , d ; a, b ) . Then g is A -equivalent to a quasi-homogeneous map germ f with type ( d i = a, d i , d i ; a, b ) , which is written as f ( x, y ) = ( x, y n + xp ( x, y ) , αy m + xq ( x, y )) , (1) for some integers n, m ≥ , α ∈ C , p, q ∈ O , p ( x,
0) = q ( x,
0) = 0 , where ( d i , d i , d i ) is a permutation of ( d , d , d ) such that d i ≤ d i . roof. Since g has corank 1, g i is a regular in x or y for some i . Without lose of generality, suppose that g is regularin x . Thus g ( x, y ) = γx + g (cid:48) ( x, y ), where γ ∈ C , γ (cid:54) = 0. We have that g is quasi-homogeneous, this implies that g (cid:48) ( x, y ) = θy a , where θ ∈ C . Also, if θ (cid:54) = 0, then b = 1. Consider the analytic isomorphisms η : ( C , → ( C ,
0) and ξ : ( C , → ( C ,
0) defined by η ( x, y ) = ( x − ( γ − θ ) y a , y ) and ξ ( X, Y, Z ) = ( γ − X, Y, Z ).Note that the map ˜ g := ξ ◦ g ◦ η is a quasi-homogeneous map germ of type ( d , d , d ; a, b ). There exist integers v , v , complex numbers α , α and polynomials p , p ∈ O such that ˜ g is written as˜ g ( x, y ) = ( x, α y v + xp ( x, y ) , α y v + xp ( x, y )),with 2 ≤ v , v . After a change of coordinates (which does not change the quasi-homogeneous type of ˜ g ), we can assumethat p ( x,
0) = p ( x,
0) = 0. So, we need to show that α i (cid:54) = 0 for some i with w ( α i y v i + xp i ( x, y )) ≤ w ( α j y v j + xp j ( x, y )),where i (cid:54) = j and i, j ∈ { , } . Since ˜ g is also finitely determined, in particular, it is finite, so either α (cid:54) = 0 or α (cid:54) = 0.Thus we have three cases.(1) α , α (cid:54) = 0.(2) α = 0 and α (cid:54) = 0.(3) α (cid:54) = 0 and α = 0.(1) Suppose that α , α (cid:54) = 0. In this case, we can suppose that α = α = 1 (applying the change of coordinates( X, Y, Z ) (cid:55)→ ( X, α − Y, α − Z )). Define n := min { v , v } and p := p i if n = v i . Define also m := v and q := p if n = v or m := v and q := p if n = v . So, ˜ g is A -equivalent to f ( x, y ) = ( x, y n + xp ( x, y ) , y m + xq ( x, y )). Note that w ( y n + xp ( x, y ) ≤ w ( y m + xq ( x, y )).(2) If α = 0 and α (cid:54) = 0, then the restriction of ˜ g to V ( x ) is v -to-1. Since ˜ g if finitely determined and singular,we have that v = 2. In this case, ˜ g ( x, y ) = ( x, xp ( x, y ) , α y + xp ( x, y )). Again, after a change of coordinates, wecan assume that α = 1. Write xp ( x, y ) = λ x k y s + · · · + λ l x k l y s l . Statement:
We have that w ( y + xp ( x, y )) ≤ w ( xp ( x, y )). Proof of Statement:
Note that p ( x, y ) (cid:54)≡
0, otherwise the set of cross-caps C (˜ g ) of ˜ g is not finite and hence ˜ g isnot finitely determined. Since p ( x,
0) = 0, we have that s i ≥ i . If s i ≥ i , then the statement isclear and after a change of coordinates, we see that ˜ g is A -equivalent to f ( x, y ) = ( x, y + xp ( x, y ) , xq ( x, y )),where p = p , q = p and w ( y + xp ( x, y )) ≤ w ( xq ( x, y )), as desired.Now, suppose that s i = 1 for all i . In this case, after a change of coordinates, we can write ˜ g as ˜ g ( x, y ) =( x, x k y, y + xp ( x, y )), for some k ≥
