A NNALES DE LA FACULTÉ DES SCIENCES DE T OULOUSE
S TEPAN Y U . O REVKOV
Riemann existence theorem and construction of real algebraic curves
Annales de la faculté des sciences de Toulouse 6
esérie, tome 12, n
o4 (2003), p. 517-531
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p. 5
Riemann existence theorem and construction of real algebraic curves
STEPAN YU. OREVKOV (1)
Annales de la Faculté des Sciences de Toulouse Vol. XII, n° 4, 2003 p
ABSTRACT. - We propose a method of construction of plane real alge-
braic curves given by
y3-p(x)y+q(x)
= 0 which has a prescribed arrange- ment on the affine plane. The construction is based on a consideration of the arrangement off -1 (RP 1 )
on CP 1 where f : CP1 ~ CP 1 is the ho- mogenized discriminant, i.e. the rational function defined byf (x)
=D/q2, D = 4p3 + 27q2.
As examples of applications, we construct some M-curves of degrees
7 and 9 on RP2 whose realizability was unknown.
RÉSUMÉ. - On propose une méthode de construction des courbes algébri-
ques réelles planes données par
y3 + p(x)y+q(x)
= 0, qui ont un arrange- ment prescrit sur le plan affine. La construction est basée sur la con-sidération de l’arrangement de
f-1(RP1)
surCP1,
où f : Cpl -1- CP1est le discriminant homogénéisé, i.e. la fonction rationnelle définie par
f(x)=D/q2,D=4p3+27q2.
Comme exemple d’applications, on construit certaines M-courbes de
degrés 7 et 9 sur RP2 dont la réalisabilité n’était pas connue.
1. Introduction
In this paper we propose a method of construction of
plane
realalge-
braic curves
given by F(x, y)
=0, degy
F = 3(trigonal curves)
which havea
prescribed arrangement
on the affineplane.
This method allows one to obtain acomplete
classification of such curves(singular
ornot)
up to fiber-wise
isotopies
of theplane (we
call anisotopy fiberwise
if theimage
of any(*) Reçu le 12 septembre 2002, accepté le 4 septembre 2003
(1) Laboratoire E. Picard, UFR MIG, Univ. Paul Sabatier, 118 route de Narbonne, Toulouse, 31062, France.
E-mail: orevkov@picard.ups-tlse.fr
vertical line at any moment is a vertical
line).
Inparticular,
this means thatthe same method may
provide
some restrictions fortrigonal
curves.This result will be
published
in the next paper. Here wejust
illustratethe method of construction
by
a realization of acomplex M-scheme(2)
ofdegree
7 and some real AI-schemes ofdegree
9 onRP2
whoserealizability
was
previously
unknown.The
proposed
method wasinspired by
a construction of extremalpoly-
nomials for the
Davenport’s
bounddeg (p3 - q2)
1 +(degp)/2
in terms ofso-called "dessins d’enfant"
(see
Sect.3).
PROPOSITION 1.1. There exists an M-curve
of degree
9 onRp2
whose real scheme is
(J
U18>
U114>
U4).
Following [3],
we say that a curve ofdegree
7 onRP2
has ajump
if itcontains 5 ovals
arranged
withrespect
to some line as it is shown inFigure
1.Figure 1 Figure 2 Figure 3
PROPOSITION 1.2. 2013 There exists an M-curve
of degree
7 onRP2
with out ajump
whosecomplex
schemeis ( J
U5+
U 4- U1+ (2+
U3_~~.
Combined with the results of
[13, 3, 9, 10, 4], Proposition
1.2provide
the classification of
complex
schemes of M-curves ofdegree
7 withoutjump (for
curves withjump,
this classification isalready completed
in thpapers cited
above).
In Sect. 5
(Proposition 5.1),
we prove therealizability
of some otheM-schemes of
degree
9. All the curves inProposition
5.1 are constructecby glueing
affine sextics into a 6-foldsingular point
of a curve ofdegree
(2) See
[12]
for the definition and notation of real and complex schemes.We choose this
example
because the sameglueing
was used in[6]
but ourmethod allows us to obtain more curves of
degree
9.2.
Preparation
To construct the curves from
Propositions
1.1 and1.2,
we first constructsingular
curvesdepicted
inFigures
2 and 3 and thenperturb
thesingularities glueing (see [13])
the affine sexticdepicted
inFigure
4(resp. quartic
inFigure 5)
into the 6-fold(resp. quadruple) point.
