Semiovals and semiarcs

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Semiovals and semiarcs

Gy. Kiss

Finite Geometry Workshop

June 12th, 2013, Szeged

gyk Semiovals and semiarcs

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Coauthors

Main collaborator: Bence Csajb´ok Other collaborators:

Daniele Bartoli Tamás Héger Stefano Marcugini Fernanda Pambianco János Ruff

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The beginning, semi-quadratic sets

Semiovals first appeared as special examples ofsemi-quadratic sets. Let Π be a projective space andQ= (P,L) be a pair consisting of a setP of points of Π,and a setL of lines of Π.A tangentto Qat P ∈ P is a line`∈ Lsuch that P is on `,and either`∩ P ={P},or `∈ L.Q is called semi quadratic set (SQS), if every point on a line ofL belongs toP, and for all P ∈ P the unionT_P of all tangents toQ atP is either a

hyperplane or the whole space Π.A lot of attempts were made to classify all SQS, but the problem is still open in general.

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Semi-ovoids

An SQSQ= (P,L) is called a semi-ovoid(or semiovalif dim Π = 2),if L=∅ andP contains at least 2 points. The complete characterization of semi-ovoids was given by J. Thas. Using elementary double counting arguments, he proved the following results.

Theorem

The only semi-ovoids of PG(3,q)are the ovoids (set of q²+ 1 points, no three of them are collinear).

In PG(n,q), n>3,there are no semi-ovoids.

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Semi-ovoids

An SQSQ= (P,L) is called a semi-ovoid(or semiovalif dim Π = 2),if L=∅ andP contains at least 2 points. The complete characterization of semi-ovoids was given by J. Thas. Using elementary double counting arguments, he proved the following results.

Theorem

The only semi-ovoids of PG(3,q)are the ovoids (set of q²+ 1 points, no three of them are collinear).

In PG(n,q), n>3,there are no semi-ovoids.

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Semiovals

In the planar case the situation is much more complicated. It is easy to see, that the following simpler definition of semiovals is equivalent to the previously given one.

Definition

Let Πq be a projective plane of order q. Asemiovalin Πq is a non-empty pointsetS₁ with the property that for every point P ∈ S₁ there exists a unique linetP such that S∩tP ={P}. This line is called the tangent toS₁ atP.

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Definition, Basic examples

Definition

LetΠq be a projective plane of order q.A non-empty pointset S_t ⊂Π_q is called a t-semiarc if for every point P ∈ S_t there exist exatly t lines`1, `2, . . . `t such that S_t∩`i ={P}for

i = 1,2, . . . ,t.These lines are called the tangents toS_t at P.

If a line`meetsS_t in 2,3 or 3<k points, then `is called bisecant, trisecant ork-secant of S_t,respectively.

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Definition, Basic examples

Definition

LetΠq be a projective plane of order q.A non-empty pointset S_t ⊂Π_q is called a t-semiarc if for every point P ∈ S_t there exist exatly t lines`1, `2, . . . `t such that S_t∩`i ={P}for

i = 1,2, . . . ,t.These lines are called the tangents toS_t at P.

If a line`meetsS_t in 2,3 or 3<k points, then `is called bisecant, trisecant ork-secant of S_t,respectively.

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Examples

Example

k-arcs,t=q+ 2−k.

Semiovals,t = 1.

Subplanes,t =q−m,wherem is the order of the subplane.

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More examples

Proposition

LetS_t be a t-semiarc inΠ_q.The followings hold:

if t =q+ 1,thenS_t is a single point,

if t =q,thenS_t is a subset of a line, and vice versa any subset of a line containing at least two points is a q-semiarc, if t =q−1,thenS_t is a set of three non-collinear points.

There existt-semiarcs for each value oft satisfying 1≤t<q−1. Example

Let`₁ and `₂ be two lines of Π_q,and let 1≤t<q−1 be an arbitrary integer. If we delete the point`₁∩`₂ andt other points from both lines, then the remaining 2(q−t) points obviously form at-semiarc.

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More examples

Proposition

LetS_t be a t-semiarc inΠ_q.The followings hold:

if t =q+ 1,thenS_t is a single point,

if t =q,thenS_t is a subset of a line, and vice versa any subset of a line containing at least two points is a q-semiarc, if t =q−1,thenS_t is a set of three non-collinear points.

There existt-semiarcs for each value oft satisfying 1≤t<q−1.

