Parametrization simple irreducible plane curve singularities in arbitrary characteristic
aa r X i v : . [ m a t h . AG ] J a n PARAMETRIZATION SIMPLE IRREDUCIBLE PLANE CURVESINGULARITIES IN ARBITRARY CHARACTERISTIC
NGUYEN HONG DUC
Abstract.
We study the classification of plane curve singularities in arbitrary char-acteristic. We first give a bound for the determinacy of a plane curve singularity withrespect to pararametrization equivalence in terms of its conductor. Then we classifyparametrization simple plane curve singularities which are irreducible by giving a con-crete list of normal forms of equations and parametrizations. In characteristic zero,the classification of parametrization simple irreducible plane curve singularities wasachieved by Bruce and Gaffney. Introduction
We classify irreducible plane curve singularities f ∈ K [[ x, y ]] which are simple withrespect to parametrization equivalence, where K is an algebraically closed field of arbi-trary characteristic. That is, the irreducible plane singularities whose parametrizationshave modality 0 up to the change of coordinates in the source and target spaces (or, left-right equivalence, see Section 2.1). The notion of modality was introduced by Arnol’d inthe seventies into the singularity theory for real and complex singularities. He classifiedsimple, unimodal and bimodal hypersurface singularities with respect to right equiva-lence, i.e. the hypersurface singularities of right modality 0,1,2 respectively [1],[2],[3].The classifications of contact simple and unimodal complete intersection singularitieswere done by Giusti [12] and Wall [21]. Classification of contact simple space curvesingularities was obtained by Giusti [12] and Fr¨uhbis-Kr¨uger [9]. In positive character-istic, the right simple, unimodal and bimodal hypersurface singularities were recentlyclassified by Greuel and the author in [14] and [20]. The classification of contact simplehypersurface singularities were achieved by Greuel-Kr¨oning [11], while classifications ofcontact unimodal and bimodal singularities are still unknown.Curve singularities can be also described by parametrisations. Two plane curvesingularities are contact equivalent if and only if their parametrizations are left-rightequivalent. The first results on classification of simple curve singularities with respectto parametrization equivalence were obtained by Bruce and Gaffney, for complex ir-reducible plane curve singularities in C { x, y } [6]. The classifications were extended to Date : January 10, 2019.The author’s research is supported by Juan de la Cierva Incorporacin IJCI-2016-29891, the ERCEAConsolidator Grant 615655 NMST and the National Foundation for Science and Technology Develop-ment (NAFOSTED), Grant number 101.04-2017.12, Vietnam. irreducible space curves by Gibson and Hobbs [10], irreducible curves of any embeddingdimension by Arnold [4], and reducible curves by Kolgushkin and Sadykov [17].In this paper, we generalize the result of Bruce and Gaffney to the singularities inarbitrary characteristic (Theorem 3.1). We give lists of normal forms of equationsand parametrizations of parametrization simple plane curve singularities which areirreducible (Tables 1,2,3 in Section 3). We first study in Section 2 the problem ofdeterminacy with respect to parametrization equivalence. The theory of determinacywas systematically studied by Mather in [18], where he defined the equivalence relations R , C , K , L and A and obtained necessary and sufficient conditions for finite determinacywith respect to them. He also gave estimates for the corresponding determinacy. Lowerestimates were provided later by Gaffney, Bruce, du Plessis and Wall. The problem ofdeterminacy in positive characteristic with respect to R , K was treated by Boubakri,Greuel and Markwig in [5] and recently by Greuel and Pham [15],[16]. We show thatreduced plane curve singularities are finitely determined with respect to parametriza-tion equivalence. Moreover, we give a lower bound for parametrization determinacy ofa plane curve singularity in terms of its conductor (Theorem 2.1). Acknowledgement.
A part of this article was done in my thesis under the supervisionof Professor Gert-Martin Greuel at the Technische Universit¨at Kaiserslautern. I amgrateful to him for many valuable suggestions.2.
Parametrization determinacy
Parametrization equivalence.
