Generalized Hyperfocused Arcs

Seconda Universit`a degli Studi di Napoli (a Caserta) Dipartimento di Matematica e Fisica

Generalized Hyperfocused Arcs

Francesco Mazzocca

(jont work with Aart Blokhuis and Giuseppe Marino)

Finite Geometry Conference and Workshop University of Szeged

10-14 June, 2013

Arcs and blocking sets

DEFINITION

Ak−arc in PG(2,q) is a set ofk points with no 3 on a line.

A line containing 1 or 2 points of ak−arc is said to be a tangent orsecant to thek−arc, respectively.

DEFINITION

LetF be a set of lines inPG(2,q).Ablocking set ofF is a set of pointsB ⊂PG(2,q) having non-empty intersection with each line inF.If this is the case, we also say that the lines in F areblocked byB.

Arcs and blocking sets

DEFINITION

Ak−arc in PG(2,q) is a set ofk points with no 3 on a line.

A line containing 1 or 2 points of ak−arc is said to be a tangent orsecant to thek−arc, respectively.

DEFINITION

LetF be a set of lines inPG(2,q).Ablocking set ofF is a set of pointsB ⊂PG(2,q) having non-empty intersection with each line inF.If this is the case, we also say that the lines in F are blocked byB.

The group associated to a conic and a line in PG (2, F )

Fany field

Let`and Γ be a line and a non singular conic in PG(2,F), respectively. An abelian group G on Γ\` can be defined in the following way:

choose a point O ∈Γ\`as the identity of the group; for any two pointsP,Q ∈Γ\`, letR⁰ be the point that the line through P,Q has in common with`;

the sum of P and Q is defined by P +Q =R, whereR is the second of the two points (counted with multiplicity) common to the line OR⁰ and Γ\`.

The group associated to a conic and a line in PG (2, F )

Fany field

Let`and Γ be a line and a non singular conic in PG(2,F), respectively. An abelian group G on Γ\`can be defined in the following way:

choose a point O ∈Γ\`as the identity of the group; for any two pointsP,Q ∈Γ\`, letR⁰ be the point that the line through P,Q has in common with`;

the sum of P and Q is defined by P +Q =R, whereR is the second of the two points (counted with multiplicity) common to the line OR⁰ and Γ\`.

The group associated to a conic and a line in PG (2, F )

Fany field

Let`and Γ be a line and a non singular conic in PG(2,F), respectively. An abelian group G on Γ\`can be defined in the following way:

choose a point O ∈Γ\`as the identity of the group;

for any two pointsP,Q ∈Γ\`, letR<sup>0</sup> be the point that the line through P,Q has in common with`;

the sum of P and Q is defined by P +Q =R, whereR is the second of the two points (counted with multiplicity) common to the line OR⁰ and Γ\`.

The group associated to a conic and a line in PG (2, F )

Fany field

Let`and Γ be a line and a non singular conic in PG(2,F), respectively. An abelian group G on Γ\`can be defined in the following way:

choose a point O ∈Γ\`as the identity of the group;

for any two pointsP,Q ∈Γ\`, letR<sup>0</sup> be the point that the line through P,Q has in common with`;

the sum of P and Q is defined by P +Q =R, whereR is the second of the two points (counted with multiplicity) common to the line OR⁰ and Γ\`.

The group associated to a conic and a line in PG (2, F )

Fany field

Let`and Γ be a line and a non singular conic in PG(2,F), respectively. An abelian group G on Γ\`can be defined in the following way:

choose a point O ∈Γ\`as the identity of the group;

for any two pointsP,Q ∈Γ\`, letR⁰ be the point that the

Dual 3−nets in PG (2, F )

Fany field

DEFINITION

Adual 3-net embedded inPG(2,F), is a triple{A,B,C} with A,B,C pairwise disjoint point-sets of size n, calledcomponents, such that every line meeting two distinct components meets each component in precisely one point.

EXAMPLE

Let`and Γbe a line and a non singular conic in PG(2,F), respectively. Let G be the abelian group associated toΓand `.

Now, given a proper subgroup A of G of finite order n and one of its cosets B (6=A), the set C of the points of `on some line intersecting both A and B has exactly n points. Thenthe triple {A,B,C} is a dual3−net of order n embedded in PG(2,F).

Dual 3−nets in PG (2, F )

Fany field

DEFINITION

Adual 3-net embedded inPG(2,F), is a triple{A,B,C} with A,B,C pairwise disjoint point-sets of size n, calledcomponents, such that every line meeting two distinct components meets each component in precisely one point.

EXAMPLE

Let`and Γbe a line and a non singular conic in PG(2,F), respectively.

Let G be the abelian group associated toΓand `.

Now, given a proper subgroup A of G of finite order n and one of its cosets B (6=A), the set C of the points of `on some line intersecting both A and B has exactly n points. Thenthe triple {A,B,C} is a dual3−net of order n embedded in PG(2,F).

Dual 3−nets in PG (2, F )

Fany field

DEFINITION

Adual 3-net embedded inPG(2,F), is a triple{A,B,C} with A,B,C pairwise disjoint point-sets of size n, calledcomponents, such that every line meeting two distinct components meets each component in precisely one point.

EXAMPLE

Let`and Γbe a line and a non singular conic in PG(2,F), respectively. Let G be the abelian group associated toΓand`.

Now, given a proper subgroup A of G of finite order n and one of its cosets B (6=A), the set C of the points of `on some line intersecting both A and B has exactly n points. Thenthe triple {A,B,C} is a dual3−net of order n embedded in PG(2,F).

Dual 3−nets in PG (2, F )

Fany field

DEFINITION

Adual 3-net embedded inPG(2,F), is a triple{A,B,C} with A,B,C pairwise disjoint point-sets of size n, calledcomponents, such that every line meeting two distinct components meets each component in precisely one point.

EXAMPLE

Let`and Γbe a line and a non singular conic in PG(2,F), respectively. Let G be the abelian group associated toΓand`.

Then the triple {A,B,C} is a dual3−net of order n embedded in PG(2,F).

