Topological type of discriminants of some special families

We will describe the topological type of the discriminant curve of the morphism $(\ell, f)$, where $\ell$ is a smooth curve and $f$ is an irreducible curve (branch) of multiplicity less than five or a branch that the difference between its Milnor number and Tjurina number is less than 3. We prove that for a branch of these families, the topological type of the discriminant curve is determined by the semigroup, the Zariski invariant and at most two other analytical invariants of the branch.


Introduction
Let f (x, y) ∈ C{x, y} irreducible. The germ of irreducible analytic curve (branch) of equation f (x, y) = 0 is denoted by C ≡ f (x, y) = 0. Observe that the curves f (x, y) = 0 and u(x, y)f (x, y) = 0 are the same, for any unit u(x, y) ∈ C{x, y}. The multiplicity of C, denoted by m(C), is by definition the order of the power series f (x, y). Suppose that C has multiplicity n. We will say that C is singular if n > 1. Otherwise C is a smooth curve. The initial form of f (x, y) is the sum of all terms of f (x, y) of degree n. Since f is irreducible its initial form is a power of a linear form. After a linear change of coordinates, if necessary, we can suppose that the initial form of f (x, y) is y n . Suppose that C has multiplicity n > 1. We denote by N * the set of positive integers. By Newton's theorem ( [Hef,Theorem 3.8]) there is α(x 1/n ) ∈ C{x} * = m∈N * C{x 1/m } with α(0) = 0 such that f (x, α(x 1/n )) = 0 and we say that α(x 1/n ) ∈ C{x} * is a Newton-Puiseux root of C. Let us denote by Zer(f ) the set of Newton-Puiseux roots of C. Let α(x 1/n ) ∈ Zer(f ). After Puiseux theorem ( [Hef,Corollary 3.12]) we have that Zer(f ) = α j := α(ω j x 1/n ) n j=1 , where ω is a nth-primitive root of the unity. Hence (1) f (x, y) = u(x, y) n j=1 y − (α(ω j x 1/n )) , where u ∈ C{x, y} is a unit. After a change of coordinates, if necessary, we can write α(x) = i≥s 1 a i x i/n , where s 1 > n and s 1 ≡ 0 mod n.
If we put x = t n , where t is a new variable, the Newton-Puiseux root α(x 1/n ) can be written as what we will call Puiseux parametrisation of C.
There are g ∈ N and a sequence (β 0 = n < β 1 = s 1 < β 2 < · · · < β g ) of nonnegative integers such that The sequence (β 0 , · · · , β g ) ⊆ N is called sequence of characteristic exponents of C. The number g is a topological invariant called genus of the branch C.
Consider the set S(C) := {i 0 (f, h) : h ∈ C{x, y}, h ≡ 0 mod f }, where i 0 (f, h) = dim C C{x, y}/(f, h) is the intersection number (or intersection multiplicity) of f (x, y) = 0 and h(x, y) = 0 at the origin. It is well-known that S(C) is a semigroup called semigroup of values of the branch C. The complementary of S(C) in N is finite. The conductor of S(C) is by definition the greatest natural number c ∈ N such that for every natural number N ∈ N, with N ≥ c, is an element of S(C).
If n > 2 we have c ≥ s 1 + 1. Let q be the number of natural numbers between s 1 and c which are not in S(C). We can verify (see [Z2, page 21] Let f, h ∈ C{x, y} be irreducible power series. After Halphen-Zeuthen formula we get Two branches C and D have the same topological type (or they are equisingular) if they are topologically equivalent as embedded surfaces in C 2 . It is well-known ( [Z2,Chapter II]) that two branches are equisingular if and only if they have the same semigroup of values or equivalently they have the same characteristic exponents. Denote by E(C) the set of branches which are equisingular to C. In the set E(C) we define the next equivalence relation: two branches D 1 and D 2 in E(C) are analytically equivalent, and we will denote it by D 1 ∼ = D 2 if there exists an analytic isomorphism T : The moduli space of the equisingularity class E(C) is the quotient space E(C)/ ∼ =. Let ν 1 < ν 2 < . . . < ν q be the integers of the set {s 1 + 1, . . . , c − 1} which are not in S(C). Zariski proved [Z2, Proposition 1.2, Chapter III] that there exists a branch C analytically equivalent to C, parametrized as follows: Let Λ := {ν(ω) : ω ∈ Ω}. If Λ\S(C) = ∅ then the number λ := min (Λ\S(C)) − v 0 is an analytical invariant of C called Zariski invariant.
