On the topology of polars of analytic plane branches · On the topology of polars of analytic plane...

On the topology of polars of analyticplane branches

Abramo Hefez

Universidade Federal Fluminense

Joint work with M. F. H. Iglesias and M. E. Hernandes

Merida - December 2014

Dedicated Pepe Seade on occasion of his 60th birthday

Our final goal in this project was to describe the variation of thetopology of the polar curve associated to a germ of irreducibleplane curve belonging to a given equisingularity class.

In the last talk I gave on this subject, two years ago, we were moreoptimistic about this program, but it turned out to be a difficultproblem, since the topology of the polar curve depends sensibly onthe analytic type of the curve and not on some other coarserattribute.

Our final goal in this project was to describe the variation of thetopology of the polar curve associated to a germ of irreducibleplane curve belonging to a given equisingularity class.

In the last talk I gave on this subject, two years ago, we were moreoptimistic about this program, but it turned out to be a difficultproblem, since the topology of the polar curve depends sensibly onthe analytic type of the curve and not on some other coarserattribute.

So, now it seems hopeless to expect to solve this problem ingeneral, but we will show the strategy we used to solve itcompletely in concrete instances.

Let f ∈ C{x , y} be a germ of an analytic function such thatf (0, 0) = 0 and let (f ) be the zero set germ of f in (C2, 0).

Since the concept of polar of a curve is somewhat subtle, becauseit depends on the equation of the curve, we start by seeing how itbehaves under some natural equivalence relations to be considered.

• The associate relation, for which f and g are equivalent, if thereexists a unit u in C{x , y} such that g = u · f .

When f and g are reduced, then one has that (f ) = (g) if andonly if f and g are associated.

• The R-equivalence, or analytic change of coordinates at thesource, for which f ∼R g if there exists an analytic localisomorphism ϕ of (C2, 0) such that g = f ◦ ϕ.

• As a combination of the associate relation with theR-equivalence relation we get the K-equivalence, for whichf ∼K g , if there exist a local analytic isomorphism ϕ of (C2, 0)and a unit u of C{x , y} such that g = u · f ◦ ϕ.

The K-equivalence on functions vanishing at the origin induces anequivalence relation ∼ among germs of curves, called the analyticequivalence, defined by

(f ) ∼ (g) ⇐⇒ f ∼K g .

• The topological equivalence, or equisingularity, in which analyticequivalence is relaxed in the following way:(f ) ≡ (g) if there exist a germ of homeomorphism ϕ of (C2, 0)and a unit in C{x , y} such that g = u · f ◦ ϕ.

We will use the following invariants to describe the equisingularityclass of a germ of plane curve:

• For a branch: its semigroup of values

Γ = {I (f , g); g ∈ C{x , y} \ 〈f 〉},

which will be represented by its minimal set of generators

Γ = 〈v0, v1, . . . , vg 〉,

where g is called the genus of (f ) and measures its topologicalcomplexity.

We will use the following invariants to describe the equisingularityclass of a germ of plane curve:

• For a branch: its semigroup of values

Γ = {I (f , g); g ∈ C{x , y} \ 〈f 〉},

which will be represented by its minimal set of generators

Γ = 〈v0, v1, . . . , vg 〉,

where g is called the genus of (f ) and measures its topologicalcomplexity.

• For a reducible curve: the semigroup of its branches and theintersection multiplicities among them.

Polars

Given f ∈ C{x , y}, a polar of f is a member of the pencil of powerseries:

L(f ) = {P(a:b)(f ) = afx + bfy ; (a : b) ∈ P1 }.

Polars

What are the common properties of the elements in L(f )?

Polars

If f is reduced, then the general member of L(f ) is reduced and thecurves associated to two general elements in L(f ) are equisingular.

Polars

If f is reduced, then the general member of L(f ) is reduced and thecurves associated to two general elements in L(f ) are equisingular.

We will call a general member P(f ) of L(f ) the polar of f and itsequisingularity class the equisingularity class of the polar of f .

This notion of polar doesn’t make sense for a curve (f ), since apolar of uf has no direct relationship with the polars of f .

RemarkOne could define the polar of (f ) as a general member of theTjurina ideal T (f ) = 〈fx , fy , f 〉. In such case the polar of (f ) wouldbe well defined. An important fact is that the topology of ageneral member of T (f ) is the same as the topology of P(f ).

Now, since T (f ) = T (uf ), it follows that the general polar of fand of uf are equisingular.

