MACIEJ BORODZIK
Abstract. In this paper we present some new results in the deformation theory of plane curve singularities. The methods rely on the study of analytic properties of linear non homogeneous ODE’s.
1. Introduction
In this paper we deal with singularities of plane curves. A plane curve singularity is meant to be a zero set of a polynomial
C = {f = 0},
such that its gradient ∇f vanishes at some point (mostly we will assume this is (0, 0) ∈ C2) and ∇f 6= (0, 0) in some punctured neighbourhood of (0, 0). This property ensures that f is a reduced polynomial and (0, 0) is an isolated singularity.
By the classical Puiseux theorem (see [Zol]) the set C intersected with a ball centered at (0, 0) is an union of sets
C1∪ C2∪ · · · ∪ Cr,
such that each Ci is homeomorphic to a disk, Ci∩ Cj = {(0, 0)} and each Ci can be given a local parametrisation
y = ci0xqi/pi+ ci1x(qi+1)/pi+ . . . , called a Puiseux expansion.
In this paper we will mostly deal with singularities with one branch, i.e. when r = 1.
In one branch case, the Puiseux expansion determines uniquely the topological type of singularity Ci. This type can be viewed as the homotopy type of the pair (Sε3, Sε3∩ Ci), where Sε3 is a sufficiently small sphere around zero in C2. Since Puiseux expansions are of vital interest to us in this paper, we recall a following definition.
Definition 1.1. Let us consider a Puiseux expansion of the singularity such that p is the multiplicity.
(1.1) y = cqxq/p+ cq+1x(q+1)/p+ cq+2x(q+2)/p+ . . . .
The sequence (p; q1, . . . , qn) is called a characteristic sequence of the singularity if the following conditions are satisfied
(1) qi is an increasing sequence. q1= q if and only if q/p is not integer.
(2) cqi 6= 0 for i = 1, . . . , n;
Date: July 22, 2007.
1991 Mathematics Subject Classification. Primary . Supported by Polish MNiSz Grant 1 P03A 015 29.
1
(3) If we define p0 = p, q0 = q and pi+1 = gcd(pi, qi) then for i ≥ 1, pi is strictly decreasing with pn+1= 1.
(4) If j is such that qi< j < qi+1 with i = 0, . . . , n then either pi+1|j or cj = 0.
It is quite clear that the characteristic sequence is uniquely defined by (1.1).
This sequence determines the topological type of the singularity (see e.g. [EiNe]).
Now we are ready to state the main definition of the article
Definition 1.2. A deformation or a specialisation of a plane curve singularity with one branch and characteristic sequence (p; q1, . . . , qn) is a polynomial map
F : D2× B → C,
where D2 is a ball in C2around zero and B = {ε ∈ C, |ε| < 1} such that
1.: For any ε the map fε: D2→ C, fε(x, y) = F (x, y, ε) defines a singularity Cεat (0, 0), i.e. Cε= fε−1(0) has singularity at (0, 0).
2.: For ε 6= 0 the singularity of Cεhas characteristic sequence of (p; q1, . . . , qn).
3.: The gradient of F does not vanish away from (0, 0) × B.
Given a deformation of a plane curve singularity we define its degeneration as the singularity of a fiber C0. The singuarity at ε 6= 0 will be called generic
Remark 1.3. In algebraic geometry the notion of degeneration is called rather specialisation. We prefer the first term since our methods are mostly analytic.
The main question we put is
Problem 1.4. What characteristic sequences (r; s1, . . . , sm) may arise as degener- ations of a singularity with a characteristic sequence (p; q1, . . . , qn)?
If the deformation in question is a deformation of cuspidal singularities and all fibers Ct, t ∈ D2, are rational then we can regard a deformation in a parametric form
xε(t) = a0(ε)tp+ a1(ε)tp+1+ . . . yε(t) = b0(ε)tq+ b1(ε)tq+1+ . . . , (1.2)
where x and y are polynomials of sufficiently high degree (or power series), the coefficients ai and bj vary analytically with ε. Sometimes, if it does not lead to confusion, we will not write explicitly the dependence of variables on ε.