Dual 3−nets in PG (2, F )

Fany field

DEFINITION

Adual 3-net embedded inPG(2,F), is a triple{A,B,C} with A,B,C pairwise disjoint point-sets of size n, calledcomponents, such that every line meeting two distinct components meets each component in precisely one point.

EXAMPLE

Let`and Γbe a line and a non singular conic in PG(2,F), respectively. Let G be the abelian group associated toΓand`.

Now, given a proper subgroup A of G of finite order n and one of its cosets B (6=A), the set C of the points of `on some line intersecting both A and B has exactly n points. Thenthe triple {A,B,C}is a dual 3−net of order n embedded in PG(2,F).

Dual 3−nets in PG (2, F )

A useful result

The following theorem follows from the main result in the paper Blokhuis A., Korchm´aros G. & M.F. - 2011

On the structure of 3-nets embedded in a projective plane,Journal of Combinatorial Theory, Series A; 0097-3165; ; Vol.118 (2011);

pp. 1228-1238.

Theorem

Let{A,B,C} be a dual3-net of order n in PG(2,F). Then, if C is contained in a line andF has positive characteristic p≥n, A∪B is contained in a conic. If this conic is irreducible then it is of type described in the previous example.

Dual 3−nets in PG (2, F )

A useful result

The following theorem follows from the main result in the paper Blokhuis A., Korchm´aros G. & M.F. - 2011

On the structure of 3-nets embedded in a projective plane,Journal of Combinatorial Theory, Series A; 0097-3165; ; Vol.118 (2011);

pp. 1228-1238.

Theorem

Let{A,B,C} be a dual3-net of order n in PG(2,F). Then, if C is contained in a line andF has positive characteristic p≥n, A∪B is contained in a conic. If this conic is irreducible then it is of type described in the previous example.

Generalized hyperfocused arcs

The definition

DEFINITION

Ageneralized hyperfocused arcHin PG(2,q) (M.Giulietti - E.Montanucci, 2006) is ak-arc with the property that the

k(k−1)/2 secants can be blocked by a set Bof k−1 points not belonging to the arc.

WhenB is contained in a line, His simply called ahyperfoused arc (W.Cherowitzo - L.Holder, 2005).

Points of the arc Hwill be calledwhite points and points of the blocking set Bblack points.

In casek >1, the secant lines toHtrough a fixed black point induce a partition into 2-sets ofH; so k must be even.

Moreover, the k−1 black points induce a factorization (i.e. a partition into

matchings)of the white k-arc.

Generalized hyperfocused arcs

The definition

DEFINITION

Ageneralized hyperfocused arcHin PG(2,q) (M.Giulietti - E.Montanucci, 2006) is ak-arc with the property that the

k(k−1)/2 secants can be blocked by a set Bof k−1 points not belonging to the arc. WhenB is contained in a line,H is simply called ahyperfoused arc (W.Cherowitzo - L.Holder, 2005).

Points of the arc Hwill be calledwhite points and points of the blocking set Bblack points.

In casek >1, the secant lines toHtrough a fixed black point induce a partition into 2-sets ofH; so k must be even.

Moreover, the k−1 black points induce a factorization (i.e. a partition into

matchings)of the white k-arc.

Generalized hyperfocused arcs

The definition

DEFINITION

Ageneralized hyperfocused arcHin PG(2,q) (M.Giulietti - E.Montanucci, 2006) is ak-arc with the property that the

k(k−1)/2 secants can be blocked by a set Bof k−1 points not belonging to the arc. WhenB is contained in a line,H is simply called ahyperfoused arc (W.Cherowitzo - L.Holder, 2005).

Points of the arc Hwill be calledwhite points and points of the blocking set Bblack points.

In casek >1, the secant lines toHtrough a fixed black point induce a partition into 2-sets ofH; so k must be even.

Moreover, the k−1 black points induce a factorization (i.e. a partition into

matchings)of the white k-arc.

Generalized hyperfocused arcs

The definition

DEFINITION

Ageneralized hyperfocused arcHin PG(2,q) (M.Giulietti - E.Montanucci, 2006) is ak-arc with the property that the

k(k−1)/2 secants can be blocked by a set Bof k−1 points not belonging to the arc. WhenB is contained in a line,H is simply called ahyperfoused arc (W.Cherowitzo - L.Holder, 2005).

Points of the arc Hwill be calledwhite points and points of the blocking set Bblack points.

In casek >1, the secant lines toHtrough a fixed black point induce a partition into 2-sets ofH; so k must be even.

Moreover, the k−1 black points induce a factorization (i.e. a partition into

matchings)of the white k-arc.

Generalized hyperfocused arcs

The definition

DEFINITION

Ageneralized hyperfocused arcHin PG(2,q) (M.Giulietti - E.Montanucci, 2006) is ak-arc with the property that the

k(k−1)/2 secants can be blocked by a set Bof k−1 points not belonging to the arc. WhenB is contained in a line,H is simply called ahyperfoused arc (W.Cherowitzo - L.Holder, 2005).

Points of the arc Hwill be calledwhite points and points of the blocking set Bblack points.

In casek >1, the secant lines toHtrough a fixed black point induce a partition into 2-sets ofH; so k must be even.

Moreover, the k−1 black

A trivial example of generalized hyperfocused arc

Fork = 2,there is only a trivial example of generalized hyperfocused arcH(in fact it is a hyperfocused arc):

Bconsists of a unique black point out ofHon the line through the two white points ofH.

A trivial example of generalized hyperfocused arc

Fork = 2,there is only a trivial example of generalized hyperfocused arcH(in fact it is a hyperfocused arc):

Bconsists of a unique black point out ofHon the line through the two white points ofH.

Generlized hyperfocused and hyperfocused arcs

Some remarks

Hyperfocused arcs only exist ifq is even (Bichara - Korchm´aros, 1980).

Although many results are known about hyperfocused arcs and their generalizations (Cherowitzo, Holder, Giulietti, Korchm´aros, Lanzone, Montanucci, Parrettini, Pasticci, Siciliano, Sonnino,...), there are still many open problems concerning them.

It is known that there exist examples of generalized

hyperfocused arcs which are not hyperfocused, providedq is even(M.Giulietti - E.Montanucci, 2006).