This allows to define the equisingularity class of the polar of acurve (f ).

Caution: one may not talk about the analytic type of the (general)polar of f , because the analytic type of P(a:b)(f ) depends sensiblyupon the direction (a : b).

All we did so far was for a fixed curve (f ).

Now, a natural question that one could ask is:

What happens to the topology of P(f ) when (f ) varies in a givenequisingularity class?

All we did so far was for a fixed curve (f ).

Now, a natural question that one could ask is:

What happens to the topology of P(f ) when (f ) varies in a givenequisingularity class?

At an early stage of the theory, mathematicians thought thatequisingular curves would have equisingular (general) polars.

This is not true, as one can see from the following simple exampledue to F. Pham:

Consider the family

fλ = y 3 − x11 + λx8y .

Consider the family

fλ = y 3 − x11 + λx8y .

The curves in the family (fλ) are all equisingular with semigroup ofvalues 〈3, 11〉.

Consider the family

fλ = y 3 − x11 + λx8y .

The general polar curve of f0 has two smooth branches intersectingwith multiplicity 5.

Consider the family

fλ = y 3 − x11 + λx8y .

The general polar of fλ, for λ 6= 0, has two smooth branchesintersecting with multiplicity 7.

Consider the family

fλ = y 3 − x11 + λx8y .

The general polar of fλ, for λ 6= 0, has two smooth branchesintersecting with multiplicity 7.

So they are not equisingular!

There are two results that describe partially the topology of thepolar for curves in a given equisingularity class of plane branches.

The first one is the following theorem due to M. Merle (1977):

TheoremLet (f ) be a germ of irreducible curve with semigroup of values〈v0, v1, · · · , vg 〉. Put ni = gcd(v0, . . . , vi−1)/gcd(v0, . . . , vi ), fori > 0, and n0 = 1. Then the generic polar P(f ) of f decomposesinto g packages h1, h2, . . . , hg , such that for each q ∈ {1, . . . , g}

I the multiplicity of hq is n1n2 · · · nq−1(nq − 1);

I each branch hq,j of hq satisfiesI (hq,j ,f )m(hq,j )

n1···nq−1.

One cannot expect to get more informations on the polars of allthe members of an equisingulary class than this theorem says,because what happens inside each package depends on the curve(f ) and not only on its equisingularity class.

n1···nq−1.

One cannot expect to get more informations on the polars of allthe members of an equisingulary class than this theorem says,because what happens inside each package depends on the curve(f ) and not only on its equisingularity class.

The second conclusion in the theorem and a classical contactformula say that the branches of hq have genus at least q − 1.

The second is a deep result, obtained by E. Casas-Alvero in the90’s:

TheoremThe topology of the polar of a general member of anequisingularity class is constant and is determined by a certainEnriques diagram constructed from data of the equisingularityclass. In the case of genus one equisingularity classes, the topologyof the polar may be described in terms of the representation of thequotient of the generators of the semigroup in continued fractions.

It is a good exercise to describe explicitly Merle’s packages for thegeneral branch in an equisingularity class by means of theseEnriques diagrams of the polar.

By doing this one can deduce some interesting things.

For instance,

1) All branches of the polar of a general member in anyequisingularity class of genus g have genus at most g .

For instance,

Recall that from Merle’s result the polar of any curve of genus ghas at least one branch of genus greater or equal than g − 1.

For instance,

2) There are very few equisingularity classes of branches of genus gsuch that the polars of their general members have no branches ofgenus g . These equisisingularity classes may be described explicitly.

For instance,

For example, when g = 2, there are two kinds of suchequisingularity classes, given by:

For instance,

• Semigroups of the form 〈2p, 2q, 2pq + d〉, with gcd(p, q) = 1and d an odd positive integer.

For instance,

• Semigroups of the form 〈2p, 2q, 2pq + d〉, with gcd(p, q) = 1and d an odd positive integer.

• Semigroups of the form 〈ep, eq, epq + 1〉, with gcd(p, q) = 1.

Equisingularity classes of the first kind are very interesting andhave been studied by Luengo and Pfister (Comp. Math. 1990) inan attempt to construct their moduli spaces with respect toanalytic equivalence.

The topology of the polar of a general member of theseequisingularity classes are completely described by the following

Equisingularity classes of the first kind are very interesting andhave been studied by Luengo and Pfister (Comp. Math. 1990) inan attempt to construct their moduli spaces with respect toanalytic equivalence.