Example

Let`₁ and `₂ be two lines of Π_q,and let 1≤t<q−1 be an arbitrary integer. If we delete the point`1∩`2 andt other points from both lines, then the remaining 2(q−t) points obviously form at-semiarc.

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Classification?

Hopeless. We have to add extra conditions.

small size,

big size, small q, bigt,

contained in the union of some lines. has some long secants,

regularity,

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Classification?

small size, big size,

small q, bigt,

regularity,

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Classification?

small size, big size, small q,

bigt,

regularity,

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Classification?

small size, big size, small q, bigt,

regularity,

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Classification?

contained in the union of some lines.

has some long secants, regularity,

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Classification?

has some long secants,

regularity,

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Classification?

has some long secants, regularity,

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Inequalities

Proposition

IfS_t is a t-semiarc in Π_q,then q−t+ 2≤ |S_t|.

Equality holds for any (q−t+ 2)-arc in Πq. Theorem (Inequality 8)

IfS_t is a t-semiarc in Π_q, then

|S_t| ≤1 +

$

q(t−1 +p

4tq−3t²+ 2t+ 1) 2t

% .

Equality holds for some values:

IfS_t is a unital, thent = 1 and|S_t|=q√ q+ 1; ifS_t is a Baer-subplane, then t =q−√

q and |S_t|=q+√ q+ 1.

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Inequalities

Proposition

IfS_t is a t-semiarc in Π_q,then q−t+ 2≤ |S_t|. Equality holds for any (q−t+ 2)-arc in Πq.

Theorem (Inequality 8)

|S_t| ≤1 +

$

q(t−1 +p

4tq−3t²+ 2t+ 1) 2t

% .

q and |S_t|=q+√ q+ 1.

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Inequalities

Proposition

IfS_t is a t-semiarc in Π_q,then q−t+ 2≤ |S_t|. Equality holds for any (q−t+ 2)-arc in Πq. Theorem (Inequality 8)

|S_t| ≤1 +

$

q(t−1 +p

4tq−3t²+ 2t+ 1) 2t

% .

q and |S_t|=q+√ q+ 1.

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Inequalities

Proposition

IfS_t is a t-semiarc in Π_q,then q−t+ 2≤ |S_t|. Equality holds for any (q−t+ 2)-arc in Πq. Theorem (Inequality 8)

|S_t| ≤1 +

$

q(t−1 +p

4tq−3t²+ 2t+ 1) 2t

% .

IfS_t is a unital, thent = 1 and|S_t|=q√ q+ 1;

ifS_t is a Baer-subplane, then t =q−√

q and |S_t|=q+√ q+ 1.

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Regular semiovals

The notion of regular semioval was introduced by de Finis.

Definition

LetS₁ be a semioval in Π_q.If all nontangent lines intersectS₁ in either 0 or a constant numberaof points, then S₁ is called regular semioval with character a.

Blokhuis and Sz˝onyi proved the following results. Theorem

LetS₁ be a regular semioval with character a inΠq.Then S₁ is an oval (thus a= 2), or a divides q−1 and the points not on S₁ are on 0 or on a tangents.

A consequence of this theorem is that the tangents ofS₁ form a regular semioval on the dual plane of Π_q.(This was proved by de Finis, too.)

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Regular semiovals

The notion of regular semioval was introduced by de Finis.

Definition

LetS₁ be a semioval in Π_q.If all nontangent lines intersectS₁ in either 0 or a constant numberaof points, then S₁ is called regular semioval with character a.

Blokhuis and Sz˝onyi proved the following results.

Theorem

LetS₁ be a regular semioval with character a in Πq.Then S₁ is an oval (thus a= 2), or a divides q−1 and the points not on S₁ are on 0 or on a tangents.

A consequence of this theorem is that the tangents ofS₁ form a regular semioval on the dual plane of Π_q.(This was proved by de Finis, too.)

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Regular semiovals

Theorem (A. Blokhuis, T. Sz˝onyi, 1992)

LetS₁ be a regular semioval with character a in PG(2,q).Then there are two possibilities:

S₁ is a unital (thus a=√ q+ 1);

a−1 and q are coprimes, and the tangents at collinear points of S₁ are concurrent.

The longstanding regular semioval conjecture in the Desarguesian planes was finally proved by G´acs.

Theorem (A. G´acs, 2006)

LetS₁ be a regular semioval in PG(2,q).Then S₁ is either an oval or a unital.

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Regular semiarcs

How can we define?

