Annales de l’institut Fourier (AIF)

Nagel, Alexander

Rosay, Jean-Pierre

Maximum modulus sets and reflection sets

Tougeron, Jean-Claude

Algèbres analytiques topologiquement noéthériennes. Théorie de Khovanskii

Casas-Alvero, Eduardo

ANNALES DE L’INSTITUT FOURIER
EDUARDO CASAS-ALVERO
Base points of polar curves
Annales de l’institut Fourier, tome 41, no 1 (1991), p. 1-10
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Ann. Inst. Fourier, Grenoble
41, 1 (1991), 1-10
BASE POINTS OF POLAR CURVES
by Eduardo CASAS-ALVERO
Introduction.
In a previous paper ([2]) we have determined the singularities of
the generic polar curves of a sufficiently general irreducible algebroid
curve 7 with prescribed characteristic exponents : we gave the (effective)
multiplicities of the generic polar curves at the (ordinary and infinitely
near) points of 7. In this paper we determine the whole set of infinitely
near base points of the system of polar curves of a such 7: this set contains
not only the multiple points of 7, but also a set of simple and free base
points lying outside of 7.
We give also two corollaries: firstly, we find a lower bound for
the Tjurina number r of irreducible curves. Secondly, many families of
continuous analytical (or formal) invariants of 7 are obtained from the set
of base points of the polar curves.
0. Notations and conventions.
Throughout this paper we are placed under the conventions and
general hypothesis of [2]. In particular we deal with algebroid curves which
Supported by CICYT-PB88 -0344-C0303.
Key-words : Polar curves - singularities of plane curves.
A.M.S. Classification : 14B05.
2 EDUARDO CASAS-ALVERO
always will be assumed to be (locally) plane curves defined over the field C
of complex numbers. Irreducible (reduced) algebroid curves will be called
branches for short. A branch 7 will be said to have general type if and only
if its generic polars go through (9(7) with effective multiplicities equal to
the virtual ones and have no singularities outside of ^(7) ([2] 11.4); such
polar curves will be called polars with general behaviour.
Let p be a point on a smooth algebraic surface S and denote by Op
the complete local ring ofpon S. A family C of algebroid curves with origin
at p will be called a linear system if and only if their equations describe the
set of non zero elements of an ideal of Op. We say that a point q infinitely
near to p is a v-fold base point of a linear system C if and only if q belongs
with (effective) multiplicity v to the generic curves of C. The (/-fold base
points with v > 0 will be also called base points.
1. The branches of a polar with general behaviour.
Let 7 be a branch with origin at p and assume that 7 has general
type. We shall use for 7 the notations of [2] §11, in particular we write
M. == {^/^}i=i,...,r for the system of characteristic exponents of 7 and
S(x)= ^ a^771
i€.I{M)
for its Puiseux series. We assume furthermore that the local coordinates
x ^ y are chosen so that the y-axis is non tangent to 7, i.e., m\ju > 1, and
take
Sk(x)= ^ a^/71.
t^I(M)
«mfc
If n§ = g.c.d. (n, m i , . . . , m^-i), let us write mk/n^ as a continued fraction
of even order
mk _ rrik-i , , fc . 1
—r — ——r— i °n i ~——————
n§ ~ »§ ' " ° ^ + 1
bt^2t(k)
In other words, with respect to the notations of [2], §11 we take, b^ = h^ for
i < s(k), 2t{k) = s(k) and b^^ = /A^x if.s(A;) is even, and 2t(k) = «s(A;)+l,
^w-i = ̂ (k) -1 and ^(k) =1 if ^W is odd-
BASE POINTS OF POLAR CURVES 3
Then we will consider the reduced fractions of the former continued
fractions:
^^^W^^ • (•?•°•t)°<l)•
' '• ?
We have :
1.1. PROPOSITION. — Jf^ ^ a polar of 7 with general behaviour,