For a plane curve singularity f , i.e. an elementin the maximal ideal m in K [[ x, y ]], there is a unique (up to multiplication with units)decomposition f = f ρ · . . . · f ρ r r , with f i ∈ m irreducible in K [[ x, y ]]. We assume, in thisnote, that f is reduced , i.e. ρ i = 1 for all i = 1 , . . . , r . The integral closure of R := R f := K [[ x, y ]] / h f i (in the total quotient ring Quot( R )) is isomorphic to ¯ R := L ri =1 K [[ t ]] (see[7], [13]). A composition K [[ x, y ]] ։ R ֒ → ¯ R = L ri =1 K [[ t ]] of the natural projection K [[ x, y ]] ։ R and a normalization R ֒ → ¯ R , is called a parametrization of f . It is anelement in the space J := Hom K ( K [[ x, y ]] , ¯ R ) of morphisms of local K -algebras. Anyelement of ψ ∈ J can be identified with the image of ψ ( x ) , ψ ( y ) in ¯ R . Hence, it is oftenwritten as a tuple of r pairs ( x i ( t ) , y i ( t )). .Two morphisms of K -algebras ψ, ψ ′ : K [[ x, y ]] → ¯ R = L ri =1 K [[ t ]] are called left-rightequivalent (or, A -equivalent), ψ ∼ A ψ ′ , if there exist an automorphism φ of ¯ R and anautomorphism Φ ∈ Aut K ( K [[ x, y ]]) such that Φ ◦ ψ = φ ◦ ψ ′ . By an automorphism of¯ R we mean a tuple of automorphisms of K [[ t ]]. Two plane curves f, g ∈ K [[ x, y ]] arecalled parametrization equivalent , denoted by f ∼ p g , if there exist a parametrization ψ of f and a parametrization ψ ′ of g such that ψ ∼ A ψ ′ . It was known that, f ∼ p g ifand only if f ∼ c g ([19, Prop. 1.2.10], see also [6, Lemma 2.2] for f irreducible). ARAMETRIZATION SIMPLE IRREDUCIBLE PLANE CURVE SINGULARITIES 3
Parametrization determinacy.
For each k = ( k , . . . , k r ) ∈ Z r ≥ , the k -jet of ψ is defined to be the composition j k ψ : K [[ x, y ]] ψ → L ri =1 K [[ t ]] → L ri =1 K [[ t ]] / ( t k i +1 ) . We call ψ parametrization k -determined if it is parametrization equivalent to every ψ ′ whose k -jet coincides with that of ψ . We say that f is parametrization finitelydetermined if one (and therefore all) of its parametrizations is parametrization k -determined for some k = ( k , . . . , k r ) ∈ Z r ≥ . A minimum k with this property iscalled a parametrization determinacy of f (or ψ ). We show, in the present note, that f is d -parametrization determined, where d is concretely given by the conductor of f .Let C := ( R : ¯ R ) := { u ∈ R | u ¯ R ⊂ R } be the conductor ideal of ¯ R in R (cf.[22]). Then C is an ideal of both R and ¯ R . So one has C = ( t c ) × · · · × ( t c r ) for some c , . . . , c r ∈ Z ≥ . We call c := c ( f ) := ( c , . . . , c r ) ∈ Z r ≥ the conductor (exponent) of f . The conductor c = ( c , . . . , c r ) of f is related to the ones of its branches and otherinvariants by the following beautiful formulas(2.1) c i = c ( f i ) + X j = i i ( f i , f j )and(2.2) | c | := c + . . . + c r = 2 δ, where δ is the delta invariant of f , defined as δ := dim K ¯ R/R .Here for g, h ∈ K [[ x, y ]], i ( g, h ) denotes the intersection multiplicity of g, h defined by i ( g, h ) := dim K K [[ x, y ]] / ( g, h ) . Note that, if h is irreducible and ψ is a parametrizationof h , then i ( g, h ) = ord ψ ( g ). Furthermore, the intersection multiplicity is additive, i.e.if h = h · . . . · h r , then i ( g, h ) = i ( g, h ) + . . . + i ( g, h r ) . Theorem 2.1.