The study of hyperfocused arcs is motivated by a relevant application to cryptography in connection with constructions of efficient secret sharing schemes (G.Simmons, 1990; L.Holder, 1997).

Generlized hyperfocused and hyperfocused arcs

Some remarks

Hyperfocused arcs only exist ifq is even(Bichara - Korchm´aros, 1980).

Although many results are known about hyperfocused arcs and their generalizations (Cherowitzo, Holder, Giulietti, Korchm´aros, Lanzone, Montanucci, Parrettini, Pasticci, Siciliano, Sonnino,...), there are still many open problems concerning them.

It is known that there exist examples of generalized

hyperfocused arcs which are not hyperfocused, providedq is even(M.Giulietti - E.Montanucci, 2006).

The study of hyperfocused arcs is motivated by a relevant application to cryptography in connection with constructions of efficient secret sharing schemes (G.Simmons, 1990; L.Holder, 1997).

Generlized hyperfocused and hyperfocused arcs

Some remarks

Hyperfocused arcs only exist ifq is even(Bichara - Korchm´aros, 1980).

Although many results are known about hyperfocused arcs and their generalizations (Cherowitzo, Holder, Giulietti, Korchm´aros, Lanzone, Montanucci, Parrettini, Pasticci, Siciliano, Sonnino,...), there are still many open problems concerning them.

It is known that there exist examples of generalized

hyperfocused arcs which are not hyperfocused, providedq is even(M.Giulietti - E.Montanucci, 2006).

The study of hyperfocused arcs is motivated by a relevant application to cryptography in connection with constructions of efficient secret sharing schemes (G.Simmons, 1990; L.Holder, 1997).

Generlized hyperfocused and hyperfocused arcs

Some remarks

Hyperfocused arcs only exist ifq is even(Bichara - Korchm´aros, 1980).

Although many results are known about hyperfocused arcs and their generalizations (Cherowitzo, Holder, Giulietti, Korchm´aros, Lanzone, Montanucci, Parrettini, Pasticci, Siciliano, Sonnino,...), there are still many open problems concerning them.

It is known that there exist examples of generalized

hyperfocused arcs which are not hyperfocused, providedq is even(M.Giulietti - E.Montanucci, 2006).

The study of hyperfocused arcs is motivated by a relevant application to cryptography in connection with constructions of efficient secret sharing schemes (G.Simmons, 1990; L.Holder, 1997).

Generlized hyperfocused and hyperfocused arcs

Some remarks

Hyperfocused arcs only exist ifq is even(Bichara - Korchm´aros, 1980).

Although many results are known about hyperfocused arcs and their generalizations (Cherowitzo, Holder, Giulietti, Korchm´aros, Lanzone, Montanucci, Parrettini, Pasticci, Siciliano, Sonnino,...), there are still many open problems concerning them.

It is known that there exist examples of generalized

hyperfocused arcs which are not hyperfocused, providedq is even(M.Giulietti - E.Montanucci, 2006).

The study of hyperfocused arcs is motivated by a relevant application to cryptography in connection with constructions of efficient secret sharing schemes (G.Simmons, 1990;

L.Holder, 1997).

A bank president problem

In a bank there area president, n first level andm second level employees. The president knows the combination to the vault and does not want to share this combination with any individual other than himself.

The problem is to find a way to give out parts (shares) of the combination (the secret) to any employees so that, combining their information, the vault can be open by

any two of the first level employees, or

any one of the first level together with any two of the second level employees, or

any three second level employees;

andthe previous ones are the only minimal combinations of employees able to open the vault.

These set of combinations is called anaccess structure and a solution to the problem is an example of asecret sharing scheme.

A bank president problem

In a bank there area president, n first level andm second level employees. The president knows the combination to the vault and does not want to share this combination with any individual other than himself.

The problem is to find a way to give out parts (shares) of the combination (the secret) to any employees so that, combining their information, the vault can be open by

any two of the first level employees, or

any one of the first level together with any two of the second level employees, or

any three second level employees;

andthe previous ones are the only minimal combinations of employees able to open the vault.

These set of combinations is called anaccess structure and a solution to the problem is an example of asecret sharing scheme.

A bank president problem

In a bank there area president, n first level andm second level employees. The president knows the combination to the vault and does not want to share this combination with any individual other than himself.

The problem is to find a way to give out parts (shares) of the combination (the secret) to any employees so that, combining their information, the vault can be open by

any two of the first level employees, or

any one of the first level together with any two of the second level employees, or

any three second level employees;

andthe previous ones are the only minimal combinations of employees able to open the vault.

These set of combinations is called anaccess structure and a solution to the problem is an example of asecret sharing scheme.

A bank president problem

In a bank there area president, n first level andm second level employees. The president knows the combination to the vault and does not want to share this combination with any individual other than himself.

The problem is to find a way to give out parts (shares) of the combination (the secret) to any employees so that, combining their information, the vault can be open by

any two of the first level employees, or

any one of the first level together with any two of the second level employees, or

any three second level employees;

andthe previous ones are the only minimal combinations of employees able to open the vault.

These set of combinations is called anaccess structure and a solution to the problem is an example of asecret sharing scheme.

A bank president problem

In a bank there area president, n first level andm second level employees. The president knows the combination to the vault and does not want to share this combination with any individual other than himself.

The problem is to find a way to give out parts (shares) of the combination (the secret) to any employees so that, combining their information, the vault can be open by

any two of the first level employees, or

any one of the first level together with any two of the second level employees, or

any three second level employees;

andthe previous ones are the only minimal combinations of employees able to open the vault.

These set of combinations is called anaccess structure and a solution to the problem is an example of asecret sharing scheme.

A bank president problem

In a bank there area president, n first level andm second level employees. The president knows the combination to the vault and does not want to share this combination with any individual other than himself.

The problem is to find a way to give out parts (shares) of the combination (the secret) to any employees so that, combining their information, the vault can be open by

any two of the first level employees, or

any one of the first level together with any two of the second level employees, or

any three second level employees;

andthe previous ones are the only minimal combinations of employees able to open the vault.

These set of combinations is called anaccess structure and a solution to the problem is an example of asecret sharing scheme.