Figure 4
(see [6])
Figure 5 Figure 6Denote
by Fn
the Hirzebruch surface ofdegree n
and letEn
be the excep- tional section(whose
self-intersection is-n).
Inparticular, Fo
=P1 P1
andFI
is theblown-up P2.
Let 03C0n :Fn - P 1
be the fibration with fiberP 1.
The surfaces
Fl, F2,
... can be obtained fromFo by
successive birational transformationsFo 03B20(p0) F1 03B21(p1) F203B22(p2) ... where /3n
is the blowup
of a
point
pn E En
followed by
the blowdown of the strict transform of the fiber
03C0-1n(03C0n(pn)).
If thepoints
Po, Pl, are real then allFn
are also real. Letus denote the set of real
points
ofFn by RFn.
We
present RFn
inpictures
as arectangle
obtainedby cutting RFn along En (horizontal edges)
and a fiber(vertical edges).
The interior of such arectangle corresponds
to an affine coordinatesystem
onFn
where ageneric
smooth curve C is definedby
apolynomial
whose Newtonpolygon
is
(0, 0)-(1
+kn, 0)-(1, k)-(0, k)
where k(resp. l)
is theintersection
of C witha fiber
(resp.
withEn).
We call(k, l)
thebidegree
of C. The action of03B2n (for
an evenn)
onRFn
is shown inFigure
6. We see in thispicture
thatRFn
is a torus for an even n and a Klein bottle for an odd n.Since
RFI
is theblown-up RP2,
the existence of the curve inFigure
2(resp. Figure 3)
follows from the existence of a curve ofbidegree (3, 6) (resp.
(3,4)) arranged
onRFI
as it is shown inFigure
7(resp. Figure 8).
Figure 7 Figure 8
If a curve C of
bidegree (k, l)
onFn
hasmultiplicity m
at thepoint
pn then the strict transform of C onFn+1
under/3n(Pn)
hasbidegree (k, l - m) . Hence, applying /31 (psi) o...o03B2~(ph)
to the curve C inFigure
7 for h = 6 and{p1
...ph}
= C nEl,
we obtain the curve ofbidegree (3, 0)
onF7
whose realpart
isdepicted
in the upperpart
ofFigure
9.Analogously,
for h =4,
weobtain
Figure
10 fromFigure
8. The isolatedpoints
inFigure
9 andFigure
10are
simple
doublepoints
withimaginary tangents (like x2
-f-y2
=0).
Thecurves in
Figure
9 andFigure
10 will be constructed in Sect. 4.Figure 9
Figure 10
Let
p(t), q(t)
EC[t], deg p
=2k, deg q
= 3k and setSuppose
r is notidentically
zero. How small can bedeg r?
Thisquestion
was
posed
in[1]
in 1965. The same yearDavenport [2]
had shown thatdeg
r k + 1 but it was unknown if the estimate issharp.
This estimateis very natural.
Indeed,
if we write p= t2k
+a1t2k-2
+ ... + a2k-1, q =t3k + blt3k-2
+... +b3k-1
with indeterminate coefficients then the conditiondeg r
k + 1imposes
5k - 2equations
that isequal exactly
to the numberof the unknowns.
However,
it is very hard to showalgebraically
that thissystem
has solutions other than p= s2,
q =s3,
s =tk
+c1tk-2
+ ... + Ck-1- Stothers[11] proved
thesharpness
ofDavenport’s
estimate for any k. His result was rediscoveredby
Zannier[14].
A. Zvonkin gave another(?) elegant proof
but he did notpublish
it because he claims that hisproof
coincideswith the Zannier’s one.
However,
hekindly permitted
us topublish
hisproof
and we do it in this section.
The main idea is to divide the both sides of
(3.1) by q2 .
Denote theobtained rational function
by f. Then f(t)
=r/q2
=p3/q2
+ 1. This meansthat
(i) f
has 3kpoles
ofmultiplicity
2 at the roots of q,(ii)
theequation f
= 1 has 2ktriple
roots at the roots of p, and ifdeg
r = k + 1 then(iii) f
has a zero ofmultiplicity
5k - 1 at t = oo.Conversely,
any rational functionf of degree
6ksatisfying (i)-(iii),
de-fines the
required
p and q.From the
topological point
of viewf
is a branchedcovering CP1~ CP1.