We shall alway assume that
(1.3) a0b06= 0, for ε 6= 0
and that the order of x at 0 for ε = 0 is equal to r.
For ε 6= 0 we can write the Puiseux expansion of yεin powers of xε in the form (1.4) y = c1(ε)xq1/p+ c2(ε)xq2/p+ c3(ε)xq3/p+ . . . .
Here the coefficients ci(ε) are locally analytic functions of ε (globally they can be multivalued).
Remark 1.5. Without additional assumptions Ctare, in general, not necessarily rational. For example, if F (x, y, ε) = x5− y7+ ε(x2− y3) then for small ε 6= 0, the curve Cεhas 22 cycles (vanishing cycles of the singularity x5− y7and therefore cannot be rational). We owe this example to H. ˙Zo l¸adek .
From now on we shall always assume that the deformation can be given in a parametric form (1.2)
Remark 1.6. In the paper we will sometimes abuse the language to make state- ments shorter and more concise, namely
(i) a singularity with a characteristic sequence (p; q1, . . . , qn) will sometimes be called simply a (p; q1, . . . , qn) singularity;
(ii) a singularity with multiplicity p will sometimes be called a p–singularity;
(iii) we will say that the limit singularity is at least (r; s1) if it is a (r; s′1, . . . , s′m) singularity with s′1≥ s1;
(iv) all limits in the article are taken with ε → 0;
(v) unless specified otherwise, bounded will mean bounded from above when ε → 0, unbounded — going to infinity for some subsequence εn → 0.
2. Classical facts
The Milnor number of a (p; q1, . . . , qn) singularity can be computed by the Milnor formula
(2.1) µ = (p − 1)(q1− 1) +
n
X
i=2
(pi− 1)(qi− qi−1), where pi= gcd(pi−1, qi−1) and p1= p (see Definition 1.1).
We have the following
Proposition 2.1. The Milnor number is upper semicontinuous with respect to deformations, namely if µ is the Milnor number in a non–degenerate case and µ0
is the Milnor number of the special fiber then µ0≥ µ
with an equality if and only if the family over the whole disk is topologically equi- singular.
The first part is classical (see [AVG] or [Zol]), while the second part is due to Teissier [Tes], [LR].
Apart of the semicontinuity of Milnor number we have also the semicontinuity of multiplicity, which can be rephrased as
Lemma 2.2. If a p−singularity specialises to r−singularity then r ≥ p.
The lemma follows from the definition of the multiplicity as the intersection index of a germ of a singular curve with a generic line passing through the singular point.
When studying the case p = r, so if the multiplicity is preserved, we can always assume that xε(t) = tp, which solves completely the problem of degenerations in this case. This statement follows from the following
Lemma 2.3. If p = r then xε(t)1/p is analytic with respect to ε around ε = 0.
Proof. As a0(0) 6= 0, it is bounded away from zero for small ε. We can then write xε(t)1/p= a0(ε)1/pt
1 +a1(ε)
a0(ε)t + · · · +aN(ε) a0(ε)tN
1/p
.
and if we apply the expansion (1 + z)1/p= 1 +1pz + . . . , the resulting power series in ε and t will be convergent in a small polydisk around |t| < δ, |ε| < δ.
3. Spectral numbers
One of the tools in studying the deformation of singular points comes from theory mixed Hodge structures. The result of Varchenko [Var] shows that the spectrum of planar singular points is semicontinuous under deformations. This can give some partial answer to the Problem 1.4.
We shall not give the precise and rather complicated definition of spectral num- bers of a singular point referring the curious reader either to [Sai] or to a wonderful book by ˙Zo l¸adek [Zol]. Two things we need are first, how to compute spectral numbers of a plane curve singularity, second: how these numbers behave under specialisation.
