# Algebraic curves and maximal arcs

## Testo completo

(1)

A. Aguglia

L. Giuzzi

G. Kor hmáros

### ∗

Abstra t

Alowerboundontheminimumdegreeoftheplanealgebrai urves

ontainingeverypointinalargepointset

### K

oftheDesarguesianplane

### PG(2, q)

is obtained. The ase where

is a maximal

### (k, n)

ar is onsidered ingreaterdepth.

1 Introdu tion

In nite geometry, plane algebrai urves of minimum degree ontaining a

given large pointset

in

### PG(2, q)

have been a useful tool to investigate

ombinatorialpropertiesof

.

When

### K

is the whole pointset of

### PG(2, q)

,a triviallower bound on the

degree of su h a plane algebrai urve is

### q + 1

. G. Tallini pointed out that

thisisattainedonlywhenthe urvesplitsinto

distin tlinesof

### PG(2, q)

,

all passing through the same point. He also gave a omplete lassi ation

of the absolutely irredu ible urves of degree

### q + 2

ontaining all points of

### PG(2, q)

;see [23, 24℄ and also[1℄. If

### K

isthe omplementary set of alinein

### PG(2, q)

,then the bound is

; see [13℄.

When

### K

onsists of all internal points to a oni

in

with

### q

odd, theabove lowerboundis

### − 1

. The analogous bound forthe set of the

external pointsto

is

### q

. These boundswere the mainingredientsfor re ent

ombinatorial hara terisationsof pointsets blo kingallexternallines to

;

see [4,12℄.

When

### K

is a lassi al unital of

, with

### q

square, the minimum

degree

### d

of an absolutely irredu ible urve

through

is

### d = √q + 1

. For

non lassi al unitals,the best known bound is

; see [18℄.

### ∗

Resear h supported by theItalian MinistryMURST, Strutture geometri he,

(2)

Our purpose is to nd similar bounds for slightly smaller, but still quite

large, pointsets

, say

with

and

, where

### c

isasuitable onstant. Sin eno ombinatorial onditiononthe onguration

of

### K

is assumed, we are relying on te hniques and results from algebrai

geometry rather than on the onstru tive methodsused in the papers ited

above.

Themain resultisthat if

### − 2α + 2

,anyplane algebrai

urve

### Γ

ontaining every point of

has degree

### ≥ 2t

. The hypothesis on

the magnitude of

an be relaxed to

whenever

### q

is

a prime.

Insome ases,thebound

### ≥ 2t

issharp,asthe followingexampleshows.

Let

be the union of

### t

disjoint ovals. If

odd, or

### q

is even and the ovals

are lassi al, then

and

### d = 2t

. The latter ase is known to

o ur when

### K

is a Denniston maximal ar [10℄ (or one of the maximal ar s

onstru ted by Mathon and others, [25, 26, 16, 20, 14℄) minus the ommon

nu leus of the ovals.

On the otherhand, somerenementof the bound isalsopossible. Let

### K

beany maximal ar of size

,that is, a

### (qt + t + 1, t + 1)

ar .

Theorem 4.2 shows that if

### q

is large enough omparing to

### t

, then no plane

algebrai urve of degree

### 2t

passes through every point of

### K

. Therefore, the

minimumdegreeisatleast

forsu h

### t

and this boundisattainedwhen

### K

is one of the above maximal ar s.

The ase

### n = 4

is onsidered in more detail. For

, the minimum

degreeis

### 7

andthisisonlyattainedwhen

### Γ

of minimum degreesplits intothree distin t oni s with the same nu leus

### ∈ K

, together with a linethrough

### N

.

2 Some ba kground on plane algebrai urves

over a nite eld

A plane proje tive algebrai urve

is dened over

### GF(q)

, but viewed asa

urve over the algebrai losure

of

### GF(q)

, if it has anane equation

, where

. The urve

### ∆

is absolutely

irredu ible if it is irredu ible over the algebrai losure

### GF(q)

. Denote by

(3)

irredu ible plane urve

of degree

### d

. From the Hasse-Weilbound

### q.

(1)

This holdstruewhensingularpointsof

lyingin

### PG(2, q)

are also ounted;

see [19℄.