then the branches of C are as follows :
For each k = 1,..., r and each t = 1,..., t(k), ^ has b^ different
branches ̂ j j = 1,..., b^ with Puiseux series
^-i
SkW+a^x' ^+...
where there are no characteristic exponents bigger than ^§^-1/^^-1
and
«,)<-^(^)<-l
f o r j ' ^ j .
Proof. — We use for the infinitely near points onjy the notations
introduced in [2], §11. By hypothesis ^ goes through ^(7) with effective
multiplicities equal to the virtual ones and no singularities outside of 3(7).
Then, by using the proximity equalities ([2] 1.4.1, for instance) it is easy to
find, for each q € 3(7), the number of (necessarily simple and free) points
on ^ in the first neighbourhood of q and outside of 9(7); there are no such
points except for the following cases :
a) q == p^_^ ^k ,t < (s(k) -j-1)/2 where we find h^ points, and
b) q = jA,. , f c , and s ( k ) odd, where we find a single point.
S^K^•ls(k) L
Each of one these points corresponds to a single branch of ^ that
contains it as a simple point. Since we know the infinitely near points each
one of these branches is going through, the claim follows from the standard
4 EDUARDO CASAS-ALVERO
relation between Puiseux series and infinitely near points on branches ([3],
IV.I.7 or [4], XI.6.1).
/< ^
--. .,' ''•- ,••••'. ,-''' ''̂ "*"^ Y y Y-Pfc —-' ^•!^ .// // ^^ s ^-^
Figure 1 : the branches ̂  for s(k) odd.
Notice that ^S^-i/71^-! ls tne I5181' characteristic exponent of ̂
if v^-i > 1. In the sequel, since no confusion may be made, we will refer
to y^^-i/m^-i (resp. to o^) as the last characteristic exponent (resp.
coefficient) of <^j even in the case in which z^_i = 1.
2. Base points.
Let 7 and ^ as before. We have :
2.1. THEOREM. — TAe base points of the system ofpolarsof^y not
belonging to 7 are, for each branch ^fc of(^, the first
k n k
^t-l ~ "fc^-l - 1
710
infinitely near points on <^ which do not belong to 7. All of them are
simple base points.
Proof. — Let us denote by ^(7) the cluster defined by adding to 9(7)
the points we claim to be base points of the system of polars, all virtually
counted once. To see that these points are base points it is enough to see
that the polar curves go through ^(7) : in that case, since C goes through
9(7) with effective multiplicities equal to the virtual ones, by [2] 3.1, the
same does a generic polar (and in fact any polar with general behaviour as
one can see directly).
Assume that the coordinates and the equation / of 7 are chosen in
such a way that the polar ^ has equation 9f/9y and denote by $ the
BASE POINTS OF POLAR CURVES 5
polar defined by Q f / Q x . Since the condition of going through a cluster is
linear ([2] 2.4), the system of polar curves is defined by the jacobian ideal
(9f/9x, 9f/9y, f) and C obviously goes through 9(7), we need only to see
that ^ and 7 also go through 9(7).
Let us denote by Vq the virtual multiplicity in 9(7) of any point
q € 9(7). We know that $ and 7 go through 9(7), hence, by using
inductively the "virtual Noether's formula" [2] 8.3 we need only to see
that, for any branch ̂  of ^,
(i) (c^.a > E e^)^ + ̂ -i - ̂ <-i -1—— ^o
qe9(^)
and
(2) (C^.7)p > E e^)^ + ̂ -i - ̂ 4-i - 1.
—- "'o
qe9(^)
We will do with 7 first. Since <^j si11^ 7 do not share points outside
of 9(7), Noether's formula ([2] 1.3.1, for instance) gives
(^r^P= E ^(0^(7),
qeO^}
so that (2) is equivalent to
(3) E ^(C^)(^(^) - ̂ ) > ̂ -i - ̂ <-i - L
— ^oge^^)
Both multiplicities 6^(7) and i/g are given in [2] §11 for all q € 9(7) from
which we obtain
€9(7) - ̂  = °
for 9 = p^, z odd and furthermore i < h^^. it i = <s(A;'), and
eg(7) - ̂  = 1
otherwise. Thus, the first member of (3) is
^E ,̂,)
q
where the summation runs on the points q == p^ with k ' < k and either i