A bank president problem

In a bank there area president, n first level andm second level employees. The president knows the combination to the vault and does not want to share this combination with any individual other than himself.

The problem is to find a way to give out parts (shares) of the combination (the secret) to any employees so that, combining their information, the vault can be open by

any two of the first level employees, or

any one of the first level together with any two of the second level employees, or

any three second level employees;

A geometric solution

to our bank president problem

Letα, β be two planes in PG(4,q) meeting in exactly one point A. Assume thatα contains a hyperfocusedk−arcH with a setB of k−1 black points on a line`. Set

A =the secret;

points of `\(B ∪A) = q−k+ 1 first level employees;

H =k second level employees.

Then any team in the access structure can get the secret.

A geometric solution

to our bank president problem

Letα, β be two planes in PG(4,q) meeting in exactly one point A.

Assume thatα contains a hyperfocusedk−arcH with a setBof k−1 black points on a line`.

Set

A =the secret;

points of `\(B ∪A) = q−k+ 1 first level employees;

H =k second level employees.

Then any team in the access structure can get the secret.

A geometric solution

to our bank president problem

Letα, β be two planes in PG(4,q) meeting in exactly one point A.

Assume thatα contains a hyperfocusedk−arcH with a setBof k−1 black points on a line`. Set

A =the secret;

points of `\(B ∪A) = q−k+ 1 first level employees;

H =k second level employees.

Then any team in the access structure can get the secret.

A geometric solution

to our bank president problem

Letα, β be two planes in PG(4,q) meeting in exactly one point A.

Assume thatα contains a hyperfocusedk−arcH with a setBof k−1 black points on a line`. Set

A =the secret;

points of `\(B ∪A) = q−k+ 1 first level employees;

H =k second level employees.

Then any team in the access structure can get the secret.

A geometric solution

to our bank president problem

Letα, β be two planes in PG(4,q) meeting in exactly one point A.

Assume thatα contains a hyperfocusedk−arcH with a setBof k−1 black points on a line`. Set

A =the secret;

points of `\(B ∪A) = q−k+ 1 first level employees;

H =k second level employees.

Then any team in the access structure can get the secret.

A geometric solution

to our bank president problem

Letα, β be two planes in PG(4,q) meeting in exactly one point A.

Assume thatα contains a hyperfocusedk−arcH with a setBof k−1 black points on a line`. Set

A =the secret;

points of `\(B ∪A) = q−k+ 1 first level employees;

H =k second level employees.

A non trivial example of generalized hyperfocused arc

and our main result

EXAMPLE

Any4-arcQ of white points, with its three black diagonal points, is a non trivial example of generalized hyperfocused arc.

Note that:

when q is even, the three black diagonal points ofQ are collinear;

when q is odd, the three black diagonal points ofQ are not collinear.

THEOREM (A.Blokhuis - G.Marino - F.M., 2013) The 4−arc is the only non trivial example of generalized hyperfocused arc, providedq is an odd prime.

A non trivial example of generalized hyperfocused arc

and our main result

EXAMPLE

Any4-arcQ of white points, with its three black diagonal points, is a non trivial example of generalized hyperfocused arc.

Note that:

when q is even, the three black diagonal points ofQ are collinear;

when q is odd, the three black diagonal points ofQ are not collinear.

THEOREM (A.Blokhuis - G.Marino - F.M., 2013) The 4−arc is the only non trivial example of generalized hyperfocused arc, providedq is an odd prime.

A non trivial example of generalized hyperfocused arc

and our main result

EXAMPLE

Any4-arcQ of white points, with its three black diagonal points, is a non trivial example of generalized hyperfocused arc.

Note that:

when q is even, the three black diagonal points ofQ are collinear;

when q is odd, the three black diagonal points ofQ are not collinear.

THEOREM (A.Blokhuis - G.Marino - F.M., 2013) The 4−arc is the only non trivial example of generalized hyperfocused arc, providedq is an odd prime.

Some Notations

From now on we setF=GF(p),p a prime 6= 2,3.

IfA= (a1,a2,a3) is a non-zero vector ofF³, we denote by [A] =h(a₁,a2,a3)i the point ofPG(2,p) with homogeneous coordinates (a₁ :a₂ :a₃).

Sometimes, abusing notation, we will just write Ainstead of [A]and, in this case, we mean that for the point [A] we are considering the coordinates (a1,a2,a3) or some other special ones, that should be clear from the context.

We writeA=^∧ B if [A] = [B], i.e. A=λB, for some non zero λin F.

Some Notations

From now on we setF=GF(p),p a prime 6= 2,3.

IfA= (a1,a2,a3) is a non-zero vector ofF³, we denote by [A] =h(a₁,a₂,a₃)i the point ofPG(2,p) with homogeneous coordinates (a₁ :a₂ :a₃).

Sometimes, abusing notation, we will just write Ainstead of [A]and, in this case, we mean that for the point [A] we are considering the coordinates (a1,a2,a3) or some other special ones, that should be clear from the context.

We writeA=^∧ B if [A] = [B], i.e. A=λB, for some non zero λin F.

Some Notations

From now on we setF=GF(p),p a prime 6= 2,3.

IfA= (a1,a2,a3) is a non-zero vector ofF³, we denote by [A] =h(a₁,a₂,a₃)i the point ofPG(2,p) with homogeneous coordinates (a₁ :a₂ :a₃).

Sometimes, abusing notation, we will just write Ainstead of [A]and, in this case, we mean that for the point [A] we are considering the coordinates (a1,a2,a3) or some other special ones, that should be clear from the context.

We writeA=^∧ B if [A] = [B], i.e. A=λB, for some non zero λin F.

Some Notations

From now on we setF=GF(p),p a prime 6= 2,3.

IfA= (a1,a2,a3) is a non-zero vector ofF³, we denote by [A] =h(a₁,a₂,a₃)i the point ofPG(2,p) with homogeneous coordinates (a₁ :a₂ :a₃).

Sometimes, abusing notation, we will just write Ainstead of [A]and, in this case, we mean that for the point [A] we are considering the coordinates (a1,a2,a3) or some other special ones, that should be clear from the context.