The Stöhr-Volo h bound depends not only on the degree

### d

, but also on

a positive integer, the Frobenius order

of

### ∆

; see [22℄. This number

is

either

or

, where

### 2

is the interse tion number

of

### ∆

with the tangent line

### ℓ

at a general point

### ∈ ∆

. It turns out that

is either

### 2

, or a power, say

### h

, of the hara teristi

### p

of the plane, and is the minimum of

, where

### Q

ranges over the nonsingular points of

and

is the

tangent to

at

. If

, then

, and either

, or

and

### p = 2

. Withthis notation, the Stöhr-Volo hboundapplied to

is

### ≤ ν(d − 3)d + d(q + 2).

(2)

The followingalgebrai ma hinery anbeusedto ompute

. Let

### P = (a, b)

beanonsingularpoint of

### ∆

su hthat the tangentlineto

at

### P

isnot the

verti al line through

### P

. The unique bran h (or pla e) entred at

### P

has a

lo alparametrisation,also alleda primitivebran h representation,

where

and

with

### ≥ 1

is a formalpower series

with oe ients in

;see [21℄. Then,

### ν

is dened tobethe smallest

integersu h that the determinant

### (ϕ(t))

does not vanish. Here

denotes the

### ν

th Hasse derivative, that is,

### + . . . .

Theaboveideastillworksifos ulating oni sareusedinpla eoftangent

lines, and, in some ases, the resulting bound improves (2). Before stating

the result, whi h is the Stöhr-Volo h bound for oni s, a further on ept

(4)

respe t to the linear system

### 2

of the oni s of the plane is the in reasing sequen e

### 5

of all interse tion numbers

of

### ∆

with oni s at a general point

. The Frobenius

### 2

order sequen e is thesubsequen e

### 4

extra ted in reasinglyfromthe

### 2

order sequen e of

### ∆

,for whi hthe following determinant doesnot vanish:

Assume that

### ≥ 3

. The Stöhr-Volo h bound for oni s, that is for

, is

### − 3)d + 2d(q + 5)].

(3)

For more onthe Stöhr-Volo h bound see [22℄.

3 Plane algebrai urves of minimum degree

through all the points of a given pointset

Inthis se tion,

standsforasetof

pointsin

,with

and

. Let

### Γ

denotea plane algebrai urve of degree

ontaining

every point

### q + 1

is the minimum degree of a

plane algebrai urve ontaining every point of

### PG(2, q)

. Thus, sin e we are

lookingforlowerboundson

### d

,wewillonlybe on erned withthe asewhere

### ≤ q

.

A straightforward ounting argumentgivesthe following result.

Lemma 3.1. If

, then

.

Proof. Sin e

### ≤ q

,thelinear omponentsof

### Γ

donot ontainallthepointsof

. Chooseapoint

### ∈ PG(2, q)

notinanyoftheselinear omponents.

Ea h of the

lines through

meets

at most

### d

distin t points. Thus,

,that is

### .

(5)

Our aim is to improve Lemma 3.1. Write

### F

for the set of all lines of

meeting

in at least

point. Set

and

.

Theorem 3.2. Let

### Γ

be an algebrai plane urve over

of minimal

degree

### d

whi h passes through all the points of

. If

(4)

then

. For prime

### q

, Condition (4) may be relaxed to

### − 1).

(5)

Proof. We prove that if

### ≤ 2t − 1

, then (4) doesnot hold. For

### q

prime we

show that also(5) is not satised. Sin e

### Γ

isnot ne essarilyirredu ible, the

followingsetup is required.

The urves

### l

are the absolutely irredu ible nonlinear ompo-nents of

dened over

### GF(q)

, respe tively of degree

;

### k

are the linear omponents of

over

;

### s

are the omponents of

### Γ

whi h are irredu ible over

but not over

### GF(q)

.

Theidea istoestimatethe number ofpointsin

### PG(2, q)

that ea hof the

above omponents an have.

Let

### i

be the number of nonsingular points of

lying in

### PG(2, q)

. Then, (2) holds for any

. Let

### (i)

denote the Frobenius order of

. If

and

,then

(6)

For

prime,

### ν = 1

; see [22℄. Sin e

an fail for

, an upper

bound on

depending on

is needed. As

, a bound on

su es. Sin e

maybeassumed,

### i

has anonsingular point

lying in

. If

isthe tangentto

at

,then

when e

. From (6),

prime.

otherwise. (7)

(6)

If

has

### i

singularpoints,fromPlü ker'stheorem

. Hen e,

### − 2) ≤ (δ − 1)(δ − 2).

(8)

The number of points of

### K

lying on linear omponents

is at most

.