Let (C, 0) ⊂ (C2, 0) be a germ of a plane curve singularity with one branch and with a characteristic sequence (p, q1, . . . , qn) as above. Let us define the Eisenbud–
Neumann diagramm of the singularity
• ⊕ 1
• w1
k1
w2⊕ k2
•
1 . . . wr⊕
kr
1 -
•
Here wi’s and ki’s are related to (p, q1, . . . , qn) as follows: we write the Puiseux expansion (1.1) in the topologically arranged form
y = xm1/k1(d1+· · ·+xm2/k1k2(d2+· · ·+xm3/k1k2k3(d3+dots+xmn/k1···kndn+. . . ) . . . )), where p = k1. . . kn and the fact that the expansion is topologically aranged means di 6= 0 and between each di and xmi+1/k1...ki+1 there are only terms with x in the fractional power xj/k1...ki. In other words k1= p/gcd(p, q), m1 = q/gcd(p, q) and, more generally ki= pi−1/gcd(pi−1, qi−1) and mi= (qi− qi−1)/gcd(pi−1, qi−1).
wi’s, on the other hand, are defined inductively by w1= m1, wi= wi−1ki−1ki+ mi.
Proposition 3.1. [Sai] The spectral numbers less then 1 of the singularity are given by the set
1
kν+1. . . kn
i kν + j
wν
+ r
kν+2. . . kn
,
where ν goes through all numbers 1 to n and 1 < i < kν, 1 < j < wν, 0 ≤ r <
kν+1. . . kn with the additional condition that kiν +wjν < 1.
The remaining half of spectral numbers (those in the interval (1, 2)) arise as the symmetrical reflection of the above set with respect to point 1. The total number of spectral numbers of a singular point is equal to its Milnor number.
Example 3.2. For the singularity (4; 6, 9) we have k1 = k2 = 2, w1 = 3 and w2= 15. We have 18 spectral numbers:
(3.1) 5
12,17 30, 7
12,19 30, 7
10,23 30,5
6, 9 10,29
30,31 30,11
10,7 6,37
30,13 10,41
30,17 12,43
30,19 12.
Example 3.3. For the singularity (5, 6) the spectral numbers are 11
30, 8 15,17
30, 7 10,11
15,23 30,13
15, 9 10,14
15,29 30, 31
30,16 15,11
10,17 15,37
30,19 15,13
10,43 30,22
15,49 30 (3.2)
Now we cite the main result concerning the semicontiuity of spectral numbers Proposition 3.4. [Var] Let us consider a deformation of a plane curve singularity as in Definition 1.2. Assume that the spectral numbers for ε 6= 0 are
{a1, a2, . . . , aµ′}, whereas for the degenerate fiber ε = 0 we have
{b1, . . . , bµ}.
Then for any α ∈ R we have
(3.3) # {i : ai< α} ≤ # {i : bi< α}
Example 3.5. Consider a singularity (32; 48, 56, 60, 62, 63) with a following Puiseux diagramm
• ⊕1
•
3
2 13⊕
2
•
1 53⊕
2
•
1 213⊕
2
•
1 853⊕
2 1 -
• Its Milnor number is 1612. The smallest spectral number is
1 3+1
2
· 1 16 = 5
96. On the other hand consider a singularity with a diagram
• β1⊕
11
•
1 β2⊕
3 1 -
•
where β1and β2 are not fixed. This is a (33; 3 · β1, β2− 30β1) singularity.
Its smallest spectral number is
1 β1
+ 1 11
·1 3, it does not depend on β2 and for
β1< 352 23 = 158
23
it is strictly larger than 965. It follows, that the singularity (32; 48, 56, 60, 62, 63) can never specialise to a singularity (33; b, . . . ) with b = 36, 39, 42, 45, independently on the Milnor number and even the codimension (see Conjecture 6.2 and above) of a degenerate singular point: we could put β2= 10000 and this will not help.
4. Case p = 2 and r = 3
This case is one of the most elaborated and is very close to be completed.
The motivation for this case is the following result proved originally by Petrov [Pet] in a different context.
Proposition 4.1. Let
x(t) = a2t2+ t3
y(t) = b2t2+ b3t3+ · · · + bw−1tw−1+ tw (4.1)
be a polynomial curve in C2with 3 6 |w. Assume that the coefficients a2, b2, . . . , bw−1 are real and the curve (4.1) has an A2k singularity at t = 0.
Then k ≤ w −w
3 − 1.
The proof, based on argument principle, can be found also in [BZ3].
Now assume that there exists a curve of the form (4.1) having the A2ksingularity.