We writeA=^∧ B if [A] = [B], i.e. A=λB, for some non zero λin F.

Some Notations

For the following, let

H={W₁ = [E1],W2 = [E2], . . . ,W2n= [E2n]}

be a generalized hyperfocused arcof order 2n in PG(2,p), with black point-set B.

We denote by Bij the unique black point on the line

<W_i,W_j > and we defineb_ij by

Since Bji =Bij we havebji = 1/bij.

Some Notations

For the following, let

H={W₁ = [E1],W2 = [E2], . . . ,W2n= [E2n]}

be a generalized hyperfocused arcof order 2n in PG(2,p), with black point-set B.

We denote by Bij the unique black point on the line

<W_i,W_j > and we defineb_ij by

Since Bji =Bij we havebji = 1/bij.

A basic relation

and a notation

Using the computational technique of Segre’s lemma of tangents one can prove that

b_ijb_jkb_ki = 1 for i 6=j 6=k 6=i.

FromB_ij =^∧ E_i +b_ijE_j,i,j 6= 1,we get

B_ij =^∧ b_1iE_i +b_1jE_j, for all i,j = 2, . . . ,2n. Taking into account that

B_1j =^∧ E1+b_1jE_j and B_i1 =^∧ E_i+b_i1E1

=∧ E1+b_1iE_i, and rescaling ourEs to b1sEs, for eachs 6= 1, we get

Bij

=∧ Ei +Ej, for all i,j = 1,2, . . . ,2n.

A basic relation

and a notation

Using the computational technique of Segre’s lemma of tangents one can prove that

b_ijb_jkb_ki = 1 for i 6=j 6=k 6=i.

FromB_ij =^∧ E_i +b_ijE_j,i,j 6= 1,we get

B_ij =^∧ b_1iE_i +b_1jE_j, for all i,j = 2, . . . ,2n.

Taking into account that

B_1j =^∧ E₁+b_1jE_j and B_i1 =^∧ E_i+b_i1E₁ =^∧ E₁+b_1iE_i,

and rescaling ourEs to b1sEs, for eachs 6= 1, we get Bij

=∧ Ei +Ej, for all i,j = 1,2, . . . ,2n.

A basic relation

and a notation

Using the computational technique of Segre’s lemma of tangents one can prove that

b_ijb_jkb_ki = 1 for i 6=j 6=k 6=i.

FromB_ij =^∧ E_i +b_ijE_j,i,j 6= 1,we get

B_ij =^∧ b_1iE_i +b_1jE_j, for all i,j = 2, . . . ,2n.

Taking into account that

B_1j =^∧ E₁+b_1jE_j and B_i1 =^∧ E_i+b_i1E₁ =^∧ E₁+b_1iE_i,

we get Bij

=∧ Ei +Ej, for all i,j = 1,2, . . . ,2n.

A basic relation

and a notation

Using the computational technique of Segre’s lemma of tangents one can prove that

b_ijb_jkb_ki = 1 for i 6=j 6=k 6=i.

FromB_ij =^∧ E_i +b_ijE_j,i,j 6= 1,we get

B_ij =^∧ b_1iE_i +b_1jE_j, for all i,j = 2, . . . ,2n.

Taking into account that

B_1j =^∧ E₁+b_1jE_j and B_i1 =^∧ E_i+b_i1E₁ =^∧ E₁+b_1iE_i, and rescaling ourEs to b1sEs, for eachs 6= 1, we get

B_ij =^∧ E_i +E_j, for all i,j = 1,2, . . . ,2n.

Intersection numbers of B

Let`be a line intersecting B in exactlym<p points, set S =`∩ B={B₁,B₂, . . . ,B_m}.

For a fixed white pointW ∈ H, define the twodisjoint sets S⁻={W₁⁻,W₂⁻, . . . ,W_m⁻} , S⁺={W =W₁⁺,W₂⁺, . . . ,W_m⁺}

where W_i⁻ is the unique white

point other than W on the line<W,B_i >

W_i⁺ the unique white point other thanW₁⁻ on the line

<W₁⁻,B_i >,i = 1, . . . ,m. Using appropriate white points,Hcan be partitioned into pairs of blocks, each block of sizem, so m must be a divisor of n.

Intersection numbers of B

Let`be a line intersecting B in exactlym<p points, set S =`∩ B={B₁,B₂, . . . ,B_m}.

For a fixed white pointW ∈ H, define the twodisjoint sets S⁻={W₁⁻,W₂⁻, . . . ,W_m⁻} , S⁺={W =W₁⁺,W₂⁺, . . . ,W_m⁺} where

W_i⁻ is the unique white point other than W on the line<W,B_i >

W_i⁺ the unique white point other thanW₁⁻ on the line

<W₁⁻,B_i >,i = 1, . . . ,m. Using appropriate white points,Hcan be partitioned into pairs of blocks, each block of sizem, so m must be a divisor of n.

Intersection numbers of B

Let`be a line intersecting B in exactlym<p points, set S =`∩ B={B₁,B₂, . . . ,B_m}.

For a fixed white pointW ∈ H, define the twodisjoint sets S⁻={W₁⁻,W₂⁻, . . . ,W_m⁻} , S⁺={W =W₁⁺,W₂⁺, . . . ,W_m⁺}

where W_i⁻ is the unique white

point other than W on the line<W,Bi >

W_i⁺ the unique white point other thanW₁⁻ on the line

<W₁⁻,B_i >,i = 1, . . . ,m. Using appropriate white points,Hcan be partitioned into pairs of blocks, each block of sizem, so m must be a divisor of n.

Intersection numbers of B

Let`be a line intersecting B in exactlym<p points, set S =`∩ B={B₁,B₂, . . . ,B_m}.

For a fixed white pointW ∈ H, define the twodisjoint sets S⁻={W₁⁻,W₂⁻, . . . ,W_m⁻} , S⁺={W =W₁⁺,W₂⁺, . . . ,W_m⁺}

where W_i⁻ is the unique white

point other than W on the line<W,Bi >

W_i⁺ the unique white point other thanW₁⁻ on the line

<W₁⁻,B_i >,i = 1, . . . ,m.