Definition

LetS_t be a t-semiarc in Π_q.If all nontangent lines intersect S_t in either 0 or a constant numberaof points, then S_t is called regular semiarc with character a.

Examples: subplanes

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Regular semiarcs

How can we define?

Definition

LetS_t be a t-semiarc in Π_q.If all nontangent lines intersect S_t in either 0 or a constant numberaof points, then S_t is called regular semiarc with character a.

Examples: subplanes

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Regular semiarcs

Another possible definition:

S_t is regular if equality holds in Inequality (8), hence|S_t|is maximal.

IfP ∈ S_t,then there are t tangents through P.IfR ∈ S/ _t,then there aren tangents through R.

Hence the tangents form a set of type (t,n) in the dual plane. There are several conditions (rememberVito’stalk yesterday!), e.g.

=⇒ n−t|q.

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Regular semiarcs

Another possible definition:

S_t is regular if equality holds in Inequality (8), hence|S_t|is maximal.

IfP ∈ S_t,then there are t tangents through P.IfR ∈ S/ _t,then there aren tangents through R.

Hence the tangents form a set of type (t,n) in the dual plane.

There are several conditions (rememberVito’stalk yesterday!), e.g.

=⇒ n−t|q.

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The small planes

Theorem

The spectrum of the sizes of semiovals in PG(2,q)is the following:

If q = 3 then|S₁| ∈ {4,6}.

If q = 5 then|S₁| ∈ {6,8,9,10,11,12}.

If q = 7 then|S₁| ∈ {8,9,12,13,14,15,16,17,18,19}.

If q = 9 then|S₁| ∈

{10,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28}.

Theorem

In PG(2,11)there are semiovals of size12,15,20,and for each s satisfying22≤s ≤34.

In PG(2,13)there are semiovals of size14,18,24,and for each integer s satisfying26≤s ≤40.

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The small planes

Theorem

In PG(2,2)each 2-semiarc S₂ consists of two or three collinear points.

In PG(2,3)each 2-semiarc S₂ is a set of 3 non-collinear points.

Theorem

InPG(2,4)there are three projectively non-equivalent2-semiarcs.

|S₂|= 4,four points in general position.

|S₂|= 6,the vertices of a complete quadrilateral.

|S₂|= 7,the points of a subplane of order 2.

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PG(2, 5)

Theorem

InPG(2,5)there are three projectively non-equivalent2-semiarcs.

|S₂|= 5,five points of a conic.

|S₂|= 6,the union of two trisecant s.

|S₂|= 9,the projective triangle.

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PG(2, 7)

Theorem

InPG(2,7)there are nine combinatorially non-equivalent 2-semiarcs (there are projectively non-equivalent subclasses in some combinatorial classes).

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PG(2, 7)

Theorem

|S₂|= 7,seven points of a conic.

|S₂|= 9,there are two types,

1 nine vertices of a3×3 grid,

2 the six vertices of two trianglesC1andC2,and the three points of intersections of the corresponding sides ofC₁ andC₂.

|S₂|= 10,there are two types,

1 the union of two 5-secants,

2 the points of a103configuration.

|S₂|= 11,there is no 5-secant, there are two types,

1 four 4-secants and four trisecants,

2 one 4-secant and ten trisecant s.

|S₂|= 12,then it has three 4-secants, there are two types

1 the 4-secants form a triangle,

2 one of the 4-secant is intersected by the two others in distinct points.

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Semiarcs contained in the union of two lines

Proposition

If a t-semiarcS_t is contained in the union of two lines `₁ and`₂ of Π_q and 1≤t<q−1,then|S_t∩`_i|=q−t for i = 1,2,andS_t does not contain the point`1∩`2.

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Semiovals contained in the union of three lines

Theorem

LetS₁ be a semioval in a projective plane Πq. If S₁ is contained in the union of three lines then

3(q−1)

2 ≤ |S| ≤3(q−1).

Theorem (Gy. K. and J. Ruff, 2006)

A semioval in PG(2,q) which is contained in the sides of a triangle and which contains one vertex of this triangle has a(q−2)-secant and two(t+ 1)-secants where t is a suitable integer. This type of semiovals exists if and only if q= 4 and t = 1,q = 8 and t = 4 or q= 32and t = 26.

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Semiovals contained in the union of three lines

Theorem (Gy. K. and J. Ruff, 2006)

If a semioval S in PG(2,q) is contained in the sides of a triangleT and does not contain any vertex ofT, then there are two

possibilities:

S₁ has two (q−1)-secants and a k-secant. Semiovals in this class exist for all 1<k<q.