1. We have that D (˜ g ) = V ( x − x (cid:48) , x k , y + y (cid:48) + x ( p ( x, y ) − p ( x, y (cid:48) )) / ( y − y (cid:48) ))which is not reduced if k ≥
2. Since ˜ g is finitely determined, by Theorem 2.5 and [6, Theorem 2.4] we have that k = 1. In this case, D (˜ g ) = V ( x, x (cid:48) , y + y (cid:48) ) ⊂ C × C is a smooth curve and ˜ g does not have any triple points. Itfollows by [5, Th. 2.14] that ˜ g is stable. Hence, ˜ g is A -equivalent to f ( x, y ) = ( x, y , xy ), which is considered withquasi-homogeneous type (1 , ,
2; 1 ,
1) and has the desired properties, that is, w ( y ) ≤ w ( xy ).Now, the analysis of case (3) is analogous. 6 emark 2.12 Some versions of Lemma are well know by specialists (see for instance [9,
Lemma 4.1 ] ). Weinclude its proof for completeness. Given a quasi-homogeneous, corank , finitely determined map germ, we willassume in the proofs throughout this paper that f is written in the normal form in (1), presented in Normal formlemma .In [12], Mond showed that if f is quasi-homogeneous then the invariants C ( f ), T ( f ) and also µ ( D ( f )) are deter-mined by the weights and the degrees of f . More precisely, he showed the following result. Theorem 2.13 ([12])
Let f : ( C , → ( C , be a quasi-homogeneous finitely determined map germ of type ( d , d , d ; a, b ) . Then C ( f ) = 1 ab (cid:18) ( d − a )( d − b ) + ( d − b )( d − b ) + ( d − a )( d − a ) (cid:19) , T ( f ) = 16 ab ( δ − (cid:15) )( δ − (cid:15) ) + C ( f )3 and µ ( D ( f )) = 1 ab ( δ − (cid:15) − a )( δ − (cid:15) − b ). where (cid:15) = d + d + d − a − b and δ = d d d / ( ab ) . J and m ( f ( D ( f ))) We note that by a parametrization of an irreducible complex germ of curve ( X, ⊂ ( C n ,
0) we mean a primitiveparametrization, that is, a holomorphic and generically 1-to-1 map germ n from ( C ,
0) to ( C n , n ( W, ⊂ ( X,
0) (see for instance [4, Section 3.1]). Before we present our main result, we will need the following lemma.
Lemma 3.1
Let f be a finitely determined, corank , quasi-homogeneous map germ of type ( d , d , d ; a, b ) . Write f as in Lemma 2.11, that is, f ( x, y ) = ( x, y n + xp ( x, y ) , αy m + xq ( x, y )) , with d ≤ d . Then(a) If V ( y ) is an irreducible component of D ( f ) , then V ( y ) ∈ IC ( D ( f )) and a = 1 .(b) D ( f ) = V ( λ ( x, y )) , where λ ( x, y ) is a quasi-homogeneous polynomial of type (cid:18) d d b − d − d + b ; a, b (cid:19) and λ ( x, y ) = x s r (cid:89) i =1 ( y a − α i x b ) , (2) where α i ∈ C are all distinct, r = ( d − b )( d − b ) − sabab ≥ and either s = 0 or s = 1 .(c) If a > d , then p ( x, y ) = 0 . That is, f ( x, y ) = ( x, y n , αy m + xq ( x, y )) .(d) If s = 1 in (2), that is, if V ( x ) is an irreducible component of D ( f ) , then it is a fold component of D ( f ) .(e) If α = 0 , then n = 2 .(f ) s = 0 if and only if α (cid:54) = 0 and gcd ( n, m ) = 1 . In other words, s = 1 if and only if either α = 0 , or α (cid:54) = 0 and gcd ( n, m ) = 2 . Proof.
