Intersection of Ovals and Unitals in Desarguesian Planes

Intersection of Ovals and Unitals in Desarguesian Planes

G´abor Korchm´aros

Workshop on Finite Geometry

June 10-15, 2013 University of Szeged

A problem on ovals and unitals

Problem 1. Determine the possible intersection patterns of an oval and a unital in PG(2,q²)

Problem 1 is open in the general case. Known results concern the classical case. Classical case:

Oval is an irreducible conic Ω, for q odd the set of all absolute points of an orthogonal polarity π of PG(2,q²),

Unital is a Hermitian curve U, i.e. set of all absolute points of a unitary polarity ω of PG(2,q²).

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

A problem on ovals and unitals

Problem 1. Determine the possible intersection patterns of an oval and a unital in PG(2,q²)

Problem 1 is open in the general case. Known results concern the classical case. Classical case:

Oval is an irreducible conic Ω, for q odd the set of all absolute points of an orthogonal polarity π of PG(2,q²),

Unital is a Hermitian curve U, i.e. set of all absolute points of a unitary polarity ω of PG(2,q²).

A problem on ovals and unitals

Problem 1. Determine the possible intersection patterns of an oval and a unital in PG(2,q²)

Problem 1 is open in the general case.

Known results concern the classical case. Classical case:

Oval is an irreducible conic Ω, for q odd the set of all absolute points of an orthogonal polarity π of PG(2,q²),

Unital is a Hermitian curve U, i.e. set of all absolute points of a unitary polarity ω of PG(2,q²).

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

A problem on ovals and unitals

Problem 1. Determine the possible intersection patterns of an oval and a unital in PG(2,q²)

Problem 1 is open in the general case.

Known results concern the classical case.

Classical case:

Oval is an irreducible conic Ω, for q odd the set of all absolute points of an orthogonal polarity π of PG(2,q²),

Unital is a Hermitian curve U, i.e. set of all absolute points of a unitary polarity ω of PG(2,q²).

A problem on ovals and unitals

Problem 1. Determine the possible intersection patterns of an oval and a unital in PG(2,q²)

Problem 1 is open in the general case.

Known results concern the classical case.

Classical case:

Oval is an irreducible conic Ω, for q odd the set of all absolute points of an orthogonal polarity π of PG(2,q²),

Unital is a Hermitian curve U, i.e. set of all absolute points of a unitary polarity ω of PG(2,q²).

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

A problem on ovals and unitals

Problem 1. Determine the possible intersection patterns of an oval and a unital in PG(2,q²)

Problem 1 is open in the general case.

Known results concern the classical case.

Classical case:

Oval is an irreducible conic Ω, for q odd the set of all absolute points of an orthogonal polarity π of PG(2,q²),

Unital is a Hermitian curve U, i.e. set of all absolute points of a unitary polarity ω of PG(2,q²).

A problem on ovals and unitals

Problem 1. Determine the possible intersection patterns of an oval and a unital in PG(2,q²)

Problem 1 is open in the general case.

Known results concern the classical case.

Classical case:

Oval is an irreducible conic Ω, for q odd the set of all absolute points of an orthogonal polarity π of PG(2,q²),

Unital is a Hermitian curveU, i.e. set of all absolute points of a unitary polarity ω of PG(2,q²).

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Previous results from Segre’s work 1965

Unital and oval in permutable position

Ω and U are in permutable positionwhen q is odd and πω =ωπ.

If Ω and U are in permutable position =⇒Ω∩ U is a classical Baer suboval, i.e. a conic of a subplanePG(2,q), =⇒

|Ω∩ U |=q+ 1 and they are tangent at each of their common points.

If Ω and U are 3-tangents andq odd then they are in permutable position.

Corollary

For q odd, a classical Baer suboval can be the complete intersection of a classical oval and a Hermitian unital. If this happens, then they are in a permutable position.

Previous results from Segre’s work 1965

Unital and oval in permutable position

Ω and U are in permutable positionwhen q is odd and πω=ωπ.

If Ω and U are in permutable position =⇒Ω∩ U is a classical Baer suboval, i.e. a conic of a subplanePG(2,q), =⇒

|Ω∩ U |=q+ 1 and they are tangent at each of their common points.