The topology of the polar of a general member of theseequisingularity classes are completely described by the following

TheoremThe general polar P(f ) of a general curve of these equisingularityclasses has one branch with semigroup 〈p, q〉 and the otherbranches are the branches of the general polar of the generalmember of the equisingularity class determined by the semigroup〈p, q〉. The branch of this last polar may be described in terms ofthe continued fractions representation of p/q.

A further step may be given by observing that if f ∼K g , then thegeneral member of L(f ) is equisingular to a general member ofL(g). So, the topology of P(f ) is a discrete analytic invariant ofthe curve (f ), hence it makes sense to talk about theequisingularity classes of the general polars of the analytic classesof germs of plane curves.

A natural thing to do in this context is to study the variation ofthis analytic invariant within a given equisingularity class of planebranches, using the analytic classification of plane branches andtheir normal forms that we developed before.

To establish a program in this direction, let us recall some facts.

Any equisingularity class of plane branches determined by asemigroup Γ = 〈v0, v1, . . . , vg 〉 may be parametrized by aconstructible set E in Cc−v1−1, whose points are the coefficients ofthe Newton-Puiseux parametrization

x(t) = tv0 , y(t) = tv1 +c−1∑

i=v1+1

ci ti ,

vhere c is the conductor of Γ, such that any element in theequisingularity class is analytically equivalent to one with aNewton-Puiseux parametrization as above.

To establish a program in this direction, let us recall some facts.

Any equisingularity class of plane branches determined by asemigroup Γ = 〈v0, v1, . . . , vg 〉 may be parametrized by aconstructible set E in Cc−v1−1, whose points are the coefficients ofthe Newton-Puiseux parametrization

x(t) = tv0 , y(t) = tv1 +c−1∑

i=v1+1

ci ti ,

vhere c is the conductor of Γ, such that any element in theequisingularity class is analytically equivalent to one with aNewton-Puiseux parametrization as above.

It was proved in [—–,Hernandes, BLMS 2011] that the parameterspace E may be decomposed into a finite union of disjointconstructible sets E = E1 ∪ · · · ∪ Er , where on each E`, the set Λ`of values of Kahler differentials on the corresponding curve, whichis an analytic invariant of the curve, is fixed. We consider E1 thegeneral stratum, that is the only one which is dense in E .

In each stratum E` there is a normal form for its element which wedescribe below.

In each stratum E` there is a normal form for its element which wedescribe below.If the set Λ` \ Γ is not empty, then the number

λ = min (Λ` \ Γ)− v0

is an analytic invariant known as the Zariski invariant of the curve.

λ = min (Λ` \ Γ)− v0

is an analytic invariant known as the Zariski invariant of the curve.We now recall the

λ = min (Λ` \ Γ)− v0

is an analytic invariant known as the Zariski invariant of the curve.We now recall the

Normal Forms Theorem If C is a curve corresponding to a pointin E`, then either C is analytically equivalent to a curve withparametrization (tv0 , tv1), when Λ` \ Γ = ∅, or to a curve with aparametrization of the form

x = tv0 , y = tv1 + tλ +∑i

ci ti ,

where the summation is over all indices i greater than λ and donot belong to the set Λ` − v0. Moreover, two curves C, with aparametrization as above, and C′ with a similar parametrizationbut with coefficients (c ′i ) instead of (ci ), are analytically equivalentif and only if there exists a complex number ζ such that ζλ−v1 = 1and for all i , one has ci = ζ i−v1c ′i .

Question: Is the topology of the general polar constant on eachstratum E`?

This would be a dream, but unfortunately, as we will see later, theanswer is NO!

But, by using the Normal Forms Theorem one can show that ineach irreducible component of E` there is an open dense Zariski setwhose elements have equisingular polars.

Part of Casas-Alvero’s theorem is the E1 part of our result.

Bad news:

Bad news: It seems to be out of reach, in general, to give anexplicit description of the equisingularity classes of the polars ofthe elements in each stratum E` for ` > 1, even for the generalmembers.

Little comfort:

Little comfort: If you give me an equisingularity class then maybe,after some work, I’ll give you back the topology of the generalpolar of all of its members. This worked in all attempts we made,giving us a stock of examples.

For example, suppose you gave me the equisisingularity classdetermined by the semigroup 〈5, 12〉. I will tell you how are thegeneral polars of all of its members.