Forevery

### i

, there exists anabsolutely irredu ible urve

### Θ

,dened over thealgebrai extension

of degree

of

in

### GF(q)

,su h

that the absolutely irredu ible omponents of

are

### Θ

and its onjugates

. Here, if

has equation

and

,

then

### w

is the urve of equation

. Sin e

,

### Θ

and the onjugates of

### Θ

pass through the same points in

### PG(2, q)

, from Bézout's

theorem, see [17, Lemma 2.24℄,

has at most

points in

where

. Notethat

. Let

### i

denotethe total numberof points (simpleorsingular)of

lying in

### PG(2, q)

. From the above argument,

(9) As,

### ,

(10)

from (7), (8), (9) and (10) it follows that

prime

otherwise.

Sin e

### ≤ 2t − 1

,the mainassertionfollows by straightforward omputation.

Remark 3.3. As

### ≤ d

, the proof of Theorem 3.2 shows that Condition (4), and, for

### q

prime, Condition (5), may be repla ed by the

somewhatweaker, butmoremanageable, ondition

(and

for

### q

prime).

Remark 3.4. As pointed out in the Introdu tion, Theorem 3.2 is sharp as

(7)

Corollary 3.5. If

### Γ

has a omponent not dened over

, then

### ≥ 2t + 1.

Proof. We use the same arguments as in the proof of Theorem 3.2,

onsid-ering that

### δ < d

. In parti ular, we have

whi h for

### ≤ 2t

proves the assertion.

Remark 3.6. Corollary3.5impliesthata planealgebrai urveof ofdegree

ontaining

### K

is always dened over

### GF(q)

.

Theorem 3.7. If the urve

### Γ

in Theorem 3.2 has no quadrati omponent,

and

(11) then

If

### q > 5

is prime,then Condition (11) may be relaxed to

### 8

(12)

Proof. Weprovethat if

### − 1)

, then(11) (and, for

### q

prime, (12))does not hold. It is su ient just a hange in the proof of Theorem 3.2. Let

### 4

be the Frobenius orders of

### i

with respe t to oni s. If

### q

is a prime greater than

, then

for

. Otherwise, set

. Sin e

and

, we have that

. From (3),

prime

### − 1)δ(δ − 3) + (q + 5)2δ,

otherwise. (13)

Using the same argumentas in(10),

(14)

(8)

prime

### qd + 1

otherwise. (15)

Then,(15)doesnotholdforany

. If

### q

isprime,also(12)isnotsatised.

4 Algebrai urves passing through the points

of a maximal ar

Remark 3.4 motivates the study of plane algebrai urves passing through

all the points of maximal

ar in

.

In this se tion

### K

always denotes a maximal

### (k, n)

ar . Re all that a

ar

### K

of a proje tive plane

is a set of

points, no

### n + 1

ollinear. Barlotti[9℄proved that

,forany

ar in

### PG(2, q)

;when

equality holds, a

### (k, n)

ar is maximal. A purely ombinatorial property

hara terising a

maximal ar

### K

is that every line of

either

meets

in

### n

points or is disjoint from it. Trivial examples of maximal

ar s in

are the

### + q + 1, q + 1)

ar given by all the points of

and the

### , q)

ar s onsisting of the points of an ane subplane

of

### PG(2, q)

. Ball, Blokhuis and Mazzo a [5℄, [6℄ have shown that

nonontrivial maximalar exists in

for

### q

odd. On the other hand,

for

### q

even, several maximal ar s exists in the Desarguesian plane and many

onstru tions are known; see [10℄, [25℄, [26℄, [16℄ [20℄, [14℄. The ar s arising

from these onstru tions, with the ex eption of those of [25℄, see also [16℄,

all onsist of the union of

### − 1

disjoint oni s together with their ommon

nu leus

### N

. In other words, these ar s are overed by a ompletelyredu ible

urveofdegree

### −1

,whose omponentsare

### −1

oni sand alinethrough

the point

### N

.

Remark 4.1. From Corollary3.5, if

### Γ

has a omponentdened over

but notover

### GF(q)

,anditpassesthrough allthe pointsof

, thenitsdegree

is atleast

### − 1

.

The following theorem shows that the above hypothesis on the

(9)

Theorem 4.2. For any

, there exists

### 2

su h that if a plane

algebrai urve

denedover

with

### 0

passesthrough allthepoints of a maximal

ar

of

then its degree

isat least

### − 1

. If

equality holds then

### Γ

has either one linear and

### − 1

absolutely irredu ible

### − 2

and one ubi omponent.

The proofdepends on the the followinglemma.