Let f ∈ m ⊂ K [[ x, y ]] be reduced, r the number of the irreduciblecomponents, c ∈ Z r ≥ its conductor, and let Z r ≥ ∋ d := if mt( f ) = 1 c + 1 if mt( f ) = 2 and r = 1 c if mt( f ) = 2 and r = 2 c − if mt( f ) > . Then f is parametrization d -determined. In particular, f is always parametrization ( c + ) -determined. The multiplicity of f , mt( f ), is defined to be the maximal of integers k for which h f i ⊂ m k . For the proof of the theorem we need the two following lemmas, which giveseveral relations between the conductor ( c ) and the maximal contact multiplicity ( ¯ β )of a reduced power series f in some concrete cases. Recall that the maximal contactmultiplicity of f is defined by¯ β ( f ) := sup { min i =1 ,...,r i ( f i , γ ) | γ regular } , NGUYEN HONG DUC where f , . . . , f r are the irreducible components of f . We omit proofs of the lemmashere and refer to [19], Lemma 2.5.4 and 2.5.5, since they are elementary. Lemma 2.2.
Let f = f · f ∈ K [[ x, y ]] be reduced such that f , f are regular. Then ¯ β ( f ) = i ( f , f ) . Lemma 2.3.
Let f ∈ K [[ x, y ]] be irreducible. (i) If mt( f ) = 2 , then c ( f ) = ¯ β ( f ) − . (ii) If mt( f ) > , then c ( f ) > ¯ β ( f ) .Proof of Theorem 2.1. Note that d + ≥ c , i.e. d i + 1 ≥ c i for all i = 1 , . . . , r . Let ψ = ( ψ , . . . , ψ r ) : K [[ x, y ]] → ¯ R be a parametrization of f and let ψ ′ : K [[ x, y ]] → ¯ R such that j d ( ψ ) = j d ( ψ ′ ). It suffices to show that ψ ∼ A ψ ′ .Indeed, we have ψ ( x ) − ψ ′ ( x ) ∈ t d + ¯ R ⊂ R and ψ ( y ) − ψ ′ ( y ) ∈ t d + ¯ R ⊂ R. Thus there exist g , g ∈ K [[ x, y ]] such that ψ ( g ) = ψ ( x ) − ψ ′ ( x ) ∈ t d + ¯ R and ψ ( g ) = ψ ( y ) − ψ ′ ( y ) ∈ t d + ¯ R. The following claim shows that, the map Φ : K [[ x, y ]] −→ K [[ x, y ]] sending x, y to x − g ( x, y ) , y − g ( x, y ) respectively, is an automorphism of K [[ x, y ]] and hence ψ ∼ A ψ ′ as required, since ψ ◦ Φ = ψ ′ . Claim 2.4. mt( g ) > (similarly, mt( g ) > ).Proof of the claim : Since the case mt( f ) = 1 is evident, we assume that mt( f ) ≥ g ) = 1. Then by thedefinition of the maximal contact multiplicity ¯ β ( f ),(2.3) min { i ( f i , g ) | i = 1 , . . . , r } ≤ ¯ β ( f ) . The following three steps comprise the proof:
Step 1: mt( f ) = 2 and r = 1. Then d = c + 1 and ψ ( g ) ∈ t d +1 K [[ t ]]. This implies i ( f, g ) = ord ψ ( g ) ≥ d + 1 = c + 2 = ¯ β ( f ) + 1 , where the last equality is due to Lemma 2.3. This contradicts to (2.3). Step 2: mt( f ) = 2 and r = 2. Then f = f · f with mt( f ) = mt( f ) = 1 and d = c .It follows from (2.1) that c = c = i ( f , f ). Since ψ ( g ) ∈ t d +1 K [[ t ]], i ( g , f ) = ord ψ ( g ) ≥ d + 1 = i ( f , f ) + 1 . Similarly, i ( g , f ) ≥ i ( f , f ) + 1 . Combining Lemma 2.2 and (2.3) we get i ( f , f ) + 1 ≤ min { i ( f , g ); i ( f , g ) } ≤ ¯ β ( f ) = i ( f , f ) , ARAMETRIZATION SIMPLE IRREDUCIBLE PLANE CURVE SINGULARITIES 5 a contradiction.
