Polar Invariants of Plane Curve Singularities: Intersection Theoretical Approach

This article, based on the talk given by one of the authors at the Pierrettefest in Castro Urdiales in June 2008, is an overview of a number of recent results on the polar invariants of plane curve singularities.


Introduction
The polar invariants (called also polar quotients) of isolated hypersurface singularities were introduced by B. Teissier in 1975 to study equisingularity problems (see [Te1975], [Te1977], [Te1980]). They are by definition, the contact orders between a hypersurface and the branches of its generic polar curve. To every polar invariant q of a given isolated hypersurface singularity one associates in a natural way an integer m q > 0 called the multiplicity of q. Teissier's collection {(q, m q )} is an analytic invariant of the singularity. Even more: it is an invariant of the "c-cosécance" which is equivalent in the case of plane curve singularities to the constancy of the local embedded topological type (see [Te1977]). The Milnor number, the Lojasiewicz exponent, the C 0degree of sufficiency and other numerical invariants can be computed in terms of Teissier's collection.
It is well-known (see [Te1976], [BriKn1986], [Te1991]) that the constancy of the local embedded topological type of plane curves is equivalent to the usual definitions of equisingularity (see Preliminaries where the definition of equisingularity in terms of intersection numbers is given).
M. Merle [Mer1977] computed Teissier's collection for a branch (irreducible analytic curve) in terms of the semigroup of the branch. Much earlier a computation of the contacts between an irreducible curve and the branches of its generic polar curve was done by Henry J. S. Smith [Sm1875] but his work fell into oblivion for long time. R. Ephraim [Eph1983] generalized the Smith-Merle result computing the polar invariants in the case of special polars and applied his result to the pencil of curves which appears when studying affine curves with one branch at infinity (see Sections 4 and 7 of this article).
C. T. C. Wall gave an account of most results obtained in the above quoted papers in his book [Wall2004] dealing with different aspects of the curve singularities.
The goal of this article is to give an overview of a number of recent results on the polar invariants of plane curve singularities.
In Section 2 we present a refinement of Teissier's invariance theorem in the case of plane curve singularities. In Section 3 we give an approach to the polar invariants based on Puiseux series developing the method due to Kuo and Lu [KuoLu1977].
Section 4 is devoted to the Smith-Merle-Ephraim theorem in the one branch case and to the irreducibility criterion obtained quite recently by García Barroso and Gwoździewicz. (Theorem 4.5 and Corollary 4.6).
In Section 5 we present explicit formulae for the polar invariants in terms of semigroup of branches and intersection multiplicities due to Gwoździewicz and P loski (Theorem 5.2). The geometric interpretation of these formulae in terms of the Newton diagrams associated with many-branched singularity is new.
In Section 6 we recall a result obtained by Lenarcik and P loski (Theorem 6.1) which gives an effective formula for the jacobian Newton diagram (see Section 2) of a nondegenerate (in the sense of Kouchnirenko) plane curve singularity. Then, we present in Section 7, some applications of the polar invariants to pencils of plane curve singularities.

Preliminaries
In this section we recall some useful notions and results that we need in this article. The references for this part are [BriKn1986], [Cas2000], [Te1991], [Wall2004].

Basic notions
Let C{X, Y } be the ring of convergent complex power series in variables X, Y . Let f ∈ C{X, Y } be a nonzero power series without constant term.
An analytic curve f = 0 is defined to be the ideal generated by f in C{X, Y }.
We say that f = 0 is irreducible (reduced) if f ∈ C{X, Y } is irreducible (f has no multiple factors). The irreducible curves are also called branches. If f = f m 1 1 . . . f mr r with non-associated irreducible factors f i then we refer to f i = 0 as the branches or components of f = 0.
Recall here that for any nonzero power series f = c αβ X α Y β we put ord f = inf{α + β : c αβ = 0} and in f = c αβ X α Y β with summation over (α, β) such that α + β = ord f . The initial form in f of f determines the tangents to f = 0.
For any power series f, g ∈ C{X, Y } we define the intersection number (f, g) 0 by putting where (f, g) is the ideal of C{X, Y } generated by f and g. If f, g are nonzero power series without constant term then (f, g) 0 < +∞ if and only if the curves f = 0 and g = 0 have no common branch. Now suppose that f = 0 is a branch and consider Y } runs over all series such that f does not divide g} .
Clearly 0 ∈ S(f ) (take g = 1) and a, b ∈ S(f ) ⇒ a + b ∈ S(f ) since the intersection number is additive. We call S(f ) the semigroup of the branch f = 0. Note that S(f ) = N if and only if ord f = 1 (we say then that f = 0 is regular or nonsingular ).
Consider two reduced curves f = 0 and g = 0. They are equisingular if and only if there are factorizations f = r i=1 f i and g = r i=1 g i with the same number r > 0 of irreducible factors f i and g i such that • S(f i ) = S(g i ) for all i = 1, . . . , r, The bijection f i → g i will be called equisingularity bijection. In particular two branches are equisingular if and only if they have the same semigroup. A function defined on the set of reduced curves is an invariant if it is constant on equisingular curves. The multiplicity ord f , the number of branches r(f ) and the number of tangents t(f ) of f = 0 are invariants.
For any analytic curve f = 0 we consider the Milnor number µ 0 (f ) = (∂f /∂X, ∂f /∂Y ) 0 . One has µ 0 (f ) < +∞ if and only if the curve f = 0 is reduced. Let us recall the following two properties: • if f = 0 is a branch then µ 0 (f ) is the smallest integer c ≥ 0 such that all integers greater than or equal to c belong to S(f ), Thus the Milnor number is an invariant. A simple proof of the above properties is given in [P l1995].