Denote the
preimage
of the realsegment [1, +~] by
r. If(i)-(iii)
hold thenr is a
graph
onCP1
whose verticesf -1 (oo)
have valence 2(white vertices)
and the vertices
f-1(1)
have valence 3(black vertices).
Thegraph
r cutsCP 1
intopolygons homeomorphic
to adisc,
one of which should have 5k-1white vertices and 5k - 1 black ones.
Figure 11
Now we are
ready
to constructf.
Let us start with anybinary
tree inS2
with k - 1triple
vertices and k + 1 ends and transform it to thegraph
0393as it is shown in
Figure
11. Define themapping
0393 ~[1, oo]
which takes the white vertices to 0o and the black vertices to1,
and extend itcontinuously
to a
mapping f : 82 -+ CP 1
which maps eachbigon homeomorphically
ontoCP1 B [1) oo]
and whose restriction onto thepolygonal component of S2B0393
isa branched
cyclic covering
ramified in asingle point to
such thatf (to )
= 0.Pull back the
complex
structure fromCP1
toS2. By
Riemann’stheorem,
the obtained surface is
isomorphic
toCP1.
Choose theisomorphism
so thatit takes
to
into oo. Thanf
becomes a rational functionsatisfying (i)-(iii).
Remark 3.1. - Given
any k
andcoprime
a,b,
similararguments
allow toconstruct
polynomials
p, q, and r =pb - qa
such thatdeg p
=ak, deg q
=bk, deg r
=(ab - a - b)k
+ 1. Like is the case a =3, b
=2,
this is the minimalpossible
value fordeg r.
4. Construction
Let us construct a curve C of
bidegree (3, 0)
onFn (n
= 7 or5)
whosereal
part
isdepicted
inFigure
9 orFigure
10respectively.
It is definedby
apolynomial
whose Newtonpolygon
is thetriangle (0, 0)-(3n, 0)-(0, 3). Killing
the coefficient of
y2,
we rewrite C in the formThe discriminant of
(4.1)
withrespect to y
isLet xo be a root of D
( "*"
inFigures
9 and10).
This means that eitherxo is the x-coordinate of a double
point
of C(then
xo is a double root ofD)
or the vertical line x = xo istangent
to C(a simple
root ofD).
LetF(y) = y3
+p(xo)y
+q(xo) = (y - y1)(y - y2)2.
Since the coefficient ofy2 vanishes,
yl and y2 haveopposite signs. Hence, q(xo)
=F(0)
> 0 when y1 y2, andq(xo) - F(0)
0 when y2 yi. This means that the real roots of q( "o"
inFigures
9 and10)
mustseparate
the root of D wherey1 y2 from those where y2 YI.
Thus,
to constructC,
we need to findpolynomials p(x), q(x),
andD(x), satisfying (4.1), (4.2)
such that the real roots of Dand q
arearranged
as inFigures
9 and 10.Now we
apply
the main idea of Sect. 3: let us divide(4.2) by q2.
Theresult is a rational function
f (x) = D/q2
=4p3/q2
+ 27 whosepoles
arethe roots of q taken with
multiplicity 2,
zeros are the roots ofD,
and thesolutions of
f
= 27 are the rootsof p
taken withmultiplicity
3. To constructf,
consider thegraph
0393c S2 depicted
in the lowerparts of Figures
9 and10
(since
0393 issymmetric,
we showonly
half ofit).
Let us map r ontoRpl according
to thecoloring
inFigure
12 and then continue thismapping
up to a branchedcovering f : S2 ~ CP 1 sending homeomorphically
each com-ponent
of,S’2 B
r onto one of thehalf-planes
ofCP 1 B RP 1
in analternating
order. The additional vertices
(which
are notmapped
onto0, 27,
oroo)
are
mapped
toarbitrarily
chosenpoints
on thecorresponding segments
ofRP2.
Then thepull-back
of thecomplex
structure makesf
to be a rationalfunction which has the needed
properties.
Due to thesymmetry principle, f
becomes real in suitable coordinates.In conclusion of this
section,
let usgive
a list of conditions on a colored embeddedgraph
0393 CS2
which are sufficient to construct a curve ofbidegree (3, 0)
onFn (these
conditions are satisfied in the constructions of Section5).