Step 3: mt( f ) >
2. Then d = c − . Let f = f · . . . · f r be an irreducible decompositionof f such that mt( f ) ≤ . . . ≤ mt( f r ). We consider the three following cases: • If mt( f r ) >
2, then i ( f r , g ) = ord ψ r ( g ) ≥ d r + 1 = c r . By Lemma 2.3 and by thedefinition of the maximal contact multiplicity of f r , one deduce that c ( f r ) > ¯ β ( f r ) ≥ i ( f r , g ) ≥ c r > c ( f r ) , a contradiction. • If mt( f r ) = 2, then r > i ( f r , g ) = ord ψ r ( g ) ≥ d r + 1 = c r . This impliesthat ¯ β ( f r ) ≥ c r . By (2.1) and the inequality i ( f , f r ) ≥ mt( f r ) = 2, c r ≥ c ( f r ) + i ( f , f r ) > c ( f r ) + 1 . It follows from Lemma 2.3 that c ( f r ) = ¯ β ( f r ) − ≥ c r − > c ( f r ) , which is acontradiction. • If mt( f r ) = 1 then mt( f ) = mt( f ) = . . . = mt( f r ) = 1 and r = mt( f ) >
2. Dueto (2.1) one has c ≥ i ( f , f ) + i ( f , f r ) ≥ i ( f , f ) + 1. Hence i ( f , g ) = ord ψ ( g ) ≥ d + 1 = c ≥ i ( f , f ) + 1 . Similarly i ( f , g ) ≥ i ( f , f ) + 1 and then i ( f , f ) + 1 ≤ min { i ( f , g ); i ( f , g ) } . Ithence follows from Lemma 2.2 that i ( f , f ) + 1 ≤ min { i ( f , g ); i ( f , g ) } ≤ ¯ β ( f · f ) = i ( f , f ) , a contradiction. This completes the theorem. (cid:3) Example 2.5.
1. Let f = x − y . Then r ( f ) = 1 and c ( f ) = 4. It is easy to see that f is not parametrization 4-determined.2. Let f = ( x − y )( x − y ). Then r ( f ) = 2, c ( f ) = (3 ,
3) and ψ : K [[ x, y ]] → K [[ t ]] ⊕ K [[ t ]] , g g ( t , t ) ⊕ g ( t , t )is a parametrization of f . It can be easily verified that f is parametrization (3 , , Parametrization simple singularities
Parametrization modality.
Consider an action of algebraic group G on a variety X (over a given algebraically closed field K ) and a Rosenlicht stratification { ( X i , p i ) , i =1 , . . . , s } of X w.r.t. G . That is, a stratification X = ∪ si =1 X i , where the stratum X i isa locally closed G -invariant subvariety of X such that the projection p i : X i → X i /G is a geometric quotient. For each open subset U ⊂ X the modality of U , G -mod( U ), isthe maximal dimension of the images of U ∩ X i in X i /G . The modality G -mod( x ) of apoint x ∈ X is the minimum of G -mod( U ) over all open neighbourhoods U of x . NGUYEN HONG DUC
Let L := Aut ( K [[ x, y ]]) resp. R := Aut ( ¯ R ) the left group resp. the right group. Theleft-right group A := R×L acts on J = Hom K ( K [[ x, y ]] , ¯ R ) by (( φ, Φ) , ψ ) Φ − ◦ ψ ◦ φ .Then, two elements ψ, ψ ′ ∈ J are left-right equivalent, if and only if they belong to thesame A -orbit.For each k ∈ Z , denoted by J k the k -jet space of J , that is, the space of morphisms K [[ x, y ]] → ¯ R k := r M i =1 K [[ t ]] / ( t k +1 ) . We may identify an element ψ in J k with the pair ( ψ ( x ) , ψ ( y )) in K [[ t ]] / ( t k +1 ) × K [[ t ]] / ( t k +1 ), and therefore J k can be identified with the variety ¯ R k ∼ = A k +1) K . Foreach element ψ ∈ J , denoted the j k ψ the image of ψ by the map induced by the projec-tion ¯ R → ¯ R k . We call ψ to be left-right k -determined if it is left-right equivalent to anyelement in J whose k -jet coincides with j k ψ . A number k is called left-right