If Ω and U are 3-tangents andq odd then they are in permutable position.

Corollary

For q odd, a classical Baer suboval can be the complete intersection of a classical oval and a Hermitian unital. If this happens, then they are in a permutable position.

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Previous results from Segre’s work 1965

Unital and oval in permutable position

Ω and U are in permutable positionwhen q is odd and πω=ωπ.

If Ω and U are in permutable position =⇒ Ω∩ U is a classical Baer suboval, i.e. a conic of a subplanePG(2,q), =⇒

|Ω∩ U |=q+ 1 and they are tangent at each of their common points.

If Ω and U are 3-tangents andq odd then they are in permutable position.

Corollary

For q odd, a classical Baer suboval can be the complete intersection of a classical oval and a Hermitian unital. If this happens, then they are in a permutable position.

Previous results from Segre’s work 1965

Unital and oval in permutable position

Ω and U are in permutable positionwhen q is odd and πω=ωπ.

If Ω and U are in permutable position =⇒ Ω∩ U is a classical Baer suboval, i.e. a conic of a subplanePG(2,q), =⇒

|Ω∩ U |=q+ 1 and they are tangent at each of their common points.

If Ω and U are 3-tangents andq odd then they are in permutable position.

Corollary

For q odd, a classical Baer suboval can be the complete intersection of a classical oval and a Hermitian unital. If this happens, then they are in a permutable position.

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Previous results from Segre’s work 1965

Unital and oval in permutable position

Ω and U are in permutable positionwhen q is odd and πω=ωπ.

If Ω and U are in permutable position =⇒ Ω∩ U is a classical Baer suboval, i.e. a conic of a subplanePG(2,q), =⇒

|Ω∩ U |=q+ 1 and they are tangent at each of their common points.

If Ω and U are 3-tangents andq odd then they are in permutable position.

Corollary

For q odd, a classical Baer suboval can be the complete intersection of a classical oval and a Hermitian unital. If this happens, then they are in a permutable position.

Previous results from Segre’s work 1965

Unital and oval in permutable position

Ω and U are in permutable positionwhen q is odd and πω=ωπ.

If Ω and U are in permutable position =⇒ Ω∩ U is a classical Baer suboval, i.e. a conic of a subplanePG(2,q), =⇒

|Ω∩ U |=q+ 1 and they are tangent at each of their common points.

If Ω and U are 3-tangents andq odd then they are in permutable position.

Corollary

For q odd, a classical Baer suboval can be the complete intersection of a classical oval and a Hermitian unital. If this happens, then they are in a permutable position.

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Previous results from Segre’s work 1965

Unital and oval in permutable position

Ω and U are in permutable positionwhen q is odd and πω=ωπ.

If Ω and U are in permutable position =⇒ Ω∩ U is a classical Baer suboval, i.e. a conic of a subplanePG(2,q), =⇒

|Ω∩ U |=q+ 1 and they are tangent at each of their common points.

If Ω and U are 3-tangents andq odd then they are in permutable position.

Corollary

For q odd, a classical Baer suboval can be the complete

If this happens, then they are in a permutable position.

Previous results from Segre’s work 1965

Unital and oval in permutable position

Ω and U are in permutable positionwhen q is odd and πω=ωπ.

If Ω and U are in permutable position =⇒ Ω∩ U is a classical Baer suboval, i.e. a conic of a subplanePG(2,q), =⇒

|Ω∩ U |=q+ 1 and they are tangent at each of their common points.

If Ω and U are 3-tangents andq odd then they are in permutable position.

Corollary

For q odd, a classical Baer suboval can be the complete intersection of a classical oval and a Hermitian unital.

If this happens, then they are in a permutable position.

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Previous results of Donati and Durante, Combinatorics2008

Intersection of a classical oval and Hermitian unital containing a classical Baer suboval

Theorem

SupposeΩandU share a classical Baer suboval Ω₀. Then one of the following occurs.

Ω∩ U is a classical Baer suboval and q odd.

Ω∩ U is the union of two classical Baer subovals,Ω₀ andΩ₁, and |Ω₀∩Ω₁| ≤2.

Remark

Proof uses counting arguments, computations with Hermitian forms plus discussions about special equations over a finite field.