(1)
Thegraph
r issymmetric
withrespect
to anequator
ofS3 (which
is included to
r)
and thecoloring
of theequator
isimposed by
thedesired
arrangement
of the realalgebraic
curve as it isexplained above;
(2)
The valence of each "·" is divisibleby 6,
and the incidentedges
arecolored
alternatively by
the colors of thesegments [0,27]
and[27, oo];
(3)
The valence of each "o" is divisibleby 4,
and the incidentedges
arecolored
alternatively by
the colors of thesegments [27, oo]
and[oo, 0] ; (4)
The valence of each "*" is even, and the incidentedges
are coloredalternatively by
the colors of thesegments [oc, 0]
and[0,27];
(5)
The valence of each non-colored vertex is even, and the incidentedges
are of the same
color;
(6)
The sum of the valences of all "*"-vertices isequal
to 12n(together
with the conditions
(2)-(5),
thisimplies
that the sums of the valences of all "o" - and "2022"-vertices are alsoequal
to12n);
(7)
Each connectedcomponent of S3 B 0393
ishomeomorphic
to an open disk whoseboundary (considered
as the set ofCarathéodory boundary elements)
is colored as acovering
ofRP1. Moreover,
the orientations ofneighbouring
disks inducedby
thecoverings
of their boundariesare
opposite.
5. Other M-curves of
degree
9PROPOSITION 5.1.
2013 (a)
There exist M-curvesof degree
9 whose real schemes are(b)
Thereexist , fiexible(3)
M-curvesof degree
9 whose real schemes areRemark 5.2. 2013 The list of the real schemes in
Proposition
5.1 isgiven
is the same format as the list in
[5;
Theorem6] (this explains,
inparticular,
such a
strange numbering
of theseries).
We do not include here the M- schemes which are listed in[5]
but we include those which are constructed in[6] (marked by
theasterisk)
and inProposition
1.1.Remark 5. 3. - We found the
following misprints
in[5].
In
[5;
Theorem6,
Series4],
there should be"(03B2,03B3) (5,7), ô
=8, 10"
instead of
"(03B2,03B3)
=(5,8), 03B4
=8, 10".
In
[5;
Theorem6,
Series5],
there should be(03B1,03B2,03B3,03B4) = (1, l, 3,15)
instead of
(1,1, 3,13).
In
[5;
Theorem7], "a, 03B2,
03B3 - even" in Series 4 means "each of cx,03B2,
03B3 is even" whereas"ce, 03B2,03B3,03B4-
even" in Series 6 means "one of 03B1,03B2,03B3
is even".We shall call central blocks the
rectangles depicted
in the upperparts
ofFigures
13.1-13.5 or theirimages
under the reflection withrespect
to a vertical or a horizontal line.(3) See
[12]
for the definition of a flexible curve.Figure 13.1 Figure 13.2 Figure 13.3
Figure 13.4 Figure 13.5
Figure 14 Figure 15
LEMMA 5.4. - Let
Bo, Boe be
two central blocks andho, hoe
the corre-sponding
valuesof
theparameter
h(indicated
in13.1-13.5).
Thenfor
any sequenceof signs
si, .... sp,p 0,
there exists a realalgebraic
curve C.
of bidegree (3, 0)
onFn,
n = p +ho
+hoo
such that thepair (Fn B En, C) (recall
thatEn
CFn, En
=-n)
is obtained(up
to afiberwise isotopy,
seethe
Introduction) by
the successivecyclic glueing (according
to thearrows) of
the blocksBo, B(s1),...,B(sp), Boo, B(-sp),...,B(-s1),
whereB(+), B(-)
are shown in the upperpart of Figure 14.
Proof. -
LetDo
andDoo
be the half-discs which are shown inFigures
13.1-13.5 under the blocks
Bo
andB~,
and letD*~
be the inversionimage
of
Doo.
LetD(+)
be the half-annulus shown in the lowerpart
ofFigure
14and
D(-)be its
mirrorimage.
Let us fill the lowerhalf-plane by
the domainsDo, D(SI), ... , D(sp), D*~ according
toFigure 15,
and dosymmetrically
theupper
half-plane.