S₁ has three(q−1−d)-secants where d is a suitable divisor of q−1.

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Three concurrent lines

Theorem

If a semiovalS₁ in PG(2,q) is contained in the union of three concurrent lines then|S|>3(q−1)/2 for q>9.

Definition

Let`1,`2 and `3 be the three concurrent lines whose union containsS. We denote by C the common point of these three lines and byL the union of`₁,`₂ and`₃. And finally, we let L_i =S ∩`i (i = 1,2,3). The semiovalS is strong, if for any point K ∈ L \(S ∪ {C}),the number of two-secants of S passing throughK is independent of K.

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Semiarcs contained in three concurrent lines

`1,`2 and`3 the three concurrent lines whose union contains S_t, V =`1∩`2∩`3,P_V : pencil of lines with carrierV,

L_i =S_t∩`_i,u_i =|L_i|.

Proposition

LetS_t be a t-semiarc inΠ_q,suppose that S_t is contained in the union of three lines ofP_V,but does not contained in the union of any two lines ofP_V.If V ∈ S_t,then there are two possibilities.

S_t consists of the six vertices of a complete quadrilateral, S_t is a Fano subplane.

In both cases t=q−2.

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Semiarcs contained in three concurrent lines

`1,`2 and`3 the three concurrent lines whose union contains S_t, V =`1∩`2∩`3,P_V : pencil of lines with carrierV,

L_i =S_t∩`_i,u_i =|L_i|.

Proposition

LetS_t be a t-semiarc inΠ_q,suppose that S_t is contained in the union of three lines ofP_V,but does not contained in the union of any two lines ofP_V.If V ∈ S_t,then there are two possibilities.

S_t consists of the six vertices of a complete quadrilateral, S_t is a Fano subplane.

In both cases t=q−2.

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Three concurrent lines II

Theorem (V ∈ S/ _t)

LetS_t be a t-semiarc inΠq,suppose that S_t is contained in the union of three lines ofP_V,but does not contained in the union of any two lines ofP_V.If V ∈ S/ _t,then there are three possibilities.

1 u₁=u₂ =u₃ =u,and 3·q−t

2 ≤ |S_t| ≤3· q+t 2−

r

qt+t² 4

! .

2 ui =uj =q−t and 2≤uk ≤t holds for{i,j,k}={1,2,3}.

The inequalities

2q−2t+ 2≤ |S_t| ≤2q−t (1) also hold in this case.

3 S_t is a 5-arc and t =q−3.

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Again an example

If Π_q contains a sublane of order t,then there exist t-semiarcs of each possible size allowed by Inequality (1).

Example

Suppose that Πq contains a subplane Πt of order t.Let`1, `2 and

`₃ be lines of Π_t.Let Z ⊆(`₃∩Π_t)\ {V} an arbitrary subset satisfying the inequalities 2≤ |Z| ≤t. Finally let

S_t:= (`₁\Π_t)∪(`₂\Π_t)∪Z.

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PG(2, q), algebraic description

S_t ⇐⇒ ordered triple (A,B,C), whereA,B,C ⊂GF(q).

The lines`1,`2, and`3 have equationsX1 = 0,X1 =X3 and X₃ = 0, respectively. V = (0,1,0).

A={a∈GF(q) : (0,a,1)∈ L/ ₁}, B ={b∈GF(q) : (1,b,1)∈ L/ ₂}, C ={c ∈GF(q) : (1,c,0)∈ L/ ₃}.

(0,a,1),(1,b,1),(1,c,0)collinear ⇐⇒a+c =b.

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PG(2, q), algebraic description

S_t ⇐⇒ ordered triple (A,B,C), whereA,B,C ⊂GF(q).

The lines`1,`2, and`3 have equationsX1 = 0,X1 =X3 and X₃ = 0, respectively. V = (0,1,0).

A={a∈GF(q) : (0,a,1)∈ L/ ₁}, B ={b∈GF(q) : (1,b,1)∈ L/ ₂}, C ={c ∈GF(q) : (1,c,0)∈ L/ ₃}.

(0,a,1),(1,b,1),(1,c,0)collinear ⇐⇒a+c =b.

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Triangular case

The line`_i has equationX_i = 0.

(0,a,1),(b,0,1),(1,c,0)collinear ⇐⇒ac =−b.

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Triangular case

The line`_i has equationX_i = 0.

(0,a,1),(b,0,1),(1,c,0)collinear ⇐⇒ac =−b.