Then we apply a change t → λt, x → λ−3x and y → λ−wy. We end up with a family
xλ(t) = a2λt2+ t3
yλ(t) = b2λw−2t2+ b3λw−3t3+ · · · + bw−1λtw−1+ tw
specialising to a curve (t3, tw). This, viewed locally in a neighbourhood of 0 ∈ C2, provides a family of A2ksingularities specialising to a singularity (3; w). If we could prove that any specialisation of an A2k singularity can be realised as a specialisation of the above form, we would have found a complete criterion for the specialisation of singularities of multiplicity 2 to multiplicity 3.
Conjecture 4.2. If a family of A2k singularities specialises to a (3; b) singularity then k ≤ b −b
3 − 1.
There is a very elementary approach to this conjecture, which could also lead to a complete answer in the case when a singularity with multiplicity p specialises to a singularity of multiplicity p + 1. Let us start with a following
Example 4.3. Let us consider a deformation x(t) = εt2+ t3
y(t) = b2(ε)t2+ b3(ε)t3+ . . . , (4.2)
Here x and y depend implicitly on ε. Aassume that bi’s are chosen in such a way that for ε 6= 0 the resulting singularity is an A10 singularity. This amounts to the fact that we have
(4.3) y = c2(ε)x + c4(ε)x2+ c6(ε)x3+ c8(ε)x4+ c10(ε)x5+ c11(ε)x11/2+ . . . . Substituting x from (4.2) into (4.3) yields
y =c2(εt2c2t3)+
+c4(ε2t4+ 2εt5+ t6)+
+c6(ε3t6+ 3ε2t7+ 3εt8+ t9)+
+c8(ε4t8+ 4ε3t9+ 6εt10+ . . . )+
+c10ε5t10+ . . . (4.4)
Now if all c2, c4, c6, c8 and c10 stay bounded (from above) while ε → 0 then, on passing to the limit ε = 0, all terms with ε in a positive power will vanish. Then, the resulting singularity would be at least (3; 11).
Some ci’s can diverge infinity. Yet, for example, if c4(ε) is unbounded, then, since b6(ε) is bounded (otherwise y would diverge), ε2c6 must be unbounded so as to cancel the term c4(ε) at t6. But then εc6 is unbounded, too. This contradicts the boundedness of b7. So c4 must be bounded. Using similar argument we can prove the following
Lemma 4.4. εc6(ε) is bounded as ε → 0.
Proof. Assume contrary. Let us take a subsequence εn→ 0 such that εc6(εn) → ∞.
To shorten the notation, we will call ck(εn) as cnk.
Let us pick n sufficiently large and consider the terms b8(ε), b9(ε) and b10(ε) written in the following way
3εn 1 0
ε4n 4ε3n 6ε2n 0 0 ε5n
cn6 cn8 cn10
=
bn8 bn9 bn10
We claim that the above 3 × 3 matrix must have non–trivial kernel. In fact, the leading terms in εcn6, ε4cn8 and ε7c10 are unbounded and must mutually cancel so that b8, b9 and b10 stay bounded as ε → 0.
The desired contradiction comes from the fact that D10:=
3 1 0 1 4 6 0 0 1
= 11 6= 0.
From this it follows that in the limit expansion b7(ε) → 0.
Corollary 4.5. If a singularity A10 specialises to a singularity (3; b) then b ≥ 8.
Remark 4.6. For n = 2k let l =n+3
4 . Consider the determinant
Dn:=
k−(l−1) l−1
k−(l−1)
l
. . . k−(l−1)2(l−1) . . . .
k−2 5−l
k−2
6−l
. . . k−24
k−1 3−l
k−1
4−l
. . . k−12
k 1−l
k
2−l
. . . k0
Conjecture 4.2 will follow once we have proved that for all even positive n we have Dn6= 0. Using computer we were able to check this up to n = 400, but no general formula for Dn has been found.
Example 4.7. The curve
x = t2+ t3 y = 3
4t6+9 4t7+ t8
Has an A10 singularity at t = 0, so by rescaling we obtain a deformation (2; 11) → (3; 8).