Using appropriate white points,Hcan be partitioned into pairs of blocks, each block of sizem, so m must be a divisor of n.

Intersection numbers of B

Let`be a line intersecting B in exactlym<p points, set S =`∩ B={B₁,B₂, . . . ,B_m}.

For a fixed white pointW ∈ H, define the twodisjoint sets S⁻={W₁⁻,W₂⁻, . . . ,W_m⁻} , S⁺={W =W₁⁺,W₂⁺, . . . ,W_m⁺}

where W_i⁻ is the unique white

point other than W on the line<W,Bi >

W_i⁺ the unique white point other thanW₁⁻ on the line

S ∪ S

∪ S

is a dual 3-net of order m

LetB =Bij be the black point on the line<W_i⁺,W_j⁻>, i,j = 1,2, . . . ,m,and fix coordinates of the black points on `so that

Bi =E₁⁺+E_i⁻ =E+E_i⁻.

Then

B =B_ij =^∧ E_i⁺+E_j⁻=^∧ B_i−E₁⁻+B_j −E₁⁺

=∧ αB_i +βB_j −(E₁⁻+E₁⁺) =αB_i +βB_j +γB₁, for some constantsα, β, γ. So B is on `∩ B=S and

S∪S⁻∪S⁺ is a dual 3-net of order m.

S ∪ S

∪ S

is a dual 3-net of order m

LetB =Bij be the black point on the line<W_i⁺,W_j⁻>, i,j = 1,2, . . . ,m,and fix coordinates of the black points on `so that

Bi =E₁⁺+E_i⁻ =E+E_i⁻. Then

B =B_ij =^∧ E_i⁺+E_j⁻=^∧ B_i −E₁⁻+B_j −E₁⁺

=∧ αB_i +βB_j −(E₁⁻+E₁⁺) =αB_i+βB_j +γB₁, for some constantsα, β, γ. So B is on `∩ B=S and

S∪S⁻∪S⁺ is a dual 3-net of order m.

S ∪ S

∪ S

is a dual 3-net of order m

LetB =Bij be the black point on the line<W_i⁺,W_j⁻>, i,j = 1,2, . . . ,m,and fix coordinates of the black points on `so that

Bi =E₁⁺+E_i⁻ =E+E_i⁻. Then

B =B_ij =^∧ E_i⁺+E_j⁻=^∧ B_i −E₁⁻+B_j −E₁⁺

=∧ αB_i +βB_j −(E₁⁻+E₁⁺) =αB_i+βB_j +γB₁, for some constantsα, β, γ. So B is on `∩ B=S and

S∪S⁻∪S⁺ is a dual 3-net of order m.

S ∪ S

∪ S

is a dual 3-net of order m

LetB =Bij be the black point on the line<W_i⁺,W_j⁻>, i,j = 1,2, . . . ,m,and fix coordinates of the black points on `so that

Bi =E₁⁺+E_i⁻ =E+E_i⁻. Then

B =B_ij =^∧ E_i⁺+E_j⁻=^∧ B_i −E₁⁻+B_j −E₁⁺

=∧ αB_i +βB_j −(E₁⁻+E₁⁺) =αB_i+βB_j +γB₁, for some constantsα, β, γ. So B is on `∩ B=S and

Classifying H

By A.Blokhuis-G.Korchm´aros-F.M. Theorem,

S⁻∪S⁺ is a set of 2m points on a conic Γ and mis the size of a subgroup of a group defined on Γ\`.

It follows that

m divides p+ 1 or p−1,

according to the fact thatΓ has 0 or 2 points in common with `, respectively.

According to this two cases, we say thatHis of elliptic orhyperbolic type, respectively.

The case|H ∩`|= 1 cannot occur, otherwisem should dividep.

Classifying H

By A.Blokhuis-G.Korchm´aros-F.M. Theorem,

S⁻∪S⁺ is a set of 2m points on a conic Γ and mis the size of a subgroup of a group defined on Γ\`.

It follows that

m dividesp+ 1 or p−1,

according to the fact thatΓ has 0 or 2 points in common with `, respectively.

According to this two cases, we say thatHis of elliptic orhyperbolic type, respectively.

The case|H ∩`|= 1 cannot occur, otherwisem should dividep.

Classifying H

By A.Blokhuis-G.Korchm´aros-F.M. Theorem,

S⁻∪S⁺ is a set of 2m points on a conic Γ and mis the size of a subgroup of a group defined on Γ\`.

It follows that

m dividesp+ 1 or p−1,

according to the fact thatΓ has 0 or 2 points in common with `, respectively.

According to this two cases, we say thatHis of elliptic orhyperbolic type, respectively.

The case|H ∩`|= 1 cannot occur, otherwisem should dividep.

Classifying H

By A.Blokhuis-G.Korchm´aros-F.M. Theorem,

S⁻∪S⁺ is a set of 2m points on a conic Γ and mis the size of a subgroup of a group defined on Γ\`.

It follows that

m dividesp+ 1 or p−1,

according to the fact thatΓ has 0 or 2 points in common with `, respectively.

According to this two cases, we say thatHis of elliptic orhyperbolic type,

A first step

Assume that the equation of` in PG(2,p) isz = 0, w.r.t.

coordinates (x:y :z).

In the hyperbolic casewe can assume that the conic Γ has equationxy =z and we may take

S⁺={(u : 1

u : 1) : u∈F^∗ and u^m= 1}. In the elliptic casewe identifyAG(2,p) =PG(2,p)\`with the field F_p2. If we denote by (a; 1), witha∈F_p2, the coordinates of affine points of PG(2,p), we can assume that the conic Γ has equation x^p+1 = 1 and we may take

S⁺={(x; 1) : x^m = 1}. Using these two representations ofS⁺ we can prove: Lemma

If a line`intersects Bin exactly m<p points, then m≤4.

A first step

Assume that the equation of` in PG(2,p) isz = 0, w.r.t.

coordinates (x:y :z).

In the hyperbolic casewe can assume that the conic Γ has equationxy =z and we may take

S⁺={(u : 1

u : 1) : u∈F^∗ and u^m= 1}.