Consider a representative f : U → V of f .((a) and (b)) Suppose that V ( y ) is an irreducible component of D ( f ). Consider the parametrization of V ( y ) givenby the map ϕ : W → U , defined by ϕ ( u ) = ( u, f ◦ ϕ : W → V , defined as7 f ◦ ϕ )( u ) = ( u, , , (3)is a parametrization of f ( V ( y )). Since f ◦ ϕ is 1-to-1, V ( y ) is an identification component of D ( f ). Since f is quasi-homogeneous and finitely determined, we have that λ ( x, y ) is a quasi-homogeneous polynomial of type (cid:18) d d b − d − d + b ; a, b (cid:19) , by [12, Prop. 1.15].The only irreducible quasi-homogeneous polynomials with w ( x ) = a and w ( y ) = b in the ring of polynomials C [ x, y ]are x, y and y a − α i x b , with α i ∈ C and α i (cid:54) = 0. Since the ring of polynomials C [ x, y ] is an unique factorization domain,each irreducible factor of λ is on the form of x, y or y a − α i x b . By Theorem 2.5, λ is reduced, hence the irreduciblefactors of λ are all distinct. So, λ can take the following form: λ ( x, y ) = x s y l r (cid:48) (cid:89) i =1 ( y a − α i x b ) , (4)where s, l ∈ { , } , r (cid:48) ≥ α i are all distinct and α i (cid:54) = 0 for all i . We note that if r (cid:48) = 0, then (cid:89) i =1 ( y a − α i x b ) = 1 (theempty product).Consider the parametrization of V ( y a − α i x b ) given by the map ϕ α i : W → U , defined by ϕ α i ( u ) = ( u a , γ i u b ),where γ i = α /ai . So, f ◦ ϕ α i : W → V , defined as( f ◦ ϕ α i )( u ) = ( u a , γ ,i u d , γ ,i u d ) , (5)is a parametrization of f ( V ( y a − α i x b )), for some γ ,i , γ ,i ∈ C .Since f ( V ( x )) ∩ f ( V ( y )) = { (0 , , } , the associated identification component of V ( y ) is a curve V ( y a − α j x b ) forsome α j (cid:54) = 0. Since, ( V ( y ) , V ( y a − α j x b )) is a pair of identification components of D ( f ), comparing (3) and (5) wesee that γ ,j = γ ,j = 0 and a = 1. Consequently, the expression (4) can be rewritten as follows: λ ( x, y ) = x s r (cid:89) i =1 ( y a − α i x b ),where s ∈ { , } , r = ( d − b )( d − b ) − sabab ≥ α i are all distinct and we allow one of the α (cid:48) i s to be zero.(c) Suppose that p ( x, y ) (cid:54) = 0. By assumption of the normal form of f , we have that p ( x,
0) = 0, hence p ( x, y ) isnot a constant. If a > d = bn , then w ( xp ( x, y )) > a > d , a contradiction.(d) Note that for all i , V ( x ) and V ( y a − α i x b ) have distinct images. Hence, V ( x ) / ∈ IC ( D ( f )). So if V ( x ) ⊂ D ( f ),then we conclude that V ( x ) ∈ F C ( D ( f )).(e) Suppose that α = 0, then restriction of f to V ( x ) is n -to-1. Since f is finitely determined, either n = 1 or n = 2. Since f has corank 1, we conclude that n = 2.(f) By (d), we have that V ( x ) is an irreducible component of D ( f ) if and only if f ◦ ϕ is generically 2-to-1, where ϕ : W → U is the parametrization of V ( x ), defined by ϕ ( u ) = (0 , u ). Suppose that V ( x ) ⊂ D ( f ). This implies thateither α = 0 or α (cid:54) = 0 and gcd ( n, m ) = 2. On the other hand, if α = 0 then by (e) we have that n = 2 and f ◦ ϕ isgenerically 2-to-1. Hence, in this case V ( x ) ⊂ D ( f ). If α (cid:54) = 0 and gcd ( n, m ) = 2, then again f ◦ ϕ is generically 2-to-1and hence V ( x ) ∈ F C ( D ( f )). 8ow, suppose that V ( x ) is not an irreducible component of D ( f ). Since f is generically 1-to-1, by (d) we havethat V ( x ) (cid:54)⊂ D ( f ) if and only if f ◦ ϕ is generically 1-to-1 if and only if α (cid:54) = 0 and gcd ( n, m ) = 1.In the following result, m ( f ( D ( f ))) denotes the Hilbert-Samuel multiplicity of the maximal ideal of the local ring O f ( D ( f )) of ( f ( D ( f )) ,