Previous results of Donati and Durante, Combinatorics2008

Intersection of a classical oval and Hermitian unital containing a classical Baer suboval

Theorem

SupposeΩandU share a classical Baer suboval Ω₀. Then one of the following occurs.

Ω∩ U is a classical Baer suboval and q odd.

Ω∩ U is the union of two classical Baer subovals,Ω₀ andΩ₁, and |Ω₀∩Ω₁| ≤2.

Remark

Proof uses counting arguments, computations with Hermitian forms plus discussions about special equations over a finite field.

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Previous results of Donati and Durante, Combinatorics2008

Intersection of a classical oval and Hermitian unital containing a classical Baer suboval

Theorem

SupposeΩandU share a classical Baer suboval Ω₀. Then one of the following occurs.

Ω∩ U is a classical Baer suboval and q odd.

Ω∩ U is the union of two classical Baer subovals,Ω₀ andΩ₁, and |Ω₀∩Ω₁| ≤2.

Remark

Proof uses counting arguments, computations with Hermitian forms plus discussions about special equations over a finite field.

Previous results of Donati and Durante, Combinatorics2008

Intersection of a classical oval and Hermitian unital containing a classical Baer suboval

Theorem

SupposeΩandU share a classical Baer suboval Ω₀. Then one of the following occurs.

Ω∩ U is a classical Baer suboval and q odd.

Ω∩ U is the union of two classical Baer subovals,Ω₀ andΩ₁, and|Ω₀∩Ω₁| ≤2.

Remark

Proof uses counting arguments, computations with Hermitian forms plus discussions about special equations over a finite field.

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Previous results of Donati and Durante, Combinatorics2008

Intersection of a classical oval and Hermitian unital containing a classical Baer suboval

Theorem

SupposeΩandU share a classical Baer suboval Ω₀. Then one of the following occurs.

Ω∩ U is a classical Baer suboval and q odd.

Ω∩ U is the union of two classical Baer subovals,Ω₀ andΩ₁, and|Ω₀∩Ω₁| ≤2.

Remark

Proof uses counting arguments, computations with Hermitian

Previous results of Donati and Durante, Combinatorics2008

Intersection of bitangent classical oval and Hermitian unital

Theorem

SupposeΩandU are tangent at A and B, with A6=B. Then one of the following occurs.

Ω∩ U ={A,B}.

Ω∩ U is a classical Baer suboval and q odd. Ω∩ U is a classical Baer suboval plus a point. Ω∩ U is the union of two classical Baer subovals.

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Previous results of Donati and Durante, Combinatorics2008

Intersection of bitangent classical oval and Hermitian unital

Theorem

SupposeΩandU are tangent at A and B, with A6=B. Then one of the following occurs.

Ω∩ U ={A,B}.

Ω∩ U is a classical Baer suboval and q odd. Ω∩ U is a classical Baer suboval plus a point. Ω∩ U is the union of two classical Baer subovals.

Previous results of Donati and Durante, Combinatorics2008

Intersection of bitangent classical oval and Hermitian unital

Theorem

SupposeΩandU are tangent at A and B, with A6=B. Then one of the following occurs.

Ω∩ U ={A,B}.

Ω∩ U is a classical Baer suboval and q odd. Ω∩ U is a classical Baer suboval plus a point. Ω∩ U is the union of two classical Baer subovals.

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Previous results of Donati and Durante, Combinatorics2008

Intersection of bitangent classical oval and Hermitian unital

Theorem

SupposeΩandU are tangent at A and B, with A6=B. Then one of the following occurs.

Ω∩ U ={A,B}.

Ω∩ U is a classical Baer suboval and q odd.

Ω∩ U is a classical Baer suboval plus a point. Ω∩ U is the union of two classical Baer subovals.

Previous results of Donati and Durante, Combinatorics2008

Intersection of bitangent classical oval and Hermitian unital

Theorem

SupposeΩandU are tangent at A and B, with A6=B. Then one of the following occurs.

Ω∩ U ={A,B}.

Ω∩ U is a classical Baer suboval and q odd.

Ω∩ U is a classical Baer suboval plus a point.

Ω∩ U is the union of two classical Baer subovals.

