Topological type of discriminants of some special families

We will describe the topological type of the discriminant curve of the morphism (ℓ,f)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\ell , f)$$\end{document}, where ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document} is a smooth curve and f is an irreducible curve (branch) of multiplicity less than five or a branch such 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.

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 [7,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's theorem [7,Corollary 3.12] we have that Zer( f ) = α j := α(ω j x 1/n ) n j=1 , where ω is an nth-primitive root of unity. Hence f (x, y) = u(x, y) n j=1 y − (α(ω j x 1/n )) , (1.1) 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 x(t) = t n , y(t) = i≥s 1 a i t i , that we will call the 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 (1. 2) The sequence (β 0 , · · · , β g ) ⊆ N is called the sequence of characteristic exponents of C. The number g is a topological invariant called the genus of the branch C.
Consider the set 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 the semigroup of values of the branch C. The complement of S(C) in N is finite. The conductor of S(C) is by definition the smallest natural number c ∈ N such that every natural number N ∈ N with N ≥ c is an element of S(C). The semigroup S(C) admits a minimal system of generators (s 0 , s 1 , . . . , s g ), where s i−1 < s i , g is the genus of C, s 0 = n = i 0 ( f , x) and s 1 = m =: i 0 ( f , y). It is a well-known property of S(C) [7, p. 88, (6.5)] that e k := gcd(s 0 , . . . , s k ) = gcd(β 0 , . . . , β k ) for 0 ≤ k ≤ g and e k−1 s k < e k s k+1 for 1 ≤ k ≤ g − 1.
If n > 2, then 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 [16, p. 21]) that q = c 2 − s 1 + s 1 s 0 + 1, for s 0 = n > 2, where z denotes the integral part of z ∈ R.
Let f , h ∈ C{x, y} be irreducible power series. Using the Halphen-Zeuthen formula we get where Zer f = {α i } i and Zer h = {γ j } j . Two branches C and D have the same topological type (or they are equisingular) if they are topologically equivalent as embedded curves in C 2 . It is well known [16,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 this by D 1 ∼ = D 2 , if there exists an analytic isomorphism T : U 1 −→ U 2 such that U i are neighbourhoods of the origin, D i is defined in U i , 1 ≤ i ≤ 2 and T (D 1 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 [16, Proposition 1.2, Chapter III] that there exists a branch C analytically equivalent to C, parametrized as follows: Using [16, Lemma 2.6, Chapter IV], we can rewrite the parametrization (1.4) in the next form: The Normal Forms Theorem (see [8,Theorem 1]) states that the plane branch C with semigroup of values S(C) = s 0 , . . . , s g and value set of differentials is either analytically equivalent to a branch with Puiseux parametrization (t s 0 , t s 1 ) or Puiseux parametrization Observe that N ( f ) = N (u f ) 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 the quotient of the length of the projection 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 2 , then f has 2 Newton-Puiseux roots of order i (see [3,Lemme 8.4.2]).
We say that f (x, y) ∈ C{x, y} is nondegenerate 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) := (i, j)∈L∩supp( f ) a i j x i y j does not have critical points outside the axes x = 0 and y = 0, or equivalently, the polynomial F L (z) := f L (1,z) z j 0 has no multiple roots, where j 0 := min{ j ∈ N : (i, j) ∈ L}. Since N ( f ) = N (u f ), for any unit u ∈ C{x, y}, the notion of nondegeneracy is extended to curves. The topological type of nondegenerate plane curves are completely determined by their Newton polygons (see [13,Proposition 4.7] and [6,Theorem 3.2]).
Let (x, y) = 0 be a smooth curve and let f (x, y) = 0 define an isolated singularity at 0 ∈ C 2 . Assume that (x, y) does not divide f (x, y) and consider the morphism (1.5) There are two curves associated with ( , f ): The topological type of the polar curve depends on the analytical type of (x, y) = 0 and f (x, y) = 0. In [9] the authors completely determine the topological type of the generic polar curve when the multiplicity of f (x, y) = 0 is less than 5.
The Newton polygon of D (u, v) in the coordinates (u, v) is called the jacobian Newton polygon of the morphism ( , f ). This notion was introduced by Teissier [14], who proved that the inclinations of this jacobian polygon are topological invariants of ( , f ) called polar invariants. After Merle [12], when f is irreducible with semigroup of values S( f ) = s 0 , s 1 , . . . , s g then the jacobian Newton polygon of ( , f ) has g compact edges . 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 e 0 s g .
In [5] the authors study the pairs ( , f ) for which the discriminant curve is nondegenerate in the Kouchnirenko sense. In particular, when f is irreducible, after [5,Corollary 4.4] the discriminant curve D(u, v) = 0 is nondegenerate if and only if the multiplicity of f (x, y) = 0 equals 2 or equals 4 and the genus equals 2. 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 ( , f ), where f is irreducible and belongs to some special families, as for example, branches of multiplicity less than 5, branches C such that the difference between its Milnor number μ(C) and Tjurina number τ (C) is less than 3, with μ(C) ∂ y and f ). For these families of plane branches we determine the topological type of their discriminant curves, in the spirit of [9]. 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 necessarily the same number of analytical invariants of the initial branch, as it happens for its generic polar curves (see [9]). Finally in Sect. 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 (1.5) (see for example [2,Section 3]). Hence in what follows we assume that (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 (1.5) for (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 [10] and [11], where the authors characterize the equisingularity classes of irreducible plane curve germs whose general members have nondegenerate general polar curves. In addition, they give explicit Zariski open sets of curves in these equisingularity classes whose general polars are nondegenerate 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 Following [4,Lemma 5.4] the discriminant curve of the morphism (x, f ) can be written as