In the elliptic casewe identifyAG(2,p) =PG(2,p)\`with the field F_p2. If we denote by (a; 1), witha∈F_p2, the coordinates of affine points of PG(2,p), we can assume that the conic Γ has equation x^p+1 = 1 and we may take

S⁺={(x; 1) : x^m = 1}. Using these two representations ofS⁺ we can prove: Lemma

If a line`intersects Bin exactly m<p points, then m≤4.

A first step

Assume that the equation of` in PG(2,p) isz = 0, w.r.t.

coordinates (x:y :z).

In the hyperbolic casewe can assume that the conic Γ has equationxy =z and we may take

S⁺={(u : 1

u : 1) : u∈F^∗ and u^m= 1}.

In the elliptic casewe identifyAG(2,p) =PG(2,p)\`with the field F_p2. If we denote by (a; 1), with a∈F_p2, the coordinates of affine points of PG(2,p), we can assume that the conic Γ has equation x^p+1 = 1 and we may take

S⁺={(x; 1) : x^m = 1}.

Using these two representations ofS⁺ we can prove: Lemma

If a line`intersects Bin exactly m<p points, then m≤4.

A first step

Assume that the equation of` in PG(2,p) isz = 0, w.r.t.

coordinates (x:y :z).

In the hyperbolic casewe can assume that the conic Γ has equationxy =z and we may take

S⁺={(u : 1

u : 1) : u∈F^∗ and u^m= 1}.

In the elliptic casewe identifyAG(2,p) =PG(2,p)\`with the field F_p2. If we denote by (a; 1), with a∈F_p2, the coordinates of affine points of PG(2,p), we can assume that the conic Γ has equation x^p+1 = 1 and we may take

S⁺={(x; 1) : x^m = 1}.

A second step

A 4-set {E_i,E_j,E_k,E_l}of withe points ofH is said to be special ifEi+Ej+Ek+El =0. This means that {E_i,Ej,E_k,E_l}is a non trivial generalized hyperfo- cused 4-arc contained inH. Lemma

Let E1,E2,E3,E4,E5,E6 be six distinct points ofH such that E₁+E₂=^∧ E₃+E₄ =^∧ E₅+E₆=^∧ B.

Then,if{E₁,E₂,E₃,E₄}and {E₁,E₂,E₅,E₆} are special, {E₃,E4,E5,E6} is not.

This lemma ensures that, ifn>2, Hcontains a non special 4−set of withe points.

A second step

A 4-set {E_i,E_j,E_k,E_l}of withe points ofH is said to be special ifEi+Ej +Ek +El =0. This means that {E_i,E_j,E_k,E_l}is a non trivial generalized hyperfo- cused 4-arc contained inH.

Lemma

Let E1,E2,E3,E4,E5,E6 be six distinct points ofH such that E₁+E₂=^∧ E₃+E₄ =^∧ E₅+E₆=^∧ B.

Then,if{E₁,E₂,E₃,E₄}and {E₁,E₂,E₅,E₆} are special, {E₃,E4,E5,E6} is not.

This lemma ensures that, ifn>2, Hcontains a non special 4−set of withe points.

A second step

A 4-set {E_i,E_j,E_k,E_l}of withe points ofH is said to be special ifEi+Ej +Ek +El =0. This means that {E_i,E_j,E_k,E_l}is a non trivial generalized hyperfo- cused 4-arc contained inH.

Lemma

Let E1,E2,E3,E4,E5,E6 be six distinct points ofH such that E1+E2

=∧ E3+E4

=∧ E5+E6

=∧ B. Then,if{E₁,E₂,E₃,E₄}and {E₁,E₂,E₅,E₆} are special, {E₃,E4,E5,E6} is not.

This lemma ensures that, ifn>2, Hcontains a non special 4−set of withe points.

A second step

A 4-set {E_i,E_j,E_k,E_l}of withe points ofH is said to be special ifEi+Ej +Ek +El =0. This means that {E_i,E_j,E_k,E_l}is a non trivial generalized hyperfo- cused 4-arc contained inH.

Lemma

Let E1,E2,E3,E4,E5,E6 be six distinct points ofH such that E1+E2

=∧ E3+E4

=∧ E5+E6

=∧ B. Then,if{E₁,E₂,E₃,E₄}and {E₁,E₂,E₅,E₆} are special,

The main result

Assumen>2 and letE₁,E₂,E₃,E₄ be a non special 4−set.

Then we have 5 different black points:

B₁₂=B₃₄,B₁₃,B₁₄,B₂₃,B₂₄.

Moreover it is possible to show that they are in a position that we may put

B₁₂=B₃₄= (0,0,1), B₁₃= (1,0,0), B₂₄= (1,0,1), B14= (0,1,0), B23= (0,1,1).

Finally, using this frame and after long and non trivial calculations, we can prove our main result.

THEOREM

Ifp is an odd prime and His a hyperfocused arc of size 2n, in PG(2,p) thenn≤2.

The main result

Assumen>2 and letE₁,E₂,E₃,E₄ be a non special 4−set.

Then we have 5 different black points:

B₁₂=B₃₄,B₁₃,B₁₄,B₂₃,B₂₄.

Moreover it is possible to show that they are in a position that we may put

B₁₂=B₃₄= (0,0,1), B₁₃= (1,0,0), B₂₄= (1,0,1), B14= (0,1,0), B23= (0,1,1).

Finally, using this frame and after long and non trivial calculations, we can prove our main result.

THEOREM

Ifp is an odd prime and His a hyperfocused arc of size 2n, in PG(2,p) thenn≤2.

The main result

Assumen>2 and letE₁,E₂,E₃,E₄ be a non special 4−set.

Then we have 5 different black points:

B₁₂=B₃₄,B₁₃,B₁₄,B₂₃,B₂₄.

Moreover it is possible to show that they are in a position that we may put

B₁₂=B₃₄= (0,0,1), B₁₃= (1,0,0), B₂₄= (1,0,1), B14= (0,1,0), B23= (0,1,1).

Finally, using this frame and after long and non trivial calculations, we can prove our main result.

THEOREM

Ifp is an odd prime and His a hyperfocused arc of size 2n, in PG(2,p) thenn≤2.