5. Analytic theory
In this section we will assume that for ε 6= 0 the Puiseux expansion of yε in powers of xε(see (1.2)) has the form
(5.1) y = c1(ε)xq1/p+ c2(ε)xq2/p+ c3(ε)xq3/p+ . . . .
Here q = q1< q2< q3< . . . and the Puiseux coefficients between xqi/p and xqi+1/p are supposed to vanish. Note that Puiseux terms xqi/pare not necessarily essential, which is contrary to the convention we have been accepting in previous sections.
The reason for this will clarify in the following.
Let us divide (5.1) by xq1/p, differentiate both parts with respect to t and mul- tiply it again by x1+q1/p (cf. [BZ2], proof of Lemma 3.2). We obtain
(5.2) ˙yx −q1
py ˙x = q2− q1
p c2(ε) ˙xxq2/p+q3− q1
p c3(ε) ˙xxq3/p+ . . . . Let us denote
(5.3) P2(ε, t) = ˙yεxε−q1
pyε˙xε. As xε∼ tp and so xqε2/p∼ tq2 we get that
ordt=0P2(ε, t) ≥ q2+ p − 1,
for ε 6= 0. The equality here holds for those ε 6= 0 such that c2(ε) = 0
But P2(ε, t) → P2(0, t) for ε → 0 uniformly in t (in some neighbourhood of 0).
Therefore
(5.4) ordt=0P2(0, t) ≥ q2+ p − 1.
Now let us put ε = 0 and regard the equation (5.3) as an ordinary differential equation for y0. Solving it we get
(5.5) y0= xq01/p
Z t 0
P2(0, s)x−q1/p−1ds + D
, with D an integration constant.
Lemma 5.1. If q1r/p is not an integer then D = 0.
Proof. In this case the r.h.s. of (5.5) is analytic in t = 0 iff D = 0. Lemma 5.2. If q1/p is integer, we can assume that D = 0.
Proof. If q1/p = n we can apply the global change of coordinates y → y − Dxn. Corollary 5.3. If either q1r/p 6∈ Z or q1/p ∈ Z then
ordt=0y0≥ q2− (r − p).
This gives some restrictions for possible Puiseux terms in the limit. We illustrate them in the following
Example 5.4. Assume that the multiplicity sequence of the singularity at ε 6= 0 is (9; 17), so that the Puiseux expansion is
y = c1x9/9+ c2x17/9+ . . . .
Assume that at ε = 0 the order of x is 10. Then, by Corollary 5.3, the order of y0
at t = 0 is at least 16. It follows that the characteristic sequence of the specialised singularity is at least (10; 16).
The Milnor number of singularity (9; 17) is 128. The Milnor number of singular- ity (10; 16, 17) is 136. But the Milnor number of (10; 15, 16) is 130 > 128. Therefore the semicontinuity of the Milnor numbers does not exclude the possibility of de- generation of (9; 17) to (10; 15, 16).
Remark 5.5. One could be tempted to do the following trick in this case. If yε = c1(ε)x9/9ε + c2(ε)x17/9ε + . . . , we apply changes y → y − c1(ε)x so that the resulting new yε has the order 17 as t = 0. What is wrong is that nothing can prevent c1 from escaping to infinity as ε → 0 (see Example 4.3). Then the new yε
cease to converge uniformly to y0. There are examples (see Proposition 5.12) that the order of y0 is precisely 16.
The above method admits further improvements. Let us take the equation (5.2).
Let us divide both sides by xq2/p˙x, differentiate them with respect to t and multiply again by xq2/p+1˙x2. We obtain
(5.6) P3(ε, t) = q3− q1
p
q3− q2
p ˙x3xq3/p+ . . . , where
(5.7) P3(ε, t) = x ˙xP2′− (q2
p ˙x2+ ¨xx)P2.
Here and in the following P2′ means ∂t∂P2(ε, t): we never differentiate w.r.t. ε.