Discriminants of branches of small multiplicities
In this section we determine the topological type of the discriminant of the morphism given in (1.5), where C ≡ f (x, y) = 0 has small multiplicity. For this we will make use the results of [16] and the analytic classification of plane branches of multiplicity less than or equal to 4, given in [8].

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 [16, Chapitre V] the moduli space of branches of multiplicity 2 have a unique point whose parametrization is given by has only one compact edge. The univariate polynomial associated with this edge is z + 1, so D(u, v) is nondegenerate.

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 is such that s 1 ≡ 0 mod 3. By [16, Chapitre V] the moduli space of branches of multiplicity 3 is completely determined by the semigroup of the branch and its Zariski invariant λ. The corresponding normal forms are as follows: where a = 0 when λ = 0 or if λ = 0 then we have a = 1 and Observe that if λ = 0 then s 1 +λ 3 is a natural number greater than 2.
+ u λ such that, after the Halphen-Zeuthen formula, the intersection multiplicity is 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 nonzero Zariski invariant, then the discriminant curve D(u, v) = 0 has no multiple irreducible branches.
Corollary 3.2 does not hold for branches of multiplicity 4 as the proof of Proposition 3.5 shows.

Discriminants of branches of multiplicity 4
Let C ≡ f (x, y) = 0 be a branch of multiplicity 4. The branch C may have genus 1 or 2.
where D 1 is a smooth branch, D 2 is a singular branch of semigroup 2, s 2 and the intersection multiplicity between the two branches is i 0 (D 1 , D 2 ) = 2s 1 .
Proof For genus 2, and after the second part of [5,Corollary 4.4] we get that D(u, v) = 0 is nondegenerate and we can determine its topological type from its Newton polygon (see [13,Proposition 4.7] and [6, Theorem 3.2]), which is the jacobian Newton polygon of (x, f ) (see Fig. 1).
Suppose now that the branch C has genus 1 and semigroup of values equals 4, s 1 . By [8], the moduli space of branches of multiplicity 4 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 After the Newton-Puiseux Theorem, the ith normal form has the equation where ω is a 4th primitive root of unity. Hence, we obtain the implicit equation for each normal form family:  3 , that is the discriminant curve is a triple smooth branch.