Lemma 4.3. Assume that

### Γ

is redu ibleand that the number of its

ompo-nents is less than

### − 1

. Then, the degree

of

satises

### q.

Proof. Weuse the samesetup asinthe proofof Theorem 3.2. This time the

HasseWeil bound is used in pla e of the Stöhr-Volo hbound. The number

of pointsof

in

is

; from(1),

(16) From (9),

(17)

If

has

omponents,then

(18) Sin e

, (18) yields

hen e,

### q

.

Proof of Theorem 4.2. Suppose

tohavedegree

. ByRemark4.1,

all omponentsof

aredenedover

. If

### Γ

isabsolutelyirredu ible,then

(1)impliesthat

ontainsatmost

points. However,

(10)

When

### Γ

has more then one omponent, denote by

### j

the number of its omponents ofdegree

. Let

### u

bethe maximum degreeof su h omponents.

Then,

and

From (1),

where

Both

and

### d

are independent from

; therefore,

(19) Hen e,

Sin e

### ≤ 2n − 2

, by Lemma4.3, for

the urve

shouldhave

at least

### − 1

omponents. This would imply that either

,

,

or

,

,

,

### = 1

. In parti ular, in both ases

### − 1

.

Remark4.4. AsmentionedintheIntrodu tion, ase

inTheorem

4.2 o urs when

### K

is a Denniston maximal ar [10℄ (or one of the maximal

ar s onstru ted by Mathon and others, [25, 26, 16, 20, 14℄). This result

may not extend toany of the other known maximal ar s; they are the Thas

maximal

ar sin

### )

arisingfromthe SuzukiTits ovoid

of

### PG(3, q)

;see[25℄. Infa t,

### 22

istheminimumdegreeofaplane urvewhi h

passes through all points of a Thas' maximal

ar in

### PG(2, 64)

; see

(11)

5 Maximal ar s of degree

## 4

From Theorem 4.2, for

### 4

, a lower bound on the degree of an algebrai

urve

### Γ

passing through all the points of a maximal ar

of degree

is

### 7

.

Our aim isto prove inthis ase the followingresult.

Theorem 5.1. Let

### K

be a maximal ar of degree

### 4

and suppose there exists

an algebrai plane urve

### Γ

ontaining allthe points of

. If

, then

### Γ

onsists of three disjoint oni s, all with the same nu leus

### N

, and a line through

### N

.

Proof. FromTheorem 4.2,the urve

### Γ

splitseitherintooneirredu ible ubi

and two irredu ible oni s or into three irredu ible oni s and one line

### r

.

These two ases are investigated separately.

Let

### C

denote any of the above oni s of nu leus

### N

. We showthat every

point of

lying in

is ontained in

### K

: in fa t, if there

were a point

### \ K

, then there would be at least

lines through

external to

### K

. All these lines would meet

in

### 4

distin tpoints, whi h, in turn, wouldnot beon

. Hen e,

### Γ

wouldhaveless than

pointsonthe

ar

Nowassumethat

splitsintoa ubi

andtwo oni s

,with

. Denote by

the nu leus of

and set

. Sin e

### |X | ≤ 2q + 4

, there exists a point

. Obviously,

### ∈ D

. Every line through

meets

### K

infourpoints;thus,there isnoline

through

meeting

both

and

in

### 2

points; otherwise,

and

### |ℓ ∩ K| ≥ 5

, a ontradi tion. Hen e, there are at least

lines through

meeting

in another point

### ′

. There are at most

### 5

bise ants to the irredu ible ubi

urve

### D

through any given point

### ∈ D

, namely the tangent in

to

### D

and,possibly,fourothertangentsindierentpointsto

passingthrough

.

Hen e,thereare

linesthrough

meeting

### D

inthreepoints. Ifthiswere

the ase,

### D

would onsist ofatleast

### − 6) + 6

points,whi his impossible.

Therefore, we may assume that

### Γ

splits into three oni s, say

,

,

, with nu lei

,

,

, and a line

### r

.

Re all that, asseen above, the nu lei of allthe oni sbelong to

### K

. Now

we show that at least one nu leus, say

### 1

, lies on the line

. Sin e

### Γ

is a urve ontaining

### K

of minimum degree with respe t to this property, there

is at least a point

on

not on

### 1

. Ea h line through

isa

se ant to

hen e, it meets

### X

in an odd number of points. If

### 1

, then the numberoflines through

meeting

### X

inanodd numberof pointsisatmost

(12)

### 3

Figure 1: Case 1 inTheorem 5.1 PSfrag repla ements

### s

Figure 2: Case 2 inTheorem 5.1

,whi hislessthan

for

### 4

,a ontradi tion. A tually,allthenu lei

lie on

### r

. In fa t, suppose that

for

. Then,

, with

and the line

joining

and

is tangentto

and

. Consequently,

meets

### s

in another point dierent from

that is, it is a

se ant to

### K

We are left with three ases, namely:

(1)

,

for any

and

.