After [Z2,Lemma 2.6,Chapter IV] we can rewrite the parametrization (4) in the next form: The Newton polygon of f , denoted by N (f ), is by definition the convex hull of supp(f ) + R 2 ≥0 . Observe that N (f ) = N (uf ) for any unit u ∈ C{x, y}. Nevertheless the Newton polygon depends on coordinates. The inclination of any compact face L of N (f ) is by definition de quotient of the length of the proyection of L over the horizonal axis by the length of its projection over the vertical axis. The Newton polygon of f gives information on the Newton-Puiseux roots of f (x, y) = 0. More precisely, if L is a compact face of N (f ) of inclination i and the length of its projection over the vertical axis is ℓ 2 then f has ℓ 2 Newton-Puiseux roots of order i (see [Ch,Lemme 8.4.2]).
We say that f (x, y) ∈ C{x, y} is non degenerate in the sense of Kouchnirenko, with respect to the coordinates (x, y), if for any compact edge L of N (f ) the polynomial f L (x, y) : does not have critical points outside the axes x = 0 and y = 0, or equivalently, the polynomial F L (z) := f L (1,z) has no multiple roots, where j 0 := min{j ∈ N : (i, j) ∈ L}. Since N (f ) = N (uf ), for any unit u ∈ C{x, y}, the notion of non degeneracy is extended to curves. The topological type of non degenerate plane curves are completely determined by their Newton polygons (see [O,Proposition 4.7] and [GB-L-P, Theorem 3.2]).
Let ℓ(x, y) = 0 be a smooth curve and f (x, y) = 0 defining an isolated singularity at 0 ∈ C 2 . Assume that ℓ(x, y) does not divide f (x, y) and consider the morphism There are two curves associated with (ℓ, f ): the polar curve ∂ℓ ∂x ∂f ∂y − ∂ℓ ∂y ∂f ∂x = 0 and its direct image D(u, v) = 0 which is called the discriminant curve of the morphism (ℓ, f ).
The topological type of the polar curve depends on the analytical type of ℓ(x, y) = 0 and f (x, y) = 0. In  the authors completely determine the topological type of the generic polar curve when the multiplicity of f (x, y) = 0 is less than five.
The Newton polygon of D(u, v) in the coordinates (u, v) is called jacobian Newton polygon of the morphism (ℓ, f ). This notion was introduced by Teissier in [T], who proved that the inclinations of this jacobian polygon are topological invariants of (ℓ, f ) called polar invariants. After Merle [M], when f is irreducible with semigroup of values S(f ) = s 0 , s 1 , . . . , s g then the jacobian Newton polygon of (ℓ, f ) has The length of the projection of E i on the vertical axis is e i−1 e i − 1 · e i−1 e 0 . The length of the projection of E i on the horizontal axis is e i−1 e i − 1 · s i . Hence the inclinations (quotient between the length of the horizontal projection and the length of the vertical projection) of the compact edges of the jacobian polygon are s 1 < e 1 e 0 s 2 < e 2 e 0 s 3 < · · · < e g−1 In [GB-Gw-L] the authors study the pairs (ℓ, f ) for which the discriminant curve is non degenerate in the Kouchnirenko sense. In particular, when f is irreducible, after [GB-Gw-L, Corollary 4.4] the discriminant curve D(u, v) = 0 is non degenerate if and only if the multiplicity of f (x, y) = 0 equals two or equals four and genus equals two. Otherwise the discriminant curve is degenerate. Our aim in this paper will be to describe the topological type of the discriminant curve D(u, v) = 0 of the morphism (ℓ, f ), where f is irreducible and belonging to some special families, as for example, branches of multiplicity less than five, branches C such that the difference between its Milnor number µ(C) and Tjurina number τ (C) is less than 3, with µ(C) = i 0 ∂f ∂x , ∂f ∂y = dim C C{x, y}/ ∂f ∂x , ∂f ∂y and τ (C) = dim C C{x, y}/ f, ∂f ∂x , ∂f ∂y ; where ∂f ∂x , ∂f ∂y (resp. ∂f ∂x , ∂f ∂y , f ) denotes the ideal of C{x, y} generated by ∂f ∂x and ∂f ∂y (resp. by ∂f ∂x , ∂f ∂y and f ). For these families of plane branches we determine the topological type of their discriminant curves, in the spirit of . We prove that the topological type of the discriminant curve D(u, v) = 0 is determined, at most, by the semigroup of values S(f ), the Zariski invariant and two other analytical invariants of the curve f (x, y) = 0. In all cases we explicitly determine such analytical invariants. Hence, in order to describe the topological type of the discriminant curve of a branch, it is necessary the same number of analytical invariants of the initial branch, as it happens for its generic polar curves (see ). Finally in Section 5 we summarize the different topological types of the discriminant curve in some tables.