sufficientlylarge for ψ , if there exists a neighbourhood U of the j k ψ in J k such that every ψ ′ ∈ J with j k ψ ′ ∈ U is left-right k -determined. We also consider the k -jet of the left-rightgroup A defined by A k := R k × L k . This group acts naturally on the k -jet space J k .The left-right modality of ψ , A -mod( ψ ), is defined to be the A k -modality of j k ψ in J k with k right sufficiently large for ψ .Let f ∈ m ⊂ K [[ x, y ]] be reduced plane curve singularity and let ψ be its parametriza-tion. By Theorem 2.1, ψ is left-right ( | c | + 1)-determined, where | c | denotes the sum c + . . . + c r for c = ( c , . . . , c r ). Note that, | c | = r X i =1 c ( f i ) + X j = i i ( f i , f j ) ! = 2 δ ( f ) . It yields that ψ is left-right (2 δ ( f ) + 1)-determined. From the upper semi-continuityof the delta function δ (see [8]), we can show, by using the same argument as in [14],that k = 2 δ ( f ) + 1 is left-right sufficiently large for ψ . The parametrization modality of f , denoted by P -mod( f ), is defined to be the left-right modality of ψ , i.e the number A k -mod( j k ψ ).A plane curve singularity f ∈ K [[ x, y ]] is called parametrization simple, uni-modal,bi-modal or r -modal if its parametrization modality is equal to 0,1,2 or r respectively.These notions are independent of the choice of a parametrization, and its sufficientlylarge number k . This may be proved in much the same way as [14, Prop. 2.6, 2.12].The simpleness can be also described by deformation theory. A plane curve singularity f ∈ K [[ x, y ]] is parametrization simple if its parametrization is of finite deformationtype, i.e. its parametrization can be deformed only into finitely many left-right classesin J .3.2. Parametrization simple irreducible plane curve singularities.
ARAMETRIZATION SIMPLE IRREDUCIBLE PLANE CURVE SINGULARITIES 7
Theorem 3.1.
Let p = char( K ) . An irreducible plane curve singularity f ∈ m ⊂ K [[ x, y ]] is parametrization simple if and only if one of its parametrizations is left-right equivalent to one of the singularities in the Tables 1, 2, 3 (where ε ∈ { , } and c k ( y ) = a + a y + . . . + a k y k ∈ K [ y ] ).Proof. The theorem follows from Propositions 3.2 and 3.4 below. (cid:3)
Name Equations Parametrizations ConditionsA k x + y k +1 ( t , t k +1 ) k ≥ x + y ( t , t )E x + y ( t , t ) p > x + y + εxy ( t , t + εt ) p = 5E k x + y k +1 + c k − ( y ) x y k +1 ( t , t k +1 + εt k + q )+2 ) 0 ≤ q ≤ k − q < k − p ∤ k + 1E k +2 x + y k +2 + c k − ( y ) x y k +1 ( t , t k +2 + εt k + q )+4 ) 0 ≤ q ≤ k − q < k − p ∤ k + 2W x + y + ax y ( t , t + εt q ) q = 6 , , q = 6 if p > x + y + c ( y ) x y ( t , t + εt q ) p = 7; q = 9 , ♯ q − ( x + y ) + c ( y ) xy q +4 ( t , t + t q +5 ) q ≥ Table 1.
Irreducible simple plane curve singularities ( p > k x + y k +1 ( t , t k +1 ) 1 ≤ k E x + y + εx y ( t + εt , t )E x + y ( t + εt ) , t W x + y + ax y ( t , t + εt q ) q = 7 , Table 2.
Irreducible simple plane curve singularities in characteristic 3.Name Equations Parametrizations ConditionsA k x + y k +1 + εxy k − q ( t k +1 , t + εt q +1 ) 1 ≤ q < k E x + y + εx y ( t + εt , t )E x + y ( t , t )E x + y + εx y ( t , t + εt ) Table 3.