even or z = .s(fc'), 5(fc') odd and t = h^ , . , and also on the points q = p^
6 EDUARDO CASAS-ALVERO
with i even and i < 2t. A straightforward computation using the proximity
relations gives
a=^t-l -1
so that (3), and hence (2), are satisfied.
In order to prove (1), recall first that former computation gives
(4) (C^.7)p = E e^)^ + ̂ -i - 1.
q€9(^)
Then assume that x = z p , y = y ( z ) is a Puiseux parameterization of <^ ,
P = nv^^/n^ being the order of <^. We have
^,^))=pg(^,(^-1^^^))^
where in fact
^(^))=o
because ̂  is a branch of (,. Thus, by equating the orders in z , we obtain
(c^.7)p-i=(c^.a+p-i,
this is,
(C^-Op-^^-^-i/nS
which with (4) proves that (1) is satisfied (and is in fact an equality).
Lastly we must show that there are no more base points outside of
7. For this let us compute first the total number of base points we have
found:
r t(k)
^-EE^ (^-1-^4-1-1)
A-I ^=1 V ^o /;=l t l v ^O
= nir — n
r t(k)
+i-EE^
k=l t=l
by using the standard properties of continued fractions.
On the other hand, let us compute
^= E ̂
<?e<9(-y)
r
=E E flk^ - ̂ + E ^+1(^+1 )2 - ̂ (2n^) - 1)
*=1 \»<<(fc)/2 • • i<s{k)l-2
BASE POINTS OF POLAR CURVES 7
where we use the values of Vq given in [2], §11 and Ck = 0 if s(k} is even,
Ek = 1 if s(k) is odd. We easily obtain
r s { k ) r t(k)
N^= E E ̂  (n? - ̂ nk - m- + n -1 + E E ̂
fc=l 1=1 fe=l t=l
which introducing Milnor's /^, gives
r t(k)
N2 = ̂ (7) -mr+n-l+^^ b^.
k=lt=l
Now assume that there is some base point outside of 7 besides that
described in the claim. By the Noether's formula any pair of generic polars,
and hence any pair of polars, must have intersection multiplicity strictly
bigger than N^ +A^ == ^(7) against the well known equality «.^)p = ^(7).
This ends the proof.
Notice that it follows in particular from 2.1 that the base points do
not depend on the polar ^. Let us state, for further reference :
2.2. COROLLARY (of the proof). — The number of base points of
the system of polar curves of 7 which do not belong to 7 is
rrir — n
r t(k)
+i-EE^.
A;==l ^=1
3. A bound for r(7).
As customary, denote by r the Tjurina number
r = r(7) = dime Op/(9f/9x^ 9f/9y^ /),
where Op is the complete local ring of S at p, and / is an equation of
7. It is well known that r(7) < ^(7), and that the equality r(7) = ^(7)
characterizes quasihomogeneous branches (cf. [5]). We have :
3.1. COROLLARY. — For any branch 7 with system of characteristic
exponents M,
r t ( k )
T(7)>^(7)+m.-^^
A-==l ^=1
8 EDUARDO CASAS-ALVERO
Proof. — Since the claimed bound depends only on the topological
type of 7, it is enough to give a proof for branches 7 with general type. Thus
assume that 7 has general type, and let H = H— be the ideal described
a(-y)
by the equations of the curves going through ^(7). Theorem 2.1 says that
H D (9f/9x,0f/9y,f), so that,
T > dime O p / H .
The last integer being given by [2] 6.1, using 2.2 we have
r> E •^)+^-»+l-i:S^
^ k=l t=l
from which the claim follows after a computation like that made for N^ in
the proof of 2.1.
It must be noticed that, although we can find non trivial examples
of branches for which the bound is reached (namely that of branches
with general type and characteristic exponent 10/3), the equality is false
in general, even for curves with general type : the curves with single
characteristic exponent 5/4 have r = 12 or r = 11 (cf. [6] V.2) whereas our
bound is 9.
4. Continuous invariants.
Assume again that 7 has general type and that C is a polar of 7 with
general behaviour, its branches and corresponding Puiseux series being
described in 1.1.
4.1. COROLLARY. — Choose k and t such that l < k < r , l < t <
t(k)). Assume furthermore that t > l i f b ^ = l and that (k,t) ^ (1,1) if