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Definitions from Additive Group Theory

Definition

Let A and B be finite, nonempty subsets of an abelian group (Z,),and let i ≥1 an integer.

Let N_i(A,B) all the elements c with at least i representations of the form c =ab with a∈Aandb ∈B.Sometimes we use the shorthand notation Ni instead ofNi(A,B).

Let stab(A) :={z ∈Z :Az =A}.This set is called the stabilizer of A.

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Theorems from Additive Group Theory

Theorem (Exact inverse sumset theorem)

Suppose that A and B are finite nonempty subsets of the abelian group Z.Then the following are equivalent.

|A+B|=|A|.

|A−B|=|A|.

Let G :=stab(A).Then G is a finite subgroup of Z,B is contained in a coset of G,and A is the union of cosets of of G.

Theorem (Pollard, 1974)

Let Z be an abelian group,|Z|=p prime, A,B ⊆G nonempty subsets, and1≤k ≤min{|A|,|B|}.Then

|N₁|+|N₂|+. . .+|N_k| ≥k·min{p,|A|+|B| −k}.

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Theorems from Additive Group Theory

Theorem (Exact inverse sumset theorem)

Suppose that A and B are finite nonempty subsets of the abelian group Z.Then the following are equivalent.

|A+B|=|A|.

|A−B|=|A|.

Let G :=stab(A).Then G is a finite subgroup of Z,B is contained in a coset of G,and A is the union of cosets of of G.

Theorem (Pollard, 1974)

Let Z be an abelian group,|Z|=p prime, A,B ⊆G nonempty subsets, and1≤k ≤min{|A|,|B|}.Then

|N₁|+|N₂|+. . .+|N_k| ≥k·min{p,|A|+|B| −k}.

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Additive Group theory II

Theorem (Grynkiewicz, 2010)

Let Z be an abelian group, A,B⊆Z finite and nonempty subsets, and k ≥1.If|A|,|B| ≥k,then either

k

X

i=1

|N_i| ≥k(|A|+|B|)−2k²+ 1,

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Additive Group theory III

Theorem

or else there exist A⁰⊆A and B⁰⊆B with l :=

A\A⁰ +

B\B⁰

≤k−1, N_k(A⁰,B⁰) =N1(A⁰,B⁰) =N_k(A,B),

k

X

i=1

|N_k| ≥k(|A|+|B|)−(k−l)(|H| −ρ)−kl ≥k(|A|+|B| − |H|), where H is the nontrivial stabilizer of Nk(A,B)and

ρ=|A⁰H| − |A⁰|+|B⁰H| − |B⁰|. In the case k = 2instead of the first inequality|N₁|+|N₂| ≥2(|A|+|B|)−4 also holds.

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Application of the Inverse Sumset Thm

Theorem

Suppose that the t-semiarcS_t in PG(2,p^r),p odd prime, belongs to the family of Case 2 of Theorem V ∈ S/ _t. Then there exists a subgroup G of E such that both A and C are union of cosets of G, and B is contained in a coset of G.

Ifφis the natural homomorphism from E to E/G,|G|=g and

|φ(C)|=h,then t =gh and|S_t|= 2p^r −2gh+|B|.

Corollary

Let p be an odd prime. Then the followings hold.

1 In PG(2,p) there is no semiarc belonging to the family of Case 2 of Theorem V ∈ S/ _t.

2 Let 1≤e <r be integers and let t =p^es,where(p,s) = 1 and t <p^r.Then PG(2,p^r) contains t-semiarcs with cardinality2p^r −2t+k for all t and k satisfying the conditions 2≤k ≤p^e.

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Semiovals

Theorem

LetS₁ be a semioval in the plane PG(2,q),q =p^r,p odd prime.

Suppose thatS₁ is contained in the union of three lines ofP_V, but does not contained in the union of any two lines ofP_V.Then

|S₁| ≥3q−3f_r(q),where

fr(q) =









 2d√

p+ 1e −2 ifr = 1, 4

qq+1 2

−4 ifr = 2, q^r−1^r +q¹^r −1 ifr ≥3.

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Strong semiovals

Theorem (B. Csajb˜ok, Gy. K., 2012)

LetS₁ be a strong semioval in PG(2,p^r),p an odd prime. Then the followings hold.