The Normal Forms Theorem, together with the algorithmsassociated to it, give us the following classification:

Normal Form Λ` \ Γ

1. (t5, t12) ∅2. (t5, t12 + t38) {43}3. (t5, t12 + t33) {38, 43}4. (t5, t12 + t28) {33, 38, 43}5. (t5, t12 + t26 + ct28), c 6= 0 {31, 38, 43}6. (t5, t12 + t26 + ct33) {31, 43}7. ((t5, t12 + t23 + ct26) {28, 33, 38, 43}8. (t5, t12 + t21 + ct23 + dt28) {26, 31, 38, 43}9. (t5, t12 + t18 + ct21 + dt26) {23, 28, 33, 38, 43}10. (t5, t12 + t16 + ct18 + dt23) {21, 26, 31, 33, 38, 43}

11. (t5, t12 + t14 + ct16 + dt18 + et23), c 6= 1312

, d 6= 4c2−13

{19, 26, 31, 33, 38, 43}

12. (t5, t12 + t14 + ct16 + ( 4c2−13

)t18 + dt23 + et28), c 6= 1312

{19, 26, 31, 38, 43}13. (t5, t12 + t14 + 13

12t16 + ct18 + dt21), c 6= 133

108{19, 28, 31, 33, 38, 43}

14. (t5, t12 + t14 + 1312

t16 + 133108

t18 + ct21 + dt23), d 6= 34c11

{19, 31, 33, 38, 43}15. (t5, t12 + t14 + 13

12t16 + 133

108t18 + ct21 + 34

11ct23 + dt28), {19, 31, 38, 43}

d 6= 81c2

32+ 5225

55987216. (t5, t12 + t14 + 13

12t16 + 133

108t18 + ct21 + 34

11ct23+ {19, 31, 43}

( 81c2

32+ 5225

559872)t28 + dt33)

17. (t5, t12 + t13 + ct14 + dt16 + et21), c 6= − 12

{18, 23, 28, , 31, 33, 38, 43}18. (t5, t12 + t13 − 1

2t14 + ct16 + dt21 + et26) {18, 23, 28, 33, 38, 43}

Any one of the curves in the families (1)− (9) has an irreduciblegeneral polar with semigroup 〈4, 11〉.

The polars of the curves in family (10) have two branches, onesmooth and the other with semigroup 〈3, 8〉 with intersectionmultiplicity 8.

Any one of the curves in the families (11)− (17) has a polar withtwo branches with semigroup 〈2, 5〉 and with intersectionmultiplicity 10.

The family (18) is the only one in which the equisingularity class ofthe polars will depend upon the parameters in E` to which itbelongs.

if c = 1, then the polar has a branch p with semigroup 〈2, 5〉 andtwo non-singular branches g1 and g2 such that I(p, gi ) = 5 andI(g1, g2) = 3.

If c 6= 1 and c 6= −54 , then the polar has two branches with

semigroup 〈2, 5〉 and with intersection multiplicity 10.

If c = −54 and d 6= − 5

16 , then the polar is irreducible of genus 2with semigroup 〈4, 10, 21〉.

If c = −54 and d 6= − 5

If c = −54 and d = − 5

16 , then the polar has two branches withsemigroup 〈2, 5〉 and with intersection multiplicity 11.

If c = −54 and d 6= − 5

If c = −54 and d = − 5

Remark The stratum E18 gives us an example in which theequisingularity class of the general polar of its members is notconstant. It also gives us a somewhat unexpected example of afamily of curves of genus 1 such that its general member has ageneral polar of genus 2.

If c = −54 and d 6= − 5

If c = −54 and d = − 5

Remark The stratum E18 gives us an example in which theequisingularity class of the general polar of its members is notconstant. It also gives us a somewhat unexpected example of afamily of curves of genus 1 such that its general member has ageneral polar of genus 2.

After we found this example, we could produce examples ofbranches of genus one whose polars have branches with arbitrarilyhigh genera.