Newton diagrams after [Te1976]
Let R + = {x ∈ R : x ≥ 0}. The Newton diagrams are some convex subsets of R 2 + . Let E ⊂ N 2 and let us denote by ∆(E) the convex hull of the set E + R 2 + . The subset ∆ of R 2 + is a Newton diagram (or polygon) if there is a set E ⊂ N 2 such that ∆ = ∆(E). The smallest set E 0 ⊂ N 2 such that ∆ = ∆(E 0 ) is called the set of vertices of the Newton diagram ∆. It is always finite and we can write In particular the Newton diagram ∆ with one vertex v = (α, β) is the quadrant (α, β) + R 2 + . According to Teissier for k, l > 0 we denote { k l } the Newton diagram with vertices (0, l) and (k, 0). We put also { k ∞ } = (k, 0)+R 2 + and { ∞ l } = (0, l)+R 2 + and call any diagram of the form { k l } an elementary Newton diagram. For any subsets ∆, ∆ ′ ⊂ R 2 + we consider the Minkowski sum ∆ + ∆ ′ = {u + v : u ∈ ∆ and v ∈ ∆ ′ }. One checks the following  For any segment S ∈ N (∆) we denote by |S| 1 and |S| 2 the lengths of the projections of S on the horizontal and vertical axes. We call |S| 1 /|S| 2 the inclination of S. If ∆ intersects both axes then ∆ = S |S| 1 |S| 2 (summation over all S ∈ N (∆)) and this representation is unique.

Theorem 1.2 (Newton-Puiseux Theorem)
For every q ∈ Q ∪ {∞} let m q be the number of roots α i (X) such that ord α i (X) = q. Then m q q (by convention 0 · ∞ = 0) is an integer or ∞ and

Nondegeneracy
According to [Kou1976] , the series f is nondegenerate if for every S ∈ N (f ) the polynomial in(f, S) has no critical points in the set C * × C * , where C * = C \ {0}. A lot of applications of the Newton diagrams are based on the following [Len2008]). Suppose that f, g ∈ C{X, Y } are reduced power series such that ∆(f ) = ∆(g). Then (ii) if f is nondegenerate but g is degenerate then f = 0 and g = 0 are not equisingular.
Let ∆ ⊂ R 2 + be a Newton diagram. It is easy to check that ∆ = ∆(f ) for a reduced nondegenerate power series f if and only if the distances from ∆ to the axes are ≤ 1. We call such diagrams nearly convenient. Every Newton diagram which intersects both axes (convenient in the sense of Kouchnirenko) is nearly convenient. If ∆ is nearly convenient then the reduced nondegenerate power series f such that ∆ = ∆(f ) form an open dense subset in the space of coefficients.
Let us consider an invariant I of equisingularity. For every nearly convenient Newton diagram ∆ we put I(∆) = I(∆(f )) where f is a nondegenerate reduced power series. According to the theorem quoted above I(∆) is defined correctly (does not depend on f ). There is a natural problem: calculate I(∆) effectively in terms of ∆. The most known result of this kind is due to Kouchnirenko [Kou1976].
Note that Kouchnirenko proved a much more general result concerning isolated singularities in n dimensions. In the case n = 2 the result is more precise: the equality µ 0 (f ) = µ(∆(f )) holds if and only if f is nondegenerate and we do not need the assumption "f is convenient". Theorem 1.3 can be easily deduced from the famous µ-constant theorem [LêR1976] and Kouchnirenko's result. One can give also a direct, elementary proof [Len2008]. Let us end this section with where w 1 , w 2 ≥ 2 are rational numbers) be a weighted homogeneous polynomial of order > 1. Then R 2 + \∆(f ) is the triangle with sides α = 0, β = 0 and α/w 1 + β/w 2 = 1. If f is nondegenerate then by Theorem 1.4 µ 0 (f ) = µ(∆(f )) = (w 1 − 1)(w 2 − 1) (the Milnor-Orlik formula).