(2)

,

,

, with

.

(3)

and

for

### 6= j

.

We are going toshowthat ases (1)and (2)do not a tually o ur.

PSfrag repla ements

(13)

Lemma 5.2. Let

,

### 2

be two oni s with a ommon point

### A

but dierent nu lei

,

. If

### 2

thenthere is a line

with

and

### 2

su h that for any point on

### )

there passes a line

with

Proof. Let

### (X, Y, Z)

denotehomogeneous oordinates of points of the plane

### PG(2, q)

. Chooseareferen esystem su h that

### A = O = (0, 0, 1)

andthe line joining

and

is the

### X

axis. We may suppose

to have equation

where

and

.

Sin e both

### i

are nondegenerate oni s, we have

and

. Furthermore,

as the nu lei

and

are distin t.

Denote by

the tra e of

over

; namely

.

If

### )) = 0

, then the two oni s

and

### 2

have more than one point in ommon, whi h is impossible. Hen e,

; inparti ular,

. A generi point

of

### \ {A}

has homogeneous oordinates

with

### ∈ GF(q) \ {0}

. Consider now a point

on the

axis,

with

. The points

,

and

### ε

are ollinearif and only if

that is

### )t = 0.

(20)

Equation (20) may be regarded as the ane equation of a ubi urve

### D

in the indeterminate

and

. Observe that

### ∈ D

. The only points

at innity of

are

,

and

(14)

Therefore,

### D

doesnot splitintothree onjugate omplexlines. Thus, by [17,

Theorem 11.34℄ and [17, Theorem 11.46℄, there is at least one ane point

on

dierent from

.

Sin e

, the line

isa

se antto

in

### (0, 0, 1)

;hen e, the point

### T

isnot onthis line.

This impliesthat, for any given

### ∈ GF(q) \ {0}

, there existat least two

distin t values

### ∈ GF(q) \ {0}

satisfying (20). Hen e,

,

and

### ε

are ollinear and the line

meets

### }

in at least three points.

From Lemma

### 5.2

,in ase (1)the set

### }

annotbe om-pleted to a maximal ar just by adding a third oni

### 3

, together with its nu leus

### 3

, sin e, in this ase, there would be at least a

se ant to

### }

. Hen e, ase (1) is ruled out.

Lemma 5.3. Given any two disjoint oni s

,

### 2

with the same nu leus

### N

, there isa unique degree

### 4

maximal ar ontaining

### 2

.

Proof. Thereis aline

in

externalto

### X

,sin e,otherwise,

would

bea

### 2

blo kingsetwithlessthan

### 2q+1

points,whi hisa ontradi tion; see [8℄.

Choose a referen e system su h that

and

is the line

at innity

. The oni s

, for

,have equation:

(21)

where

and

.

Sin e both

### i

are nondegenerate,

### 6= 0

. We rst show that,

, as

and

### 2

are disjoint. We argue by ontradi tion. If it were

### 2

, then we ould assume

; in fa t, if

and

### 2

, the linear system generated by

and

### 2

would ontain the line

### Y = 0

; thus their interse tionwould not be empty. Letnow

hen e,

and the line

### X = γY

would meetthe two oni s inthe same point

(15)

We also see that

and

### 1

, sin e, otherwise, the points

### , 1)

would lie onboth oni s.

Letnow

### 3

be the oni with equation

Set

### .

The ollineation H of

### PG(2, q)

given by the matrix

where

and

maps the oni s

,

, to

and

to

If it were

, then

and

### 2

would share some pointin ommon on the lineat innity. Hen e,

. In parti ular,

,

and

### 2

together with their ommonnu leus

, forma degree

maximalar

### K

of Denniston

type; see [10℄, [2℄and [20, Theorem 2.5℄.

It remains to show the uniqueness of

### K

. We rst observe that

is a

### (0, 2, 4)

set with respe t to lines of the plane. No point lying ona

se ant to

to get a degree

maximal ar .