Let us deal with the obtainedgraph
F in the same way as in Sect. 4(in
the case13.4,
to construct themapping
of theregion A,
oneshould take a double
covering
branched at asingle point).
0Example. -
We show inFigure
16 how the curve inFigure
9 can beobtained from the central blocks 13.1 and 13.3
by applying
Lemma 5.4followed
by
acontracting
of an oval into an isolated doublepoint (see
Lemma5.7
below).
COROLLARY 5.5. - Let
(s1,...,sp), p 1,
si =+ 1 ,
be anarbitrarg
sequence
of signs
such that s, = 1. Let al, ..., ap benon-negative integers
such that
Then there exists a curve
of bidegree (3, 0)
onFp+2
arranges as inFigure
17with bi
=2ai + 1, i =1,...p.
Proof.
- SetBo
=Boo = [the
block inFigure 13.1]
in Lemma 5.4. ~COROLLARY 5.6. There exist curves
of bidegree (3, 0) arranged
onF7 as in Fgiure 17 with p = 5 and (b1,...,b5) = (15,1,1,1,1), (11,5,1,1,1),
- 528 - Figure 17
LEMMA 5.7. - Let A C
Fp+2 be a
curve constructed in Lemma5.4 (or
in Corollaries
5.4
and5.6).
Then there exist a nodal curve A’ CFp+2 of
thesame
bidegree
obtainedfrom
Aby applying of
any numberof transformations
shown in
Figure
18.Figure 18 Figure 19
Proof.
-Apply
the transformations inFigure
19 to thegraph
h. DRemark 5.8. 2013
Corollary
5.5 can beeasily proved by
Viro’s methodusing
the subdivision of thetriangle (0, 0)-(3p+ 6, 0)-(0, 3)
into n+ 2triangles (3p, 0)-(3p + 3, 0)-(0, 3). However,
it is not clear how to prove Lemma 5.7 in this way.COROLLARY 5.9. - There exist curves
of degree
9 with asimple
6-fold singular point
and 13 ovals distributed as inFigure
20 where a andb are the number
of
ovals in thecorresponding regions,
the exterior ovalsare not
shown,
andSi = t(a, b) |a
+b
10 and a, b areoddl, S2
=S1~{(1,11)}B{(5,5)}.
Proof.
-Apply
Lemma 5.7 and the transformation03B21(p1)
...03B26(p6) (see
Sect.2)
to the curves from Lemma 5.4. The curves fromCorollary
5.6provide
the upper two rows ofFigure
20.1. In the other cases, one should choose the central blocks in Lemma 5.4 in thefollowing
way.The curves in the lower row in
Figure
20.1. The left:(13.1
and13.3)
or(13.1
and13.5);
the middle:(13.1
and13.4);
theright: (13.2
and13.2).
The curves in
Figure
20.2 and 20.3. The upper left curve in each ofFigure
20.2 and 20.3:(13.1
and13.4);
the other curves:(13.1
and13.2).
~Remark 5.10. - It is clear from the construction that any collection of the
tangents
at thesingular point
is realizable in the case of the left curvein the lower row of
Figure
20.1(it
is markedby
theasterisk). Unfortunately,
in the other cases this is not so.
Figure 20.1
Figure 20.2
Figure 20.3
Remark 5.11. -
Using
the braid-theoretical methods(the
Garside nor-mal form of the braid from
[7]),
one can prove thatFigure
20.1-20.3 containall the
isotopy types
of curves ofdegree
9 which have asimple
six-foldpoint
and whose
perturbation
canprovide
an M-curve ofdegree
9.Proof of Proposition
5.1.- (a).
Maximaldissipation (see [13])
of a sim-ple
6-foldsingular point
are described in[6].
Two moredissipations B2(1, 8,1 )
and
A3(0, 5, 5)
are constructed in[8]. Applying
all thedissipations
of theseries A
(resp.
B orC)
to all the curves inFigure
20.1(resp. Figure
20.2 orFigure 20.3)
one obtains all therequired algebraic
curves.(b).
Flexibledissipations
of thetypes A4(1, 4, 5), B2 (1, 4, 5),
andC2 (1, 3, 6)
are constructed in
[7;
Sect.7.2]. Applying
them to thesingular point
of the curves inFigure 20.1-Figure 20.3,
one obtains all therequired
flexiblecurves.
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