From (2.3) we conclude that the discriminant curve is the union of three smooth curves
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 have three subcases: for some nonzero complex number a and where w i is a cubic root of 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) . In 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 ) = 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, Since z 3 − 2z = a k , we get q(z) = −3z 4 + 4z 2 + 2 and if z r = z l then q(z r ) = q(z l ). Hence D(u, v) = i (z + u s 1 + q(z i )u 2(s 1 − j) + · · · ) and the discriminant curve D(u, v) = 0 is the union of three smooth branches D l (x, y) = 0 such that i 0 (D l , D r ) = 2(s 1 − j).
. The polynomial p(z) has z 1 = 2 2 3 as a simple root and z 2 = − 2 3 as a double root. If γ i ∈ Zer( ∂ f 5 ∂ y ) corresponds to z i then where r 2 , r 3 ∈ C are different. Observe that q(z 1 ) = q(z 2 ). Hence, if s 1 − 2 j is odd then the discriminant curve is the union of a smooth branch D 1 (u, v) = 0 and a singular branch then the topological type of the discriminant curve D(u, v) = 0 is as in In emphatic way.
Hence, in the next step of the Newton procedure it is enough to consider the polynomial The topological type of the discriminant curve will depend on the relation between s and s 1 − 2 j: C.3.2.1 Suppose that s 1 − 2 j > s. We obtain Hence where c := √ −a k+s 4 √ 6 , and l(z) = −4z 2 z − 1. As a consequence, when s is odd then the discriminant curve D(u, v) = 0 is the union of a smooth branch D 1 (u, v) = 0 and a singular branch with semigroup 2, 4(s 1 − j)+3s , with i 0 (D 1 , D 2 ) = 4(s 1 − j). Otherwise, if s is even then the discriminant curve D(u, v) = 0 is the union of three smooth braches D l (u, v) = 0 such that i 0 (D 1 , D l ) = 2(s 1 − j) for 2 ≤ l ≤ 3, and... and i 0 (D 2 , D 3 ) = 2(s 1 − j) + 3 2 s. C.3.2.2 Suppose that s 1 − 2 j < s. After (3.11) and for the next step of the Newton procedure we only need the polynomial whose roots are ± √ 8 9 . Let d = √ 8 9 . We obtain Hence

+ · · ·
Since 4 and s 1 are coprime then 7s 1 − 10 is odd and the discriminant curve D(u, v) = 0 is the union of a smooth branch D 1 (u, v) = 0 and a singular branch with semigroup 2, 7s 1 − 10 j , where i 0 (D 1 , D 2 ) = 4(s 1 − j). C.3.2.3 Suppose that s 1 − 2 j = s. After (3.11), in order to obtain the next term in the power series γ i , it is enough to consider the polynomial H (Z ) = 12z 2 Z 2 − 4(z 2 a 2 k + a k+s + a k ). The topological type of the discriminant will depend on the value of a k+s : C. 3 We conclude that , whose roots t 1 , t 2 are simple. Hence We conclude that  If s 1 − 2 (respectively s 0 ) and 3 are coprime then the discriminant curve is given by and D 2 (u, v) is a branch of the semigroup 3, (s 1 − 2)s 0 and, after the Halphen-Zeuthen formula, the intersection multiplicity is i 0 (D 1 , D 2 ) = 3s 1 . Otherwise D(u, v) is the product of D 1 (u, v) s 0 −4 and three smooth branches D k (u, v), 2 ≤ k ≤ 4, where i 0 (D 1 , D k ) = s 1 for 2 ≤ k ≤ 4 and i 0 (D l , D r ) = (s 1 −2)s 0 3 , for 2 ≤ l = r ≤ 4.

Remark 4.2
Observe that in Theorem 4.1 Case (B) with s 0 = 4 coincides with case σ 3 in (3.2) for j = 2. Hence this case was studied in Proposition 3.4.

Table 9
Discriminants of branches with r  Table 11 Case Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.
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