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Previous results of Donati and Durante, Combinatorics2008

Intersection of bitangent classical oval and Hermitian unital

Theorem

SupposeΩandU are tangent at A and B, with A6=B. Then one of the following occurs.

Ω∩ U ={A,B}.

Ω∩ U is a classical Baer suboval and q odd.

Ω∩ U is a classical Baer suboval plus a point.

Ω∩ U is the union of two classical Baer subovals.

Classification of Ω ∩ U

Theorem (Donati-Durante-Korchm´aros, 2009)

In PG(2,q²) with q >3, the intersection pattern of a classical ovaleΩand a Hermitian unital U is one of the following.

(I) Ω∩ U =∅;

(II) Ω andU have only one common point and q odd;

(III) Ω andU have exactly two common points and are tangent at those points;

(IV) Ω∩ U is a classical Baer suboval and q odd;

(V) Ω∩ U is the union of two classical Baer subovals sharing either zero, or two, points, or, for q odd, one point; (VI) |Ω∩ U |=k with k ∈ {q,q+ 1,q+ 2}

(VII) |Ω∩ U |=k with (√

q−1)²+ 1≤k ≤(√

q+ 1)²+ 1

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Classification of Ω ∩ U

Theorem (Donati-Durante-Korchm´aros, 2009)

In PG(2,q²) with q >3, the intersection pattern of a classical ovaleΩand a Hermitian unitalU is one of the following.

(I) Ω∩ U =∅;

(II) Ω andU have only one common point and q odd;

(III) Ω andU have exactly two common points and are tangent at those points;

(IV) Ω∩ U is a classical Baer suboval and q odd;

(V) Ω∩ U is the union of two classical Baer subovals sharing either zero, or two, points, or, for q odd, one point; (VI) |Ω∩ U |=k with k ∈ {q,q+ 1,q+ 2}

(VII) |Ω∩ U |=k with (√

q−1)²+ 1≤k≤(√

q+ 1)²+ 1

Classification of Ω ∩ U

Theorem (Donati-Durante-Korchm´aros, 2009)

In PG(2,q²) with q >3, the intersection pattern of a classical ovaleΩand a Hermitian unitalU is one of the following.

(I) Ω∩ U =∅;

(II) Ω andU have only one common point and q odd;

(III) Ω andU have exactly two common points and are tangent at those points;

(IV) Ω∩ U is a classical Baer suboval and q odd;

(V) Ω∩ U is the union of two classical Baer subovals sharing either zero, or two, points, or, for q odd, one point; (VI) |Ω∩ U |=k with k ∈ {q,q+ 1,q+ 2}

(VII) |Ω∩ U |=k with (√

q−1)²+ 1≤k≤(√

q+ 1)²+ 1

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Classification of Ω ∩ U

Theorem (Donati-Durante-Korchm´aros, 2009)

In PG(2,q²) with q >3, the intersection pattern of a classical ovaleΩand a Hermitian unitalU is one of the following.

(I) Ω∩ U =∅;

(II) Ω andU have only one common point and q odd;

(III) Ω andU have exactly two common points and are tangent at those points;

(IV) Ω∩ U is a classical Baer suboval and q odd;

(V) Ω∩ U is the union of two classical Baer subovals sharing either zero, or two, points, or, for q odd, one point; (VI) |Ω∩ U |=k with k ∈ {q,q+ 1,q+ 2}

(VII) |Ω∩ U |=k with (√

q−1)²+ 1≤k≤(√

q+ 1)²+ 1

Classification of Ω ∩ U

Theorem (Donati-Durante-Korchm´aros, 2009)

In PG(2,q²) with q >3, the intersection pattern of a classical ovaleΩand a Hermitian unitalU is one of the following.

(I) Ω∩ U =∅;

(II) Ω andU have only one common point and q odd;

(III) Ω andU have exactly two common points and are tangent at those points;

(IV) Ω∩ U is a classical Baer suboval and q odd;

(V) Ω∩ U is the union of two classical Baer subovals sharing either zero, or two, points, or, for q odd, one point; (VI) |Ω∩ U |=k with k ∈ {q,q+ 1,q+ 2}

(VII) |Ω∩ U |=k with (√

q−1)²+ 1≤k≤(√

q+ 1)²+ 1

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Classification of Ω ∩ U

Theorem (Donati-Durante-Korchm´aros, 2009)

In PG(2,q²) with q >3, the intersection pattern of a classical ovaleΩand a Hermitian unitalU is one of the following.