The jacobian Newton polygon
The following lemma is well-known (see, for example [Del1991] or [P l2004]).
Lemma 2.1 Let f, g ∈ C{X, Y } be two power series without constant term.
Let J(f, g) = (∂f /∂X)(∂g/∂Y ) − (∂f /∂Y )(∂g/∂X) be the Jacobian of the pair (f, g). Then The right side of the above equality is finite if and only if the left is too.
Assume that l = 0 is a regular curve. Let f = 0 be a reduced curve such that J(f, l)(0, 0) = 0. If l = 0 is not a branch of f = 0 then we call J(f, l) = 0 the polar curve of f = 0 relative to l = 0. It depends on the power series f and l.
If l = bX − aY is a nonzero linear form then and we speak about the polar curve relative to the direction (a : b) ∈ P 1 (C). Using Lemma 2.1 it is easy to check the following two properties. We assume J(f, l)(0, 0) = 0. In the sequel we assume that f = 0 is a reduced curve and that the regular curve l = 0 is not a branch of f = 0.
Recall that J(f, l)(0, 0) = 0 and let J(f, l) = h 1 · · · h s be the decomposition of J(f, l) into irreducible factors. Then the rational numbers are called the polar invariants of f = 0 relative to l = 0. Let Q(f, l) be the set of polar invariants. If J(f, l)(0) = 0 then we put Q(f, l) = ∅. For every polar invariant q ∈ Q(f, l) we put Thus We call m q = (l, J q ) 0 the multiplicity of the polar invariant q. Using Lemma 2.1 we check Then Moreover if and only if there exists exactly one polar invariant of f = 0 relative to l = 0.
From the above property it follows that a regular plane curve f = 0 has exactly one polar invariant, equal to 1 relative to any nontransverse regular curve l = 0. In the sequel we assume that f = 0 is a singular reduced curve. Following Teissier [Te1980] we define the jacobian Newton polygon by putting It is easy to see that Property 2.6 The jacobian Newton polygon intersects the axes at points (0, (f, l) 0 −1) and (µ 0 (f )+(f, l) 0 −1, 0). All faces of Q(f, l) have inclinations strictly greater than 1.
The above property follows from Property 2.4 and from the following formula Let f = 0 and f ′ = 0 be reduced singular curves and let l = 0 and l ′ = 0 be regular branches such that l = 0 (resp. l ′ = 0) is not a component of f = 0 (resp. f ′ = 0). We will say that the pairs f = 0, l = 0 and f ′ = 0, l ′ = 0 are equisingular if there is an equisingularity bijection of the set of branches The following result is a refinement of Teissier's theorem on invariance of the jacobian Newton polygon [Te1977] in the case of plane curve singularities.
The computations of the jacobian Newton polygons in the next two examples were done using Theorem 6.1.
We apply the Newton-Puiseux Theorem (see Preliminaries) to D(X, T ) ∈ C{X, T }.

Polar invariants and Puiseux series
The following lemma due to Kuo and Lu (see [KuoLu1977], Lemma 3.3) is crucial for the approach to the polar invariants based on Puiseux series (see [Egg1982], [GwP l2002], [Wall2003]). Remark 3.2 In [KuoLu1977] the following property is stated:
Note also that property (*) does not hold under the assumption added in [Gar2000] that f (X, 0)f (0, Y ) = 0. To get an example it suffices to replace the series f (X, Y ) considered above by the series f (X, Y − X).
Thus #Z f = n. The tree over Z f will be denoted T (f ) and called the Kuo-Lu tree model of f (see [KuoLu1977] where the balls are called bars and h(B) is called height of B).
The balls are represented by points situated on different levels corresponding to the heights h ∈ O(Z f × Z f ). We join every ball by continuous lines with its successors.
The above quoted theorem is implicit in [KuoLu1977]. Part (i) was proved in [GwP l2002]. A short proof of (i) and (ii) is given in [GarGw2008].
Theorem 4.1 (Smith-Merle-Ephraim) Suppose that f = 0 is a singular branch and l = 0 a regular curve. Letb 0 , . . . ,b h be the (f, l) 0 -minimal system of generators of the semigroup S(f ). Then with the notation introduced above By convention the empty product which appears for k = 1 is equal to 1. The sequence of generators can be characterized in purely arithmetical terms. Let us recall (see [Bre1972], [Zariski1973], [Del1994], [GwP l1995]).  (2) Q(f ) is a complete invariant of the branch f = 0.
Theorem 4.5 [GarGw2008] Let f = 0 and g = 0 be two reduced curves such that Q(f ) = Q(g). Suppose that f = 0 is an irreducible curve. Then g = 0 is also irreducible.