We can repeat this procedure of dividing, differentiating and multiplying several times. The reader may easily verify the following formula valid for n > 2
(5.8) Pn(ε, t) =
n−1Y
k=1
qn− qk
p · cn(ε) ˙x2n−3xqn/p+ . . . , where Pn are defined inductively by the formula
(5.9) Pn+1(ε, t) = x ˙xPn′ − qn
p ˙x2+ (2n − 3)¨xx
Pn. An analogue of equation (5.4) is
(5.10) ordt=0Pn(ε, t) ≥ qn+ (2n − 3)(p − 1).
The inequality for orders is valid for ε 6= 0 by virtue of (5.8) (since ordt=0 ˙x2n−3= (2n − 3)(p − 1) and ordt=0 ˙xqn/p= qn). It holds also for ε = 0 because Pn(ε, t) → Pn(0, t) uniformly in t (in some neighbourhood of t = 0) if ε → 0.
As before we can treat the equation (5.9) as the ordinary differential equation with the known function Pn+1(0, t) and unknown Pn(0, t). We get the following solution
(5.11) Pn(0, t) = xq0n/p˙x2n−30
Z s 0
Pn+1(0, s) ˙x(s)−2n+2x(s)−qn/p−1ds + D
. The condition that Pn(0, t) is analytic near t = 0 implies the following
Lemma 5.6. If qnr/p is not an integer then D = 0. Moreover, if qn/p is integer, we can still perform a change of coordinates so that D = 0.
Proof. Only the second part of the proof requires some comments. If qn/p = k ∈ Z, we apply the change y → y − ˜Dxk. Such a change induces, by virtue of formulae (5.3) and (5.9) the change Pl→ Pl− Dlxk˙x2l−3, where Dl depends linearly on ˜D.
It is now clear that picking a suitable ˜D we can ensure that D = 0.
Proposition 5.7. Assume that for all i = 1, . . . , n either qi/p ∈ Z or qir/p 6∈ Z.
Then we have
ordt=0y0≥ max
i=2,...,nqi− (2i − 3)(r − p).
Proof. If D = 0 in (5.11) then
ordt=0Pn(t, 0) ≥ ordt=0Pn+1(t, 0) − 2(r − p).
The statement follows now from an easy induction on n. Remark 5.8. These (non)integrality assumptions are automatically satisfied if r and p are coprime.
In some cases Proposition 5.7 gives better estimates than the semicontinuity of Milnor number.
Example 5.9. In this example the main issue is that r − p > 1. The numbers occurring here are quite large, but the bound is about 20% better than the bound that could be obtained from semicontinuity of Milnor numbers.
Let the singularity at ε 6= 0 be (32; 112, 152, 5196, 5201). Its Milnor number is 39364. Moreover in the expansion (5.1) we have
q1= 32, q2= 64, . . . , q8= 152, . . . ,
q638= 5192, q639= 5196, q640= 5200, q641= 5201.
Suppose, that the multiplicity of singularity increses to r = 35. Therefore, by Proposition 5.7, the order of y0at t = 0 is greater or equal q638−3·(2·638−3) = 1373.
Note, in passing, that q639− 3 · 1275 < 1373. Hence the characteristic pair of the singularity at ε = 0 is (35; b, . . . ) with b ≥ 1373.
The Milnor number of the singularity (35; 1373) is equal to 46648. Yet even the singularity (35; 1158) has Milnor number greater than 39364. Hence the bound here is stronger than the Milnor bound.
Remark 5.10. The semicontinuity of spectral numbers gives here weaker result, namely b ≥∼ 1177. This result is close to the Milnor number bound.
Remark 5.11. In Example 5.9 we had that q639− 3 · 1275 ≤ q638− 3 · 1273. This means that our method provides the same bound for the order of y0 in the case of singularity (32; 112, 152, 5196, 5201) as in the case of (32; 112, 152, 5196, 5197). In general, the method gives the better results, the larger is the difference qi+1− qi
compared to 2(r − p). In particular in case of a 2−singularity degenerating to any r−singularity with r ≥ 3, this method does not give anything interesting.
We shall prove one more proposition and give some example that will shed light on some phenomena occuring here.
Proposition 5.12. Assume that q2− q1> r − p. Then there exists a deformation as in (1.2) such that the order of y0 is precisely q2− (r − p).