Equation of the discriminant curve
An analytic change of coordinates does not affect the discriminant curve of the morphism defined in (5) (see for example [Ca,Section 3]). Hence in what follows we assume that ℓ(x, y) = x. Then ∂f ∂y = 0 is the polar curve of the morphism (x, f ).
In this paper we will determine the topological type of the discriminant curve of the morphism (5) for ℓ(x, y) = x and f (x, y) ∈ C{x, y} irreducible belonging to some special families. The corresponding study relative to the polar curves was done in  and [HI], where the authors characterize the equisingularity classes of irreducible plane curve germs whose general members have non degenerate general polar curves. In addition, they give explicit Zariski open sets of curves in these equisingularity classes whose general polars are non degenerate and describe their topology.
Suppose that the Newton-Puiseux factorizations of f (x, y) and ∂f ∂y (x, y) are of the form If f is irreducible of order n then n is the smallest natural number such that {α i (x)} i ⊂ C{x 1/n }. Moreover if we fix α i (x 1/n ) then α j (x 1/n ) = α i (ωx 1/n ) for any 1 ≤ j ≤ n, where w is a nth-root of the unity.

Discriminants of branches of small multiplicities
In this section we determine the topological type of the discriminant of the morphism given in (5), where C ≡ f (x, y) = 0 has small multiplicity. For this we will make use the results of [Z2] and the analytic classification of plane branches of multiplicity less than or equal to four, given in [Hef-Her].
3.1. Discriminants of branches of multiplicity 2. Let C ≡ f (x, y) = 0 be a branch of multiplicity 2.
The minimal system of generators of the semigroup of C is 2, s 1 , where s 1 is an odd natural number. By [Z2, Chapitre V] the moduli space of branches of multiplicity two have a unique point which parametrization is given by (t 2 , t s 1 ), that is the branch y 2 − x s 1 = 0. Then f y (x, y) = 2y whose Newton-Puiseux root is y = 0. Hence, after (8) The Newton polygon of D(u, v) has only one compact edge. The univariate polynomial associated with this edge is z + 1, so D(u, v) is non degenerate.
3.2. Discriminants of branches of multiplicity 3. Let C ≡ f (x, y) = 0 be a branch of multiplicity 3.
The minimal system of generators of the semigroup of C is 3, s 1 , where s 1 ∈ N such that s 1 ≡ 0 mod 3. By [Z2, Chapitre V] the moduli space of branches of multiplicity three is completely determined by the semigroup of the branch and its Zariski invariant λ. The corresponding normal forms are : Observe that if λ = 0 then s 1 +λ 2 is a natural number greater than 2.
Proposition 3.1. Let C ≡ f (x, y) = 0 be a branch of semigroup 3, s 1 and Zariski invariant equals λ. The discriminant curve D(u, v) = 0 is degenerate and its topological type is determined by (3, s 1 , λ) in the next way: (1) If λ = 0 then the discriminant is the double smooth branch (v + u s 1 ) 2 = 0.
The Newton polygon of D(u, v) is elementary (it has only one compact edge) and the univariate polynomial associated with its compact edge is (z + 1) 2 . Hence the discriminant D(u, v) = 0 is degenerate.
Corollary 3.2. If C is a branch of multiplicity 2 or 3 and non-zero Zariski invariant then the discriminant curve D(u, v) = 0 has not multiple irreducible branches.
Corollary 3.2 does not hold for branches of multiplicity 4 as the proof of Proposition 3.5 shows.
3.3. Discriminants of branches of multiplicity 4. Let C ≡ f (x, y) = 0 be a branch of multiplicity 4. The branch C may have genus one or two.
where D 1 is a smooth branch, D 2 is a singular branch of semigroup 2, s 2 and the intersection multiplicity between both branches is i 0 (D 1 , D 2 ) = 2s 1 .