Polar invariants in many branched case
Let ϕ, ψ ∈ C{X, Y } be irreducible power series. The contact coefficient (in the sense of Hironaka) with respect to a regular curve l = 0 is the rational number h(ϕ, ψ; l) = (ϕ, ψ) 0 (l, ψ) 0 .
If l = 0 and ψ = 0 are transverse then h(ϕ, ψ; l) = (ϕ, ψ) 0 /ord ψ and we write h(ϕ, ψ) instead of h(ϕ, ψ; l). Let f = 0 be a reduced curve with r > 1 branches. To describe the contacts of f i = 0 with the branches f j = 0, j = i let us consider the following diagram Note that the diagram H i (f, l) lies above horizontal axis and has vertices (0, (l, f ) 0 ) and ((f i , f /f i ) 0 , (l, f i ) 0 ). The distance from H i (f, l) to the horizontal axis is equal to (l, f i ) 0 .
We omit the simple proof of the following

Now let
for τ > 0 and i = 1, . . . , r. According to Lemma 5.1 the function q i is determined by the diagram H i (f, l) and has an obvious geometric interpretation. The functions q i are piecewise linear, continuous and strictly increasing. The following explicit formula for polar quotients of a many-branched curve is due to [GwP l2002].
Theorem 5.2 Let f = f 1 . . . f r be a reduced power series with r > 1 irreducible factors. Then We call the elements of q i (Q(f i , l) ∪ H i (f, l)) polar invariants associated with the branch f i = 0. A polar invariant can be associated with more than one branch.
The polar invariants associated with the branch f i = 0 can be interpreted in terms of the Newton diagram H i (f, l) and the jacobian Newton polygon Q(f i , l) of the branch f i = 0. To this end call a line supporting H i (f, l) distinguished if it extends a face of H i (f, l) or is parallel to a face of Q(f i , l). Then the polar invariants associated to the branch f i = 0 are exactly the quotients of the form p d i where (p, 0) is the point of intersection of a distinguished supporting line with the horizontal axis and d i = (l, f i ) 0 is the distance from H i (f, l) to this axis.
Let us calculate η(f, l) = sup Q(f, l). Using the fact that the functions q i are increasing we get For the applications of the above formula see [GarKP2005].
To construct a complete invariant of the pair f = 0, l = 0 the notion of partial polar quotient introduced in [Egg1982] is useful. E. García Barroso characterized the type of equisingularity of the curve by matrices of partial polar quotients (see [Gar2000]).

Polar invariants and the Newton diagram
We want to calculate the jacobian Newton polygon of a nondegenerate singularity f = 0 in terms of the Newton diagram ∆(f ). To formulate the result we need some notions. Let f ∈ C{X, Y } be a nonzero power series without constant term. The segment S ∈ N f is principal if |S| 1 = |S| 2 . If a principal segment exists it is unique. Put N ′ f = N f \ {principal segment}. For every segment S ∈ N ′ f we put m(S) = min(|S| 1 , |S| 2 ) − 1 if 1 ≤ |S| 1 < |S| 2 and S has a vertex on the vertical axis or if 1 ≤ |S| 2 < |S| 1 and S has a vertex on the horizontal axis. Moreover we let m(S) = min(|S| 1 , |S| 2 ) for all remaining cases.
Recall that t(f ) is the number of tangents to f = 0. If f is nondegenerate then t(f ) can be read from the Newton diagram ∆(f ). We have the following result due to [LenP l2000] (see also [LenMaP l2003]). We put by convention { 0 0 } = R 2 + (the zero Newton diagram).
Hence the set of weights is an invariant of f = 0.

Application to pencils of plane curve singularities
When studying the singularities at infinity of polynomials in two complex variables of degree N > 1 one considers the pencils of plane curves of the form f t = f − tl N , t ∈ C where f, l ∈ C{X, Y } are coprime and a regular curve l = 0 is not a component of the local curve f = 0 (see [Eph1983], [GarP l2004], [LenMaP l2003], [P l2004]). Let U ⊂ C be a Zariski open subset of C. We say that the pencil (f t : t ∈ U) is equisingular if the Milnor number µ 0 (f t ) is constant for t ∈ U. This means by µ-constant theorem for pencils [Cas2000] that for any t 1 , t 2 ∈ U the curves f t 1 = 0 and f t 2 = 0 are equisingular.  (2) the pencil (f − tl N : t ∈ C) is equisingular if and only if η(f, l) = sup Q(f, l) < N.