Generalizing further

The Ball cylinder conjecture

Generalized hyperfocused arcs were introduced as a

generalization of hyperfocused arcs,but this is not how we hit upon them.

We see them as a special case of a configuration whose study is motivated by the (strong) cylinder conjecture.

The strong cylinder conjeture (Ball, 2011)

A setC ofq² points in AG(3,q) that intersects every plane in 0 modq points must be a cylinder, i.e. the union ofq parallel lines. The ordinary conjecture states the same thing forq a prime.

Generalizing further

The Ball cylinder conjecture

Generalized hyperfocused arcs were introduced as a

generalization of hyperfocused arcs,but this is not how we hit upon them.

We see them as a special case of a configuration whose study is motivated by the(strong) cylinder conjecture.

The strong cylinder conjeture (Ball, 2011)

A setC ofq² points in AG(3,q) that intersects every plane in 0 modq points must be a cylinder, i.e. the union ofq parallel lines. The ordinary conjecture states the same thing forq a prime.

Generalizing further

The Ball cylinder conjecture

Generalized hyperfocused arcs were introduced as a

generalization of hyperfocused arcs,but this is not how we hit upon them.

We see them as a special case of a configuration whose study is motivated by the(strong) cylinder conjecture.

The strong cylinder conjeture (Ball, 2011)

A setC ofq² points in AG(3,q) that intersects every plane in 0 modq points must be a cylinder, i.e. the union ofq parallel lines.

Generalizing further

The Ball cylinder conjecture

In an attempt to prove the ordinary cylinder conjecture we found in PG(2,p) configurations consisting of

a set W of k ≤(p+ 1)/2 ’white’ points, a multisetBof k−1 ’black’ points,

with the property thatevery line containing m>0 white points contains exactlym−1 black points (counted with multiplicity).

Generalizing further

The Ball cylinder conjecture

In an attempt to prove the ordinary cylinder conjecture we found in PG(2,p) configurations consisting of

a setW of k ≤(p+ 1)/2 ’white’ points,

a multisetBof k−1 ’black’ points,

with the property thatevery line containing m>0 white points contains exactlym−1 black points (counted with multiplicity).

Generalizing further

The Ball cylinder conjecture

In an attempt to prove the ordinary cylinder conjecture we found in PG(2,p) configurations consisting of

a setW of k ≤(p+ 1)/2 ’white’ points, a multisetBof k−1 ’black’ points,

with the property thatevery line containing m>0 white points contains exactlym−1 black points (counted with multiplicity).

Generalizing further

The Ball cylinder conjecture

In an attempt to prove the ordinary cylinder conjecture we found in PG(2,p) configurations consisting of

a setW of k ≤(p+ 1)/2 ’white’ points, a multisetBof k−1 ’black’ points,

with the property thatevery line containing m>0 white points contains exactlym−1 black points (counted with multiplicity).

Configurations of withe and black points

and their connection with Ball cylinder conjecture

Let C be a set ofq² points in AG(3,q) that intersects every plane in 0 mod q points.

Take a pointP ∈AG(3,q)\C and projectC fromP,i.e. consider the quotient geometry π=PG(2,q) inP (points and lines in the quotient are lines and planes trough P in PG(3,q),respectively).

The projection of C is a multisetX of q² points in PG(2,q) with a multiple ofq points on every line.

Let the weight of a lineα in π be k−1,ifα contains kq points of C as plane of PG(3,q).

All points of a line of weight −1 do not occur in the multiset X,that is they have weight 0 as points of X.

Configurations of withe and black points

and their connection with Ball cylinder conjecture

Let C be a set ofq² points in AG(3,q) that intersects every plane in 0 mod q points.

Take a pointP ∈AG(3,q)\C and projectC fromP,i.e.

consider the quotient geometry π=PG(2,q) inP (points and lines in the quotient are lines and planes trough P in PG(3,q),respectively).

The projection of C is a multisetX of q² points in PG(2,q) with a multiple ofq points on every line.

Let the weight of a lineα in π be k−1,ifα contains kq points of C as plane of PG(3,q).

All points of a line of weight −1 do not occur in the multiset X,that is they have weight 0 as points of X.

Configurations of withe and black points

and their connection with Ball cylinder conjecture

Let C be a set ofq² points in AG(3,q) that intersects every plane in 0 mod q points.

Take a pointP ∈AG(3,q)\C and projectC fromP,i.e.

consider the quotient geometry π=PG(2,q) inP (points and lines in the quotient are lines and planes trough P in PG(3,q),respectively).

The projection of C is a multisetX of q² points inPG(2,q) with a multiple ofq points on every line.

Let the weight of a lineα in π be k−1,ifα contains kq points of C as plane of PG(3,q).

All points of a line of weight −1 do not occur in the multiset X,that is they have weight 0 as points of X.

Configurations of withe and black points

and their connection with Ball cylinder conjecture

Let C be a set ofq² points in AG(3,q) that intersects every plane in 0 mod q points.

Take a pointP ∈AG(3,q)\C and projectC fromP,i.e.

consider the quotient geometry π=PG(2,q) inP (points and lines in the quotient are lines and planes trough P in PG(3,q),respectively).

The projection of C is a multisetX of q² points inPG(2,q) with a multiple ofq points on every line.

Let the weight of a lineα in π be k−1,ifα contains kq points of C as plane ofPG(3,q).

All points of a line of weight −1 do not occur in the multiset X,that is they have weight 0 as points of X.

Configurations of withe and black points

and their connection with Ball cylinder conjecture

Let C be a set ofq² points in AG(3,q) that intersects every plane in 0 mod q points.

Take a pointP ∈AG(3,q)\C and projectC fromP,i.e.

consider the quotient geometry π=PG(2,q) inP (points and lines in the quotient are lines and planes trough P in PG(3,q),respectively).

The projection of C is a multisetX of q² points inPG(2,q) with a multiple ofq points on every line.

Let the weight of a lineα in π be k−1,ifα contains kq points of C as plane ofPG(3,q).

All points of a line of weight −1 do not occur in the multiset X,that is they have weight 0 as points of X.