(I) Ω∩ U =∅;

(II) Ω andU have only one common point and q odd;

(III) Ω andU have exactly two common points and are tangent at those points;

(IV) Ω∩ U is a classical Baer suboval and q odd;

(V) Ω∩ U is the union of two classical Baer subovals sharing either zero, or two, points, or, for q odd, one point; (VI) |Ω∩ U |=k with k ∈ {q,q+ 1,q+ 2}

(VII) |Ω∩ U |=k with (√

q−1)²+ 1≤k≤(√

q+ 1)²+ 1

Classification of Ω ∩ U

Theorem (Donati-Durante-Korchm´aros, 2009)

In PG(2,q²) with q >3, the intersection pattern of a classical ovaleΩand a Hermitian unitalU is one of the following.

(I) Ω∩ U =∅;

(II) Ω andU have only one common point and q odd;

(III) Ω andU have exactly two common points and are tangent at those points;

(IV) Ω∩ U is a classical Baer suboval and q odd;

(V) Ω∩ U is the union of two classical Baer subovals sharing either zero, or two, points, or, for q odd, one point;

(VI) |Ω∩ U |=k with k ∈ {q,q+ 1,q+ 2} (VII) |Ω∩ U |=k with (√

q−1)²+ 1≤k≤(√

q+ 1)²+ 1

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Classification of Ω ∩ U

Theorem (Donati-Durante-Korchm´aros, 2009)

In PG(2,q²) with q >3, the intersection pattern of a classical ovaleΩand a Hermitian unitalU is one of the following.

(I) Ω∩ U =∅;

(II) Ω andU have only one common point and q odd;

(III) Ω andU have exactly two common points and are tangent at those points;

(IV) Ω∩ U is a classical Baer suboval and q odd;

(V) Ω∩ U is the union of two classical Baer subovals sharing either zero, or two, points, or, for q odd, one point;

(VII) |Ω∩ U |=k with (√

q−1)²+ 1≤k≤(√

q+ 1)²+ 1

Classification of Ω ∩ U

Theorem (Donati-Durante-Korchm´aros, 2009)

In PG(2,q²) with q >3, the intersection pattern of a classical ovaleΩand a Hermitian unitalU is one of the following.

(I) Ω∩ U =∅;

(II) Ω andU have only one common point and q odd;

(III) Ω andU have exactly two common points and are tangent at those points;

(IV) Ω∩ U is a classical Baer suboval and q odd;

(V) Ω∩ U is the union of two classical Baer subovals sharing either zero, or two, points, or, for q odd, one point;

(VI) |Ω∩ U |=k with k ∈ {q,q+ 1,q+ 2}

(VII) |Ω∩ U |=k with (√

q−1)²+ 1≤k≤(√

q+ 1)²+ 1

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Proof of the classification

Setup

C:= conic of equation X₁² =X₀X₂;

Ω :={P_t = (1,t,t²)|t ∈GF(q²)} ∪ {P∞= (0,0,1)} GF(q²) =GF(q)(i) with

i² =s with nonsquare s ∈GF(q) when q odd; i²+i =δ with δ∈GF(q),Tr(δ) =1, when q even. U := Hermitian unital associated to the unitaryU. For p >2,

U=





a11 a12+b12i a13+b13i a12−b12i a22 a23+b23i a₁₃−b₁₃i a₂₃−b₂₃i a₃₃





For p = 2, U=





a₁₁ a₁₂+b₁₂i a₁₃+b₁₃i a₁₂+b₁₂(i+ 1) a₂₂ a₂₃+b₂₃i a13+b13(i+ 1) a23+b23(i+ 1) a33





Proof of the classification

Setup

C:= conic of equation X₁² =X₀X₂;

Ω :={P_t = (1,t,t²)|t ∈GF(q²)} ∪ {P∞= (0,0,1)} GF(q²) =GF(q)(i) with

i² =s with nonsquare s ∈GF(q) when q odd; i²+i =δ with δ∈GF(q),Tr(δ) =1, when q even. U := Hermitian unital associated to the unitaryU. For p >2,