Take

and denoteby

with

,the numberof

se antsto

through

### P

, that is the number of lines meeting

in

### i

points. The lines through

### P

whi h areexternalto

are alsoexternal

### K

and the onverse also

holds. Therefore, when

,wehave

. From

(16)

also

### q

. Hen e, no point

### X

to obtain a maximal ar of degree

### 4

.

Finally,Lemma5.3 shows that ase (2) doesnot o ur.

Referen es

[1℄ V. Abatangelo, G. Kor hmáros; A generalization of a theorem of B.

Segre on regular points with respe t to an ellipse of an ane Galois

plane, Ann. Mat.Pura Appl. 72 (1997),87102.

[2℄ V. Abatangelo, B. Larato; A hara terization of Denniston's maximal

ar s, Geom. Dedi ata30 (1989), 197203.

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ar s in

### )

, Results Math., toappear.

[4℄ A. Aguglia,G.Kor hmáros;Blo king sets of externallines to a oni in

,

### q

odd , Combinatori a 26 (2006),no. 4,379394.

[5℄ S.Ball,A.Blokhuis, F.Mazzo a;Maximal ar s inDesarguesian planes

of odd order do not exist, Combinatori a 17 (1997), no. 1,3141.

[6℄ S. Ball, A. Blokhuis; An easier proof of the maximal ar s onje ture,

Pro . Amer. Math. So . 126 (1998),no. 11,33773380.

[7℄ S.Ball, A.Blokhuis; The lassi ation of maximal ar s in small

Desar-guesian planes, Bull. Belg. Math. So . Simon Stevin, 9 (2002), no. 3,

433445.

[8℄ S. Ball, A. Blokhuis; On the size of a double blo king set in

### PG(2, q)

,

Finite FieldsAppl. 2 (1996),no. 2, 125137.

[9℄ A. Barlotti; Sui

### (k, n)

ar hi di un piano libero nito, Boll. Un. Mat.

Ital. (3) 11 (1956), 553556.

[10℄ R.H.F. Denniston; Some maximal ar s in nite proje tive planes, J.

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hara teristi , Manus riptaMath. 59 (1987), no. 4, 457-469.

[12℄ M. Giulietti; Blo king sets of external lines to a oni in

,

### q

even, European J. Combin 28 (2007), no. 1, 3642.

[13℄ V.D. GoppaGeometry and Codes, Kluwer(1988).

[14℄ N.Hamilton,R.Mathon;Moremaximalar s in Desarguesianproje tive

planes and their geometri stru ture, Adv.Geom. 3 (2003), 251261.

[15℄ N. Hamilton, T. Penttila; A Chara terisation of Thas maximal ar s in

translation planesof square order, J. Geom.51 (1994), no. 12, 6066.

[16℄ N. Hamilton, T. Penttila; Groups of maximal ar s, J. Combin. Theory

Ser. A 94 (2001),no. 1, 6386.

[17℄ J.W.P. Hirs hfeld; Proje tive geometries over nite elds, Se ond

edi-tion, OUP (1998).

[18℄ J.W.P. Hirs hfeld, G. Kor hmáros; Ar s and urves over nite elds,

Finite FieldsAppl. 5 (1999),no. 4, 393408.

[19℄ D.B. Leep, C.C. Yeomans; The number of points on a singular urve

over a nite eld, Ar h. Math. (Basel), 63 (1994),no. 5,420426.

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The-ory Ser. A, 97 (2002), no. 2,353368.

[21℄ A. Seidenberg; Elements of the theory of algebrai urves, Addison

Wesley (1968).

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Pro . LondonMath. So .(3) 52 (1986),no. 1, 119.

[23℄ G. Tallini; Sulle ipersuper ie irridu ibili d'ordine minimo he

on-tengono tutti i punti di uno spazio di Galois

### r,q

Rend. Mat. e Appl. (5) 20 (1961),431479.

[24℄ G. Tallini; Le ipersuper ie irridu ibili d'ordine minimo he invadono

uno spazio di Galois, Atti A ad. Naz. Lin ei Rend. Cl. S i. Fis. Mat.

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ometriae Dedi ata,3 (1974), 6164.

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Authors' addresses: Angela AGUGLIA Dipartimentodi Matemati a Polite ni odi Bari ViaOrabona 4 70125Bari (Italy) Email: a.agugliapoliba.it Lu a GIUZZI Dipartimentodi Matemati a Polite ni odi Bari Via Orabona 4 70125 Bari(Italy) Email: l.giuzzipoliba.it GáborKORCHMÁROS Dipartimentodi Matemati a

Università della Basili ata