Irreducible simple plane curve singularities in characteristic 2.
Proposition 3.2.
Let f ∈ m ⊂ K [[ x, y ]] be an irreducible plane curve singularity andlet ( x ( t ) , y ( t )) be its parametrization with m = ord x ( t ) = mt( f ) < ord y ( t ) = n . Then f is not parametrization simple if either NGUYEN HONG DUC (i) m > or (ii) m = 4 and p = 2 or (iii) m = 4 and n > or (iv) m = 4 and n = 7 and p = 7 , or (v) m ≥ , n ≥ and p = 3 or (vi) m = 3 and n ≥ and p = 2 . For the proof of these theorems we need the following lemma which is deduced fromCorollaries A.4, A.9, A.10 of [14] (see [19, Prop. 3.2.4, Cor. 3.3.4 and Cor. 3.3.6] formore details).
Lemma 3.3.
Let the algebraic groups G resp. G ′ act on the varieties X resp. X ′ . Let h : Y → X a morphism of varieties and let h ′ : Y → X ′ an open morphism such that h − ( G · h ( y )) ⊂ h ′− ( G ′ · h ′ ( y )) , ∀ y ∈ Y. Then for all y ∈ Y we have G - mod( h ( y )) ≥ G ′ - mod( h ′ ( y )) ≥ dim X ′ − dim G ′ . Proof of Proposition 3.2.
We give a proof for (iv), since the others are similar andsimpler. Let k be left-right sufficiently large for the paramterization ( x ( t ) , y ( t )) of f .Let X := J k , G := A k . We denote Y := { ψ ∈ J k | ψ ( x ) = at , a = 0 , ord ψ ( y ) ≥ } , X ′ := { ψ ∈ J | ψ ( x ) = at , a = 0 , ord ψ ( y ) ≥ } ∼ = ( A \ { } ) × A ,G ′ := R × L ′ with dim G ′ = 4 and define h : Y → X and h ′ : Y → X ′ to be the naturalinclusion and projection respectively, where L ′ := (cid:8)(cid:0) a x + m (3) , b y + b x + m (3) (cid:1) ∈ L (cid:9) . Here and below, for each k ≥ m ( k ) resp. O ( k ) stands for a series of multiplicity resp.of order at least k . We are going to show that h − ( G · h ( y )) ⊂ h ′− ( G ′ · h ′ ( y )) , ∀ y ∈ Y. Indeed, for a given ψ = ( at , b t + b t + b t + b t + O (11)) ∈ Y , assume that ψ ′ = ( a ′ t , b ′ t + b ′ t + b ′ t + b ′ t + O (11)) ∈ Y such that h ( ψ ′ ) ∈ G.h ( ψ ). That is,there exist φ = c t + c t + . . . ∈ R k ,Φ = ( a x + a y + m (2) , b x + b y + m (2)) ∈ L k such that ψ ′ = Φ ◦ ψ ◦ φ . This implies that c = c = 0 and therefore a ′ = a c a , b ′ i = b c b , b ′ = b c b + b a , b ′ = b c b , b ′ = b c b . Putting φ ′ := c t ∈ R and Φ = ( a x, b y + b x ) ∈ L ′ we get that ψ ′ = Φ ′ ◦ ψ ◦ φ ′ ,i.e. h ′ ( ψ ′ ) ∈ G ′ .h ′ ( ψ ). It follows from Lemma 3.3 that A -mod( ψ ) ≥ dim X ′ − dim G ′ = 1 . (cid:3) Proposition 3.4.