Suppose now that the branch C has genus 1 and semigroup of values equals 4, s 1 . By [Hef-Her] the moduli space of branches of multiplicity four and genus 1 has five families of normal forms: where λ i is the Zariski invariant of the ith-normal form family. More precisely we have (1) (λ 1 , σ 1 (t)) = (0, t s 1 ), After the Newton-Puiseux Theorem, the ith-normal form admits the equation where ω is a 4th primitive root of the unity. Hence, we obtain the implicit equation for each normal form family: Observe that NF 4.5: f 5 (x, y) = y 4 + P (x)y 2 + Q(x)y + R(x), where Proof. Suppose λ i = 0. By (11) the implicit equation of the normal form is f 1 (x, y) = y 4 − x s 1 . Then (f 1 ) y (x, y) = 4y 3 whose Newton-Puiseux root is y = 0, with multiplicity three. Hence, after (8) From the implicit equations f i (x, y), 2 ≤ i ≤ 4, given in (12) and from the inequality (13) we get that the Newton polygon of (f i ) y (x, y) has only one compact edge whose vertices are (0, 3) and (s 1 − j, 0). All the parametrisations of (f i ) y (x, y) = 0, for 2 ≤ i ≤ 4, have the same order and we can write them by γ r (u) = εu s 1 −j 3 + · · · , where ε is a 3th-root of the unity. From (8) and for a fix i ∈ {2, 3, 4}, we have (v − f i (u, γ r (u))), and considering the development of f i (u, γ r (u)) we obtain: (v + u s 1 + 3εu 2s 1 +λ i 3 + · · · } ε 3 =1 and ord(η l − η r ) = 2s 1 +λ i 3 , for 1 ≤ l = r ≤ 3. The topological type of D(u, v) = 0 is determined by the semigroup of the branch C ≡ f i (x, y) = 0 and its Zariski analytical invariant λ i = 2s 1 − 4j for 2 ≤ j ≤ [ s 1 4 ]. We distinguish two cases: if 3 and 2s 1 + λ i are coprime then the discriminant D(u, v) = 0 is a branch of semigroup 3, 2s 1 + λ i . Otherwise, by (2), we conclude that the discriminant curve is the union of three different smooth branches D l (u, v) = 0 with intersection multiplicity i 0 (D l , D r ) = 2s 1 +λ i 3 for l = r.
In all cases the Newton polygon of D(u, v) is elementary with vertices (0, 3) and (3s 1 , 0). The polynomial associated with its compact edge is (z + 1) 3 , so the discriminant curve D(u, v) = 0 is degenerate.
Proposition 3.5. Let C ≡ f (x, y) = 0 be a branch belonging to the family NF 4.5. Then the discriminant curve D(u, v) = 0 is degenerate and its topological type is determined by the semigroup S(f ) = 4, s 1 , the Zariski invariant λ 5 and at most two other analytical invariants of C.

We distinguish different cases:
Case A. If a i = 0 in σ 5 (t) for all i : then we have that (f 5 ) y (x, y) = 4y 3 − 8x s 1 −j y = 4y(y 2 − 2x s 1 −j ) whose Newton-Puiseux roots are 0, ± √ 2x Case B. If a k = 0 and a k+l = 0 in σ 5 (t) for l > 0: Then So the Newton polygon of (f 5 ) y (x, y) depends on the position of the point M = (s i − j, 1) with respect to the line passing by E = (0, 3) and F = (s 1 − j + [ s 1 4 ] + k, 0). We get three situations: ]+k then N ((f 5 ) y ) has only one compact edge of vertices E and F . Hence the order of the Newton-Puiseux roots {γ i } 3 i=1 of (f 5 ) y (x, y) = 0 equals for some nonzero complex number a and where w i is a cubic root of the unity. If gcd(3, (s 1 −j +[ s 1 4 ]+k)) = 1 then the discriminant curve is irreducible with semigroup 3, 4(s 1 − j + [ s 1 4 ] + k) . On the other case, we get gcd(3, (s 1 − j + [ s 1 4 ] + k)) = 3 and the discriminant curve is the union of three smooth curves D l (x, y) = 0 such that i 0 (D l , D r ) = 4(s 1 −j+[ If s 1 − j is odd then the discriminant curve is the union of a smooth branch D 1 (u, v) = 0 and a singular branch D 2 (x, y) = 0 with semigroup 2, 3(s 1 − j) + 2([ s 1 4 ] + k) . Moreover i 0 (D 1 , D 2 ) = 4(s 1 − j). Otherwise, if s 1 − j is even then the discriminant curve is the union of three smooth branches D l (x, y) = 0 such that i 0 (D 1 , D l ) = 2(s 1 − j) for 2 ≤ l ≤ 3 and i 0 ( ]+k then N ((f 5 ) y ) has only one compact edge of vertices E and F and M is an interior point of this edge. The polynomial in one variable, associated with the compact edge of N ((f 5 ) y ), is p(z) = z 3 − 2z − a k . After the z-discriminant of p(z) we have that the roots of p(z) are simple if and only if a k = ± 4 √ 6 9 . We will study both cases: . Denote by z i the three different roots of the polynomial p(z). For any γ i (x) ∈ Zer((f 5 ) y ) we have: f (u, γ i (u)) = −u s 1 + q(z i )u 2(s 1 −j) + · · · , where q(z) = z 4 − 4z 2 − 4a k z + 2.