Proof. Let d = q2− q1and q = q1. Consider a vector space V = Vx⊕ Vy of pairs of polynomials
x(t) = a0tp+ a1tp+1+ a2tp+2+ · · · + ad+1tp+d y(t) = b0tq+ b1tq+1+ b2tq+2+ · · · + bd+1tq+d.
To simplify notation let us denote
hm,n= m − nq p.
The conditions that ordt=0P2(t) = q2+ p − 1 translate into a set of equations (5.12) F1(a, b) = · · · = Fd−1(a, b) = 0 6= Fd(a, b),
where
(5.13) Fl(a, b) = X
i+j=l
hijaibj. For k ≤ d define
Σk= {(a, b) ∈ V : F1(a, b) = · · · = Fk(a, b) = 0}.
Lemma 5.13. Both Σk’s are smooth away from the set {a0= 0}.
Proof of Lemma 5.13. The matrix of partial derivatives of function F = (F1, . . . , Fk) with respect to b variables is
h10a1 h01a0 0 . . . . h20a2 h11a1 h02a0 0 . . . . h30a3 h21a2 h12a1 h03a0 0 . . . . . . . .. . ..
hk,0ak hk−1,1ak−1 . . . h0,ka0
This is a k × (k + 1) matrix. If a0 6= 0 it is obivious that its rows are linearly independent. The lemma follows from the implicit function theorem.
The next thing we need is the structure of the set Nd−1= Σd−1∩ {a0= 0}.
Lemma 5.14. The set Nd−1is an union of sets Nd−1,0, . . . , Nd−1,d−1, where Nd−1,k= {a0= a1= · · · = ak = 0 = b0= · · · = bd−2−k}.
In particular set Nd−1 is a codimension one subset in Σd−1. Proof. We shall prove slightly more. Namely let
Nl,k= {a0= · · · = ak= 0 = b0= · · · = bl−1−k}.
Then the set Nl= Σl∩ {a0= 0} is a sum of Nl,k’s for k = 0, . . . , l.
For l = 1 the statement is trivial. Assume it has been proved for l − 1. Consider the equation Fl = 0 on Nl−1,k. From the definition of this space we infer (see (5.13)) that the only monomial in Flthat does not vanish identically is
hk+1,d−k−1ak+1bd−k−1.
In fact in all other aibj’s either i ≤ k or j ≤ l − k − 2. It follows that (5.14) Nl−1,k∩ {Fl= 0} = Nl,k∩ Nl,k+1,
and the induction step is proved.
Corollary 5.15. The set Nd−1= Σd−1∩ {a0= 0} is not contained in Σd. Proof. From (5.14) we infer that Fd does not vanish identically on any subset
Nd−1,k.
The rest of the proof is easy. Consider the set Nd−1,r−p−1in Σd−1. There exists a small topological disk D in Σd−1omitting Σdand intersecting Nd−1,r−p−1in one point disjoint from Σd. Hence ar−p6= 0 at this point (otherwise Fd = 0).
Then this disk represents a specialisation of a singularity y = c1xq1/p+ c2xq2/p+ . . .
to the singularity with order of x equal to r and order of y equal to q2− (r − p) (because if (a, b) ∈ Nd−1,r−p−1 then b0 = b1 = · · · = bq2−q1−(r−p)−1 = 0 so the
order of y jumps to q2− (r − p)).
The above result answers one questions and makes us ask another one: when is the bound in Proposition 5.7 sharp? The example below will hopefully shed some light on the phenomena occuring in the problem.
Let g be some unknown constant (independent of ε). Consider a deformation x = ε2tp+ εgtp+1+ tp+2
y = b0tq+ b1tq+2+ . . . such that
P2(ε, t) = h22tq+p+3.
According to Corollary 5.3 we should have ordt=0y0≥ q + 2.
Let us write down equations for y and see what happens h01ε2b1+ h10εgb0= 0 h02ε2b2+ h11εgb1+ h20b0= 0 h03ε2b3+ h12εgb2+ h21b1= 0 h04ε2b4+ h13εgb3+ h22b2= h22
(5.15)
Since all functions bi(ε) are supposed to be bounded as ε → 0, from the last equation we deduce that b2= 1 + O(ε).