1 If r = 2l,then S₁ contains 3(p^2l−p^l) points.

2 If r = 2l+ 1and p>7,then there is no strong semioval in PG(2,p^r).

3 If r = 2l+ 1and p= 3,5 or 7,thenS₁ contains 3(p^2l+1−p^l⁺¹)points.

Conjecture

The projective plane PG(2,q),q odd, contains strong semiovals if and only if q is a square.

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Strong semiovals

Theorem (B. Csajb˜ok, Gy. K., 2012)

LetS₁ be a strong semioval in PG(2,p^r),p an odd prime. Then the followings hold.

1 If r = 2l,then S₁ contains 3(p^2l−p^l) points.

2 If r = 2l+ 1and p>7,then there is no strong semioval in PG(2,p^r).

3 If r = 2l+ 1and p= 3,5 or 7,thenS₁ contains 3(p^2l+1−p^l⁺¹)points.

Conjecture

The projective plane PG(2,q),q odd, contains strong semiovals if and only if q is a square.

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2-semiarcs

Theorem

LetS₂ be a 2-semiarc in PG(2,q),q=p^r,p odd prime. Suppose thatS₂ belongs to the family of Case 1 of Theorem V ∈ S/ _t.Then

|S₂| ≥3q−3fr(p),where

f_r(p) =









 2d√

2p+ 4e −4 if r = 1, 4

q p²+⁷₂

−8 if r = 2,

14,37,66 if r = 3 and p = 3,5,7, p²+ 2p+ 2 if r = 3 and p ≥11, p^r⁻¹+ 2p−2 if r ≥4.

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The next example

Example

LetE be the direct sum of the subgroupsX andY.Then the sets A=B =C :=X ∪Y define a 2-semiarcS₂.This semiarc has 3q−3(|X|+|Y| −1) points, hence if q=p^r and |X|=p^r−1,

|Y|=p,then |S₂|= 3q−3(p^r−1+p−1).

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Semiarcs with long secants

Proposition

LetS_t be a t-semiarc and` be a line in Π_q.Suppose that

|S_t∩`|=k.Then

k ≤q+ 1−t, (2)

k+q−t ≤ |S_t| ≤k+(q+ 1−k)q

t . (3)

These inequalities are sharp.

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Semiarcs with long secants

Proposition

LetS_t be a t-semiarc and` be a line in Π_q.Suppose that

|S_t∩`|=k.Then

k ≤q+ 1−t, (2)

k+q−t ≤ |S_t| ≤k+(q+ 1−k)q

t . (3)

These inequalities are sharp.

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Long secants

Theorem (B. Csajb´ok)

InPG(2,q), a t-semiarc with a (q+ 1−t)-secant exists if and only if t≥(q−1)/2.

Proposition (no. 44)

LetΠq be a projective plane of order q. If S_t is a t-semiarc of size 2(q−t) with a(q−t)-secant `, thenS_t consists of the symmetric difference of two lines, with t further points removed from each line.

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Semiarcs with two long secants

Lemma (B. Csajb´ok, 2012)

LetΠ_q be a projective plane of order q,1<t<q an integer and S_t be a t-semiarc in Π_q. Suppose that there exist two lines `₁ and

`2 such that|`₁\(S_t∪`2)|=n and|`₁\(S_t∪`2)|=m. If

`₁∩`₂ ∈ S/ _t, then either n=m=t or q≤n+ 2nm/(t−1).

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t-semiarcs in PG(2, q) with two (q − t)-secants

If`₁ and`₂ are (q−t)-secants, then each point inS_t\(`₁∪`₂) is a center of a perspectivity mapping`1\ S_t to`2\ S_t.The

description of perspective pointsets was given by Korchm´aros and Mazzocca. Applying their result Csajb´ok proved the folliwing.

Theorem (B. Csajb´ok, 2012)

LetS_t be a t-semiarc inPG(2,q), q =p^h, p prime, with two (q−t)-secants such that the point of intersection of these secants is not contained inS_t. Then the following hold.

1 If gcd(q,t) = 1, thenS_t is contained in a vertexless triangle.

2 If gcd(q,t) = 1 and gcd(q−1,t−1) = 1, thenS_t consists of the symmetric difference of two lines, with t further points removed from each line.

3 If gcd(q−1,t) = 1, then S_t is contained in a vertexless triangle, or in the union of three concurrent lines with their common point removed.

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Small semiarcs with long secants

The following example shows the existence oft-semiarcs of size k+q−t with threek-secants for odd values oft.