0) (or equivalently, the multiplicity of f ( D ( f )) at 0). Also, J denotes the number of tacnodesthat appears in a stabilization of the transversal slice curve of f ( C ) (see [7, Def. 3.7]).Note that if ϕ : W ⊂ C → V ⊂ C , ϕ ( u ) = ( u m , ϕ ( u ) , ϕ ( u )) is a Puiseux parametrization of a reduced curvein C , then its multiplicity is m (see for instance [1, page 98]). We remark that given a germ of reduced curve( C, ⊂ ( C n , X ,
0) = ( X ∪ X ,
0) is a germ of reduced curve in ( C , X ,
0) is not reduced at 0.Suppose that f : ( C , → ( C ,
0) is finitely determined. So, D ( f ) is a reduced curve, by Theorem 2.5. It followsby [6, Th. 4.3] that f ( D ( f )) is also a reduced curve. However, given an irreducible component f ( D ( f ) i ) of f ( D ( f )),it may contain a (embedded) zero dimensional component, and therefore may not be reduced. If this is the case, wesay that f ( D ( f ) i ) is a generically reduced curve. Recently, the author and Snoussi showed in [16, Lemma 4.8] thatif ( C,
0) is a germ of generically reduced curve and ( | C | ,
0) is its associated reduced curve, then the multiplicities of( C,
0) and ( | C | ,
0) at 0 are equal. Hence, we also can calculate the multiplicity of f ( D ( f ) i ) considering its reducedstructure and using a corresponding Puiseux parametrization for it. We are now able to present our main result. Theorem 3.2
Let f be a finitely determined, corank , quasi-homogeneous map germ. Write f as in Lemma 2.11,that is, f ( x, y ) = ( x, y n + xp ( x, y ) , αy m + xq ( x, y )) and it is of type ( d = a, d , d ; a, b ) such that d ≤ d . Then m ( f ( D ( f )) = 12 ab (cid:18) ( d − b )( d − b ) c + sab ( d − c ) (cid:19) and J = 12 ab (cid:18) ( d − b )( d − b )( c − b ) + b ( δ − (cid:15) − a )( δ − (cid:15) − b ) + b ( (cid:15) − δ )( δ − (cid:15) ) + sab ( d − c ) − ab (cid:19) where (cid:15) = d + d − b , δ = d d /b , c = min { a, d } and s = (cid:26) if α (cid:54) = 0 and gcd ( n, m ) = 1 , otherwise. Proof.
Take a representative f : U → V of f . By Lemma 3.1 (b), we have that D ( f ) = V ( λ ( x, y )), where λ ( x, y ) = x s r (cid:89) i =1 ( y a − α i x b ), s = 0 or 1, α i ∈ C are all distinct and r = ( d − b )( d − b ) − sabab .Set C α i := V ( y a − α i x b ). As in the proof of Lemma 3.1, consider a parametrization ϕ α i : W → U of C α i definedby ϕ α i ( u ) = ( u a , γ i u b ), where W is an open neighbourhood of 0 in C and γ i := α /ai . So, if C α i is an identificationcomponent of D ( f ), then the mapping ˜ ϕ α i := f ◦ ϕ α i : W → V , defined by˜ ϕ α i := ( u a , γ ,i u d , γ ,i u d ) , (6)is a parametrization of f ( C α i ), for some γ ,i , γ ,i ∈ C . On the other hand, if C α i is a fold component of D ( f ), thenthe mapping ϕ (cid:48) α i : W → V , defined by ϕ (cid:48) α i ( u ) := ( u a/ , γ (cid:48) ,i u d / , γ (cid:48) ,i u d / ) , (7)9s a parametrization of f ( C α i ), for some γ (cid:48) ,i , γ (cid:48) ,i ∈ C . Set c := min { a, d } . Note that if a > d , then γ ,i , γ (cid:48) ,i (cid:54) = 0, byLemma 3.1 (c). It follows by (6) and (7) that m ( f ( C α i )) = (cid:26) c if C α i ∈ IC ( D ( f )) ,c/ if C α i ∈ F C ( D ( f )) . Set C := V ( x ). If C ⊂ D ( f ), then by Lemma 3.1 (d) we have that it is a fold component of D ( f ). In this case,the map ϕ : W → V defined by ϕ ( u ) = (0 , u n/ , αu m/ ) (8)is a parametrization of C . It follows by (8) that m ( C ) = n/