The third equation implies that b1= −gh12
h21
ε + O(ε2).
Next, from the second equation we infer that b0=h11h12g2− h02h21
h20h21 ε2+ O(ε3).
But now, unless
(5.16) h10h11h12g3− (h10h02h21+ h01h20h12)g = O(ε).
the first equation cannot hold.
If it does not, we get a contradiction. The only assumption we have made is that bi(ε) remain bounded. So, in general, this assumption is false. It means that whenever we consider the yεdefined by
yε= xqε1/p Z t
0
P2(ε, s)x−qε 1/p−1ds,
the function xε(t) and P2(ε, t) must be chosen so that yε→ y0 uniformly in some neighbourhood of t = 0. Without additional assumptions this turns out not to be true. These additional assumption could, for example, be that the order of P2(ε, t) increases as ε → 0. This would lead to the increment of the order of y0at t = 0. We
were not able to find in literature satisfactory criteria for the analytic dependence of solutions to (5.2) or (5.9) at ε = 0.
Remark 5.16. There is no contradiction between this example and Proposition 5.7.
In fact, if g does not satisfy (5.16) and for a given xε we want to find yεsuch that (xε, yε) ∈ N3then the analysis of (5.15) implies that yεwill pass to infinity, (xε, yε) will approach a point in N4.
6. Conjectures
In papers [Or] and [BZ3] a new invariant of the singularity has been introduced, namely the codimension of the signular point. Let us recall briefly its definition.
Definition 6.1. Let (C, 0) ⊂ C be a germ of a plane curve singularity. Let π : X → C2be the minimal resolution of the singularity of the curve C. We denote by E the reduced exceptional locus of π; by ˜C the strict transform of C and by K the canonical divisor, i.e. the divisor associated to the holomorphic form π∗dx ∧ dy, where x and y are the coordinates on C2.
The codimension of the singular point is the quantity ν = K · (K + ˜C + E).
It has been shown in [Or] for unibranched singularity and in [BZ3] in general case, that the codimension is the number of conditions for the given singularity in a suitably defined parameter space of curves. Given this interpretation we state the following
Conjecture 6.2. The codimension is semi-continuous with respect to deformation of singular points. More precisely, given a deformation like in Definition 1.2 if νε denotes the codimension of the singularity Cε and ν0 the codimension of the singularity C0 then
νε≤ ν0,
with the equality if and only if the deformation is equisingular on the whole disc.
This conjecture is motivated by the definition of the invariant ν: if we are given parameter space Vd of curves of degree d in CP2, its dimension is equal to d+22 .
For a singularity with codimension ν, with ν small compared to d, the space of degree d curves having among their singularities the given one, has codimension ν in Vd. The ”number of conditions” should increase during the specialisation. But we have by now no idea of a possible proof of this conjecture.
Remark 6.3. A singularity A2k has codimension k. The singularity (t3, tb) with 3 6 |b has codimension b − ⌊b3⌋. The results and conjectures in Section 4 stay in agreement with Conjecture 6.2.
Conjecture 6.2 apart of its own interest could have many applications. First of all, in case of p = 2 it would allow a linear bound for limit cycles bifurcating from origin in case of Li´enard system, the one precisely conjectured by Christopher and Lynch [ChLy].
On the other hand, if we consider a polynomial curve in C2with bidegree (m, n) with (m.n) coprime then such a curve, by a parameter change t → λ, x → λ−mx, y → λ−ny (see discussion after Proposition 4.1), deforms to the curve (tm, tn).
Now the codimension of the (m; n) singularity is equal to m + n −n
m − 3. The semicontinuity of codimensions would lead to a very strong bound for the sum of
codimensions of polynomial curves, similar to the conjectured one in [BZ1]. Similar bound is known to hold (see [BZ3]), but the sum of codimensions is bounded by the sum of degrees plus the number of self–intersections at finite distance, which itself can be proportional to the product mn.
Acknowledgements. The author wishes to thank M. Koras, A. N´emethi, P. Russell, A. Sathaye and H. ˙Zo l¸adek for fruitful discussions on the subject.
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Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
E-mail address: mcboro@mimuw.edu.pl