Example (no. 47)

LetC denote the set of non-squares inGF(q),q odd. The pointset{(0 : 1 :s),(s : 0 : 1),(1 :s : 0) : −s ∈C} is a semioval inPG(2,q) of size 3(q−1)/2 with three (q−1)/2-secants. If we deleter <(q−1)/2−1 points from each of the (q−1)/2-secants, then the remaining pointset is at-semiarc of sizek+q−t with threek-secants, wherek = (q−1)/2−r andt = 2r+ 1.

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Long secants, t even

A (q+t,t)-arc of type (0,2,t),t 6= 0,2, in Π_q is a setT of q+t points inPG(2,q) for which each line `meetsT in 0, 2 ort points. Korchm´aros and Mazzocca proved that (q+t,t)-arcs of type (0,2,t) exist inPG(2,q) only if q is even and t|q.

Example

LetT be a (q+τ, τ)-arc of type (0,2, τ) inPG(2,q). Delete r< τ −1 points from each of theτ-secants of T. The remaining k+q−t points form at-semiarc withq/τ+ 1k-secants, where k=τ−r andt=rq/τ.

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Long secants

Example (no. 49)

Let Π^√_q be a Baer subplane in the projective plane Π_q,q ≥9, and let`be a line of Π^√_q. LetP be a set of t points in Π^√_q\`such that no line intersectsP in exactly √

q−1 points. For example a (t,√

q−2)-arc is a good choice for P. Let T be a set of t points in`\Π^√_q. Then the pointsetS_t := (Π^√_q4`)\(T ∪ P) is a t-semiarc of sizek+q−t with ak-secant, where k =q−√

q−t.

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Long secants and the direction problem

The so-called direction problem is closely related tot-semiarcs of sizek+q−t having ak-secant. Consider

PG(2,q) =AG(2,q)∪`∞. LetU be a set of points inAG(2,q).

A pointP of `∞ is called a direction determined by U if there is a line throughP that contains at least two points of U.

Theorem

LetS_t be a t-semiarc inPG(2,q) q=p^h, p prime, of size k+q−t and let` be a k-secant of S_t. Then the conditions

t = 1, q >4 and k >(q−1)/2, or 2≤t ≤α√

q and k > α(q+ 1)for some 1/2≤α≤p

(p−1)/p if p is an odd prime and 1/2≤α≤√

3/2 if p= 2

imply thatS_t is either a semiarc described in Example 49, or k=q−t and S_t is the semiarc described in Proposition 44.

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Long secants

Corollary

LetS₁ be a semioval of size k +q−1 inPG(2,q), q=p^h, p prime, h≤2, and let `be a k-secant of S₁. Then we have the following.

1 If h = 1and k >(p+ 4)/3, then there are two possibilities:

k =q−1 andS1is the semioval described in Proposition 44, S1is the semioval described in Example 47.

2 If h= 2 and k >(p²−p)/2, then there are four possibilities:

k =q−1 andS₁is the semioval described in Proposition 44, S1is the semioval described in Example 47,

S1is the semioval described in Example 49, p= 2 andS1 is an oval inPG(2,4).

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Semiovals with a long secants

Theorem (Gy. K., 2004)

LetS₁ be a semioval in the Desarguesian plane PG(2,q). If there exist integers1≤r and−1≤k such that|S₁|= 2q−r+k, r+ 4k+ 4<q,2(r+k)<q andS₁ has a(q−r)-secant, then the tangent lines at the points of the(q−r)-secant are concurrent.

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A much more general result

Lemma (B. Csajb´ok, T. H´eger, Gy. K., 2013)

LetS_t be a set of points inPG(2,q), let `be a k-secant of S_t with k≤q and let1≤t ≤q−3 be an integer. Suppose that through each point of`∩ S_t there pass exactly t tangent lines to S_t. Denote by s the size ofS_t and let s =k+q−t+. Let A(n) be the set of those points in`\ S_t through which there pass at most n skew lines ofS_t. Then the following hold.

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A much more general result

Lemma

If t = 1 and

1 < ^k₂−1, then the k tangent lines at the points ofS1∩`and the skew lines through the points of A(2)belong to a pencil (hence A(2)\A(1)is empty),

2 if < ^2k₃ −2,then the k tangent lines at the points ofS1∩` either belong to two pencils or they form a dual k-arc. If k <q, then the skew lines through the points of A(2)belong to the same pencils or dual k-arc.