fcS = 1. Then the v^-i -powers of the ratios between the last characteristic
coefficients of the branches (k , namely
H.,^-
^k
^3' .
for j -f- /, j, j ' = 1,..., &^, are analytic (or formal) invariants of 7.
Proof. — For s(k) odd we have b^. = 1, i.e., a single branch < ^ ) ,
and in this case the claim is empty. Hence we may assume that t < t{k)
for s(k) odd.
BASE POINTS OF POLAR CURVES 9
An easy computation shows that the number of base points on ̂  is
positive unless k = 1, t = 1 and b^ = 1, which is excluded in the claim.
Thus, under our hypothesis, the first point on each branch ̂  outside
of 7, say qf,, is a base point of the system of polars. All these points, for
j = 1,..., 6^, belong to the first neighbourhood E^ ofp^_^ ̂  , the last
point shared by 7 and the branches C^j.
Since we assume t < t(k) for s(k) odd, we have fc^_i = &^-i and
hence, in particular, P^_i ̂  / P^i because of the hypothesis. Then the
point p ^ . . .k is always a satellite point and we know ([1] §4) that an
-" ^'-'St—l
absolute projective coordinate may be taken in its first neighbourhood E^
in such a way that :
a) The two satellite points in E^ have coordinates 0 and oo (depending
on which point besides p^. < i. each of them is proximate to), and
-"t—l'/l2t-l
b) each point q^j has coordinate (o^)^*"1.
On the other hand ([1] §5), any analytic (or formal) transformation
(^ defined at p induces a linear projectivity between E^ and the first
neighbourhood of ^ ( p ^ i ̂  ) under which the points with coordinate
0 are correspondent, as well as that with coordinate oo. This projectivity
must take the form a —> \a for certain fixed A if we use in both first
neighbourhoods the absolute coordinate mentioned above. Since the base
points of the system of polars of 7 must be transformed in the base points
of the system of polars of (^(7), the claim follows.
Remark. — For k = 1, using the way on which the characteristic
coefficients a\ may be computed from the Newton polygon of the polar
curve <^, one can easily find an equivalent set of analytic invariants which
are rational functions of the coefficients of the equation / of 7. Let us show
this by means of an example :
Example. — Consider branches with characteristic exponent 12/5
and equation
/= ]C ^^y3'
5z+12j>60
For generic coefficients the rr-polar has two branches with characteristic
exponent 5/2 and its Newton polygon has a single side. The monomials on
this side give rise to the equation
5ao,5<^4 + 3a5^a2 + aio,i = 0
10 EDUARDO CASAS-ALVERO
whose roots give the characteristic coefficients of the branches. It follows
from 4.1 that aio,o^o,5/^j 3 is an invariant.
BIBLIOGRAPHIE
[1] E. CASAS-ALVERO, Moduli of algebroid plane curves, Algebraic Geometry, La
Rabida 1981, Lecture Notes in Math 961, Springer Verlag, 1982, 32-83.
[2] E. CASAS-ALVERO, Infinitely near singularities and singularities of polar curves,
Math. Ann., 287 (1990), 429-454.
[3] F. ENRIQUES, 0. CHISINI, Teoria Geometrica delle Equazioni e delle Funzioni
Algebriche, N. Zanichelli, Bologna, 1915.
[4] J.G. SEMPLE, G.T. KNEEBONE, Algebraic Curves, Oxford University Press, Lon-
don, 1959.
[5] 0. ZARISKI, Characterization of plane curves whose module of differentials has
maximal torsion, Proc. Nat. Acad. of Science, U.S.A.,56,3 (1966), 781-786.
[6] 0. ZARISKI, Le probleme des modules pour les branches planes, Centre de Math.,
Ecole Polytechnique, Paris, 1973.
Manuscrit recu le 31 mai 1990.
Eduardo CASAS-ALVERO,
Universidad de Barcelona
Facultat de Matematiques
Departament d'Algebra i Geometria
Gran Via 585
08007 BARCELONA (Espagne).


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