2. Hence, we have that m ( f ( D ( f ))) = (cid:18) r i ( D ( f ))2 (cid:19) c + ( r f ( D ( f )) − s ) (cid:18) c (cid:19) + s (cid:18) n (cid:19) .By Lemma 3.1 (b), we have that r i ( D ( f )) + r f ( D ( f )) − s = r = ( d − b )( d − b ) − sabab . Also, note that n = d /b .Hence, m ( f ( D ( f ))) = (cid:18) ( d − b )( d − b ) − sabab (cid:19)(cid:18) c (cid:19) + s (cid:18) d b (cid:19) = 12 ab (cid:18) ( d − b )( d − b ) c + sab ( d − c ) (cid:19) . (9)It follows by Propositions 3.5 and 3.10 and Corollary 4.4 of [7] that J = 12 (cid:18) µ ( D ( f )) − C ( f ) − (cid:19) − T ( f ) + m ( f ( D ( f )) . (10)Now, by Theorem 2.13 and the expressions (9) and (10), we have that J = 12 ab (cid:18) ( δ − (cid:15) − a )( δ − (cid:15) − b ) + ( (cid:15) − δ )( δ − (cid:15) ) − d − b )( d − b ) − ab (cid:19) + 12 ab (cid:18) ( d − b )( d − b ) c + sab ( d − c ) (cid:19) = 12 ab (cid:18) ( d − b )( d − b )( c − b ) + b ( δ − (cid:15) − a )( δ − (cid:15) − b ) + b ( (cid:15) − δ )( δ − (cid:15) ) + sab ( d − c ) − ab (cid:19) ,where (cid:15) = d + d − b and δ = d d /b . When we look to the formulas in Theorem 3.2, we identify four cases. More precisely, we identified four situationsdepending on the values that c and s assume. In this section, we present examples illustrating these situations. Example 4.1 (a) ( c = a and s = 0 ) Consider the F -singularity of Mond’s list [11] , given by f ( x, y ) = ( x, y , y + x y ). We have that f is quasi-homogeneous of type (4 , ,
15; 4 , . In this case c = 4 and s = 0 . By Theorem we havethat m ( f ( D ( f ))) = 2 and J = 3 .(b) ( c = a and s = 1 ) Consider the map germ f ( x, y ) = ( x, y , y + x y − x y + 4 xy ),10 hich is quasi-homogeneous of type (1 , ,
6; 1 , . In this case c = 1 and s = 1 . Again by Theorem we have that m ( f ( D ( f ))) = 9 and J = 39 . We remark that f is from [13, Example 5.5 ] , where finite determinacy is proved.(c) ( c = d and s = 0 ) Consider the H -singularity of of Mond’s list, given by f ( x, y ) = ( x, y , y + xy ), which is quasi-homogeneous of type (4 , ,
5; 4 , . Using Theorem we have that m ( f ( D ( f ))) = 3 and J = 2 .(d) ( c = d and s = 1 ) Consider the map germ f ( x, y ) = ( x, y , x y − xy ), which is quasi-homogeneous of type (4 , ,
9; 4 , . We have that D ( f ) = V ( x ( x − y )) which is reduced. So, by Theorem we have that f is finitely determined. Using Theorem we have that m ( f ( D ( f ))) = 2 and J = 4 . Example 4.2
Inspired in Example (a) and (c) we finish this work presenting in Table values for m ( f ( D ( f )) and J for every map germ in Mond’s list using the formulas of Theorem We also include in Table the values of r i ( D ( f )) and r f ( D ( f )) . Table 1: Quasi-homogeneous map germs in Mond’s list [11].Name f ( x, y ) = Quasi-Homogeneous type r i ( D ( f )) r f ( D ( f )) m ( f ( D ( f ))) J Cross-Cap ( x, y , xy ) (1 , ,
2; 1 ,
1) 0 1 1 0 S k , k ≥ x, y , y + x k +1 y ) (1 , k + 1 , k +1)2 ; 1 , k +12 ) 2 0 1 0 S k , k ≥ x, y , y + x k +1 y ) (2 , k + 2 , k + 3; 2 , k + 1) 0 1 1 0 B k , k ≥ x, y , y k +1 + x y ) ( k, , k + 1; k,
1) 2 0 2 kB k , k ≥ x, y , y k +1 + x y ) ( k, , k + 1; k,
1) 0 2 2 kC k , k ≥ x, y , xy + x k y ) (1 , k − , k − ; 1 , k − ) 2 1 2 2 C k , k ≥ x, y , xy + x k y ) (2 , k − , k −
1; 2 , k −
1) 0 2 2 2 F ( x, y , y + x y ) (4 , ,
15; 4 ,
3) 0 1 2 3 H k ( x, y , y k − + xy ), k ≥ k − , , k −
1; 3 k − ,
1) 2 0 3 2 T ( x, y + xy, y ) (2 , ,
4; 2 ,
1) 2 1 3 3 P ( x, y + xy, cy + xy ) ∗ (2 , ,
4; 2 ,
1) 2 1 3 3 ∗ c (cid:54) = 0 , / , , / Acknowlegments:
We would like to thank Jawad Snoussi and Guillermo Pe˜nafort-Sanchis for many helpfulconversations, suggestions and comments on this work. The author would like to thank CONACyT for the financialsupport by Fordecyt 265667 and UNAM/DGAPA for support by PAPIIT IN 113817.11 eferences [1] Chirka, E.M.:
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