If t ≥2 and k >q−^q_t + 1, then

3 if < _t+1^k −^t₂, then the kt tangent lines at the points of S₁∩`and the skew lines through the points of A(t+ 1) belong to t pencils whose carriers are not on`(hence A(t+ 1)\A(t)is empty),

4 if < _t+1^k −1and t ≤√

q, then the kt tangent lines at the points ofS1∩`belong to t+ 1pencils whose carriers are not on`. If k <q, then the skew lines through the points of A(t+ 1)belong to the same pencils.

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Some corollaries

Corollary

LetS₁ be a semioval in PG(2,q)and let `be a k-secant of S₁.If

|S₁|<q+^3k₂ −2,then the k tangent lines at the points ofS₁∩` belong to a pencil. If|S₁|<q+^5k₃ −3,then the k tangent lines at the points ofS₁∩`either belong to two pencils or they form a dual k-arc.

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Semiovals and blocking sets

Lemma

Let k ≤q and 1≤t ≤q−3 be integers. Let S_t be a set of k+q−t+points in Π_q such that the line `is a k-secant of S_t. Let A(n) be the set of those points in `\ S_t through which there pass at most n skew lines ofS_t. Suppose that through each of the k points of`∩ S_t there pass exactly t tangent lines toS_t, and also suppose that these kt tangent lines and the skew lines through the points of A(n) belong to n pencils. Let P be the set of carriers of these pencils and assume thatP ∩`=∅. Define the pointset B_n(S_t, `) in the following way:

B_n(S_t, `) := (`\(A(n)∪ S_t))∪(S_t\`)∪ P.

ThenB_n(S_t, `)has size 2q+ 1 ++n−t−k− |A(n)|and it is an affine blocking set in the affine planeΠ_q\`or a blocking set inΠ_q. In the latter case the points of`∩ B_n(S_t, `) are essential points.

We distinguish two cases, as the results on blocking sets in PG(2,q) are stronger if q is a prime.

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Semiovals and blocking sets

Lemma

Let k ≤q and 1≤t ≤q−3 be integers. Let S_t be a set of k+q−t+points in Π_q such that the line `is a k-secant of S_t. Let A(n) be the set of those points in `\ S_t through which there pass at most n skew lines ofS_t. Suppose that through each of the k points of`∩ S_t there pass exactly t tangent lines toS_t, and also suppose that these kt tangent lines and the skew lines through the points of A(n) belong to n pencils. Let P be the set of carriers of these pencils and assume thatP ∩`=∅. Define the pointset B_n(S_t, `) in the following way:

B_n(S_t, `) := (`\(A(n)∪ S_t))∪(S_t\`)∪ P.

ThenB_n(S_t, `)has size 2q+ 1 ++n−t−k− |A(n)|and it is an affine blocking set in the affine planeΠ_q\`or a blocking set inΠ_q. In the latter case the points of`∩ B_n(S_t, `) are essential points.

We distinguish two cases, as the results on blocking sets in PG(2,q) are stronger if q is a prime.

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q prime

Theorem

LetS_t be a t-semiarc in PG(2,p), p prime, and let`be a k-secant ofS_t.

1 If t = 1,p ≥5 and k ≥ ^p−1₂ , then

S1is contained in a vertexless triangle and has two (p−1)-secants, or

S1is projectively equivalent to Example 47, or

|S1| ≥min{^3k₂ +p−2,2k+^p+1₂ }.

2 If t = 2,p ≥7 and k ≥ ^p+3₂ , then

S2consists of the symmetric difference of two lines, with two further points removed from each line, or

|S2| ≥min{^4k₃ +p−3,2k+^p−1₂ }.

3 If3≤t <√

p,p ≥23 and k >p−^p_t + 1, then St is contained in a vertexless triangle and has two (p−t)-secants, or

|St| ≥k^t+2_t+1+p−t−1.

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q is a prime pover

Theorem

LetS_t be a t-semiarc inPG(2,q), q =p^h, h>1 if p is an odd prime and h≥6if p= 2. Let the size of the smallest minimal non-trivial blocking set inPG(2,q) be denoted by q+ 1 +b(q), and suppose thatS_t has a k-secant`with k ≥q−b(q)−t. Then the following hold.

1 If h = 2d ,|S_t| ≤2k+b(q) and t <Φ(√

q−1), then

S_t has two(q−t)-secants, having no common point in S_t, or S_t is as in Example 49.

2 If h = 2d + 1,|S_t|<2k+b(q) and t <q^1/3−3/2 (or t <(2q)^1/3−2,when p = 2,3), thenS_t has two (q−t)-secants whose common point is not in S_t.

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