Let f ∈ m ⊂ K [[ x, y ]] be an irreducible plane curve singularity andlet ( x ( t ) , y ( t )) be its parametrization with m = ord x ( t ) < ord y ( t ) = n . ARAMETRIZATION SIMPLE IRREDUCIBLE PLANE CURVE SINGULARITIES 9 (i) If m = 2 and c ( f ) = 2 k then f is parametrization equivalent to a singularity oftype A k . (ii) If m = 3 and c ( f ) = q ( q = 6 , if p = 3 ; q = 6 , , if p = 2 ), then f isparametrization equivalent to a singularity of type E q . (iii) If m = 4 , n = 6 and c ( f ) = q , then f is parametrization equivalent to a singu-larity of type W q . (iv) If m = 4 , n = 6 and c ( f ) = 2 q + 14 , then f is parametrization equivalent to asingularity of type W ♯ q − .Proof. We prove only for the case W ♯ , the other cases are proved completely similar.Assume that m = 4 , n = 6 and c ( f ) = 16. Since m = 4 is not divisible by p , we mayassume that x ( t ) = t . Let k be the smallest odd exponent with non zero coefficient in y ( t ). By [19, Prop. 2.3.9],16 = c ( f ) = 2 δ ( f ) = (6 − −
2) + ( k − − , and hence k = 7. It is easy to see that ( x ( t ) , y ( t )) is left-right equivalent to ( x ( t ) , y ( t ))of form (cid:0) t , t + t + O (8) (cid:1) . We shall show that, ( x ( t ) , y ( t )) is left-right equivalent to a singularity of type W ♯ . ByTheorem 2.1, it suffices to prove that there exist φ ∈ R and Φ ∈ L such thatΦ ( x ( φ ( t )) , y ( φ ( t ))) = (cid:0) t + 0(16) , t + t + 0(16) (cid:1) . We construct a sequence of equivalent elements ( x i ( t ) , y i ( t )) , i = 1 , . . . , φ i ∈ R and automorphisms Φ i ∈ L , i = 1 , . . . , x i ( t ) , y i ( t )) = Φ i ( x i − ( φ i ( t )) , y i − ( φ i ( t ))) , ∀ i = 1 , . . . , x ( t ) , y ( t )) = (cid:0) t + 0(16) , t + t + 0(16) (cid:1) . Indeed, we may write( x ( t ) , y ( t )) = (cid:0) t , t + t + b t + b t + 0(10) (cid:1) , for some b , b ∈ K , and define φ ( t ) = t + ct + ct , Φ ( x, y ) = ( x − cy, y )with c = ( b − b ) /
7. Then( x ( t ) , y ( t )) = (cid:0) t + O (8) , t + t + O (10) (cid:1) . Since 4 is not divisible by p , there exists an automorphism φ ∈ R such that x ( φ ( t )) = t . Putting Φ ( x, y ) = ( x, y ) one has( x ( t ) , y ( t )) = (cid:0) t , t + t + b t + b t + O (12) (cid:1) , for some b , b ∈ K . We define φ ( t ) = t + ( b − b ) t , Φ ( x, y ) = ( x, y − (7 b − b ) xy ) . Then( x ( t ) , y ( t )) = (cid:0) t + a t + a t + 0(16) , t + t + b t + b t + O (14) (cid:1) , for some a , a , b , b ∈ K . The automorphism φ ( t ) = t and the automorphismΦ ( x, y ) = (cid:18) x − a x + (2 a − a ) x , y − b y + ( b − b ) x (cid:19) yield that ( x ( t ) , y ( t )) = (cid:0) t + 0(16) , t + t + b t + b t + O (16) (cid:1) . Applying φ ( t ) = t + ( b − b ) t , Φ ( x, y ) = (cid:0) x − b − b ) x , y − (7 b − b ) x y (cid:1) to ( x ( t ) , y ( t )) we obtain( x ( t ) , y ( t )) = (cid:0) t + 0(16) , t + t + O (16) (cid:1) as desired. (cid:3) Proposition 3.5.
The singularities in Tables 3 are parametrization simple.Proof.
It follows directly from Proposition 3.4 and the upper semicontinuity of themultiplicity m and of the conductor c . For instance, a singularity of type W ♯ q − ( p > A k , E k , W , W , W ♯ q − with k ≤ q + 14. (cid:3) References [1] V. I. Arnol’d,
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E-mail address : [email protected] † TIMAS, Thang Long University,Nghiem Xuan Yem, Hanoi, Vietnam.
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