Branches of multiplicity four

Based on a previous analytic classification of such branches done in[—–,Hernandes,JSC 2009] we have the following

Branches of multiplicity four

Based on a previous analytic classification of such branches done in[—–,Hernandes,JSC 2009] we have the following

Normal Forms for the genus one case:

Normal form Λ\ < 4,m >1. y(t) = tm ∅2. y(t) = tm + t3m−4j + a1t

2m−4(j−[ m4

]−1)+ · · · {3m − 4s; 1 ≤ s ≤ j − 1}

+aj−[ m4

]−2t2m−8 2 ≤ j ≤ [ m

3. y(t) = tm + t2m−4j + ak t3m−(4[ m

4]+j+1−k)

+ · · · {2m − 4s; 1 ≤ s ≤ j − 1} ∪+aj−[ m

4]−2t

3m−4([ m4

]+3−k) {3m − 4s; 1 ≤ s ≤ [ m4

] + 1− k}ak 6= 0 , 2 ≤ j ≤ [ m

4] , 1 ≤ k ≤ [ m

4]− j

4. y(t) = tm + t2m−4j + a[ m4

]−j+1t3m−8j {2m − 4s; 1 ≤ s ≤ j − 1} ∪

+a[ m4

]−j+2t3m−4(2j−1) + · · · + a[ m

4]−1t

3m−4(j+2) {3m − 4s; 1 ≤ s ≤ j}

]−j+1 6=3m−4j

2m, 2 ≤ j ≤ [ m

5. y(t) = tm + t2m−4j + 3m−4j2m

t3m−8j+ {2m − 4s; 1 ≤ s ≤ j − 1} ∪∑[ m4

i=[ m4

]−j+2ai t

3m−4([ m4

]+j+1−i) {3m − 4s; 1 ≤ s ≤ j − 1}

2 ≤ j ≤ [ m4

Normal Forms for the genus two case:

Normal form Λ\ < 4, v1, v2 >

y(t) = tv1 + tv2−v1 + a1tv2−4[

]v2 + v1 − 4s;

+a2tv2−4([

]−1)+ · · · + a

]−1tv2−8, 1 ≤ s ≤ [

v2 > 2v1

Multiplicity four, genus one

First Normal Form (monomial curves)

In this case, the equation of the curve is y 4 − xm = 0, so its polaris 4by 3 − amxm−1, that has d = gcd(3,m − 1) smooth branches

with mutual intersection multiplicity equal to 3(m−1)d2 .

Second Normal Form

y = tm + t3m−4j + a1t2m−4(j−[ m

4]−1)

+ · · · + aj−[ m4

]−2t2m−8; 2 ≤ j ≤ [ m

∃ j ; aj 6= 0, k = min{j ; aj 6= 0}For ak 6=

±4√

3the polar has three smooth branches p1, p2, p3, and

2m−j

= 1[ m

I(pi , pr ) = m−j2

The polar has one branch, with semigroup 〈3,m − j + [ m4

] + k〉,2

m−j< 1

]+kif gcd(3,m − j + [ m

4] + k) = 1; otherwise it has three smooth branches

p1, p2, p3 with I(pi , pr ) =m−j+[ m

The polar has one branch p1, with semigroup 〈2,m − j〉 and2

m−j> 1

]+kone smooth branch p2, with I(p1, p2) = m − j , if gcd(2,m − j) = 1;

otherwise it has three smooth branches p1, p2, p3, with I(pi , pr ) = m−j2

a1 = a2 = · · · = aj−[ m4

]−2 = 0

2m−j

= 1j−1

The polar has three smooth branches p1, p2, p3, with I(pi , pr ) = j − 1.

The polar has one branch with semigroup 〈3,m − 1〉,2

m−j< 1

j−1if gcd(3,m − 1) = 1; otherwise it has three smooth branches

p1, p2, p3, with I(pi , pr ) = m−13

The polar has one branch p1 with semigroup 〈2,m − jrangle and2

m−j> 1

j−1one smooth branch p2, with I(p1, p2) = m − j , if gcd(2,m − j) = 1;

otherwise it has three smooth branches p1, p2, p3, with I(pi , pr ) = m−j2

Third to Fifth Normal forms

gcd(3,m − j) = 1 One branch with semigroup 〈3,m − j〉.gcd(3,m − j) = 3 Three smooth branches with mutual intersection numbers m−j

Third to Fifth Normal forms

gcd(3,m − j) = 1 One branch with semigroup 〈3,m − j〉.gcd(3,m − j) = 3 Three smooth branches with mutual intersection numbers m−j

Multiplicity four and genus twoThe polar of any branch of genus 2 and multiplicity 4 has onebranch with semigroup 〈2, v1

2 〉 and one smooth branch, withintersection multiplicity v1.

FELIZ CUMPLEANOS PEPE!!!

On the topology of polars of analytic plane branches · On the topology of polars of analytic plane...

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