U=





a11 a12+b12i a13+b13i a12−b12i a22 a23+b23i a₁₃−b₁₃i a₂₃−b₂₃i a₃₃





For p = 2, U=





a₁₁ a₁₂+b₁₂i a₁₃+b₁₃i a₁₂+b₁₂(i+ 1) a₂₂ a₂₃+b₂₃i a13+b13(i+ 1) a23+b23(i+ 1) a33





G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Proof of the classification

Setup

C:= conic of equation X₁² =X₀X₂;

Ω :={P_t = (1,t,t²)|t ∈GF(q²)} ∪ {P∞= (0,0,1)}

GF(q²) =GF(q)(i) with

i² =s with nonsquare s ∈GF(q) when q odd; i²+i =δ with δ∈GF(q),Tr(δ) =1, when q even. U := Hermitian unital associated to the unitaryU. For p >2,

U=





a11 a12+b12i a13+b13i a12−b12i a22 a23+b23i a₁₃−b₁₃i a₂₃−b₂₃i a₃₃





For p = 2, U=





a₁₁ a₁₂+b₁₂i a₁₃+b₁₃i a₁₂+b₁₂(i+ 1) a₂₂ a₂₃+b₂₃i a13+b13(i+ 1) a23+b23(i+ 1) a33





Proof of the classification

Setup

C:= conic of equation X₁² =X₀X₂;

Ω :={P_t = (1,t,t²)|t ∈GF(q²)} ∪ {P∞= (0,0,1)}

GF(q²) =GF(q)(i) with

i² =s with nonsquare s ∈GF(q) when q odd;

i²+i =δ with δ∈GF(q),Tr(δ) =1, when q even.

U := Hermitian unital associated to the unitaryU. For p >2,

U=





a11 a12+b12i a13+b13i a12−b12i a22 a23+b23i a₁₃−b₁₃i a₂₃−b₂₃i a₃₃





For p = 2, U=





a₁₁ a₁₂+b₁₂i a₁₃+b₁₃i a₁₂+b₁₂(i+ 1) a₂₂ a₂₃+b₂₃i a13+b13(i+ 1) a23+b23(i+ 1) a33





G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Proof of the classification

Setup

C:= conic of equation X₁² =X₀X₂;

Ω :={P_t = (1,t,t²)|t ∈GF(q²)} ∪ {P∞= (0,0,1)}

GF(q²) =GF(q)(i) with

i² =s with nonsquare s ∈GF(q) when q odd;

i²+i =δ with δ∈GF(q),Tr(δ) =1, when q even.

U := Hermitian unital associated to the unitaryU.

For p >2, U=





a11 a12+b12i a13+b13i a12−b12i a22 a23+b23i a₁₃−b₁₃i a₂₃−b₂₃i a₃₃





For p = 2, U=





a₁₁ a₁₂+b₁₂i a₁₃+b₁₃i a₁₂+b₁₂(i+ 1) a₂₂ a₂₃+b₂₃i a13+b13(i+ 1) a23+b23(i+ 1) a33





Proof of the classification

Setup

C:= conic of equation X₁² =X₀X₂;

Ω :={P_t = (1,t,t²)|t ∈GF(q²)} ∪ {P∞= (0,0,1)}

GF(q²) =GF(q)(i) with

i² =s with nonsquare s ∈GF(q) when q odd;

i²+i =δ with δ∈GF(q),Tr(δ) =1, when q even.

U := Hermitian unital associated to the unitaryU.

For p >2, U=





a11 a12+b12i a13+b13i a12−b12i a22 a23+b23i a₁₃−b₁₃i a₂₃−b₂₃i a₃₃





For p = 2, U=





a₁₁ a₁₂+b₁₂i a₁₃+b₁₃i a₁₂+b₁₂(i+ 1) a₂₂ a₂₃+b₂₃i a13+b13(i+ 1) a23+b23(i+ 1) a33





G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Proof of the classification

Setup

C:= conic of equation X₁² =X₀X₂;

Ω :={P_t = (1,t,t²)|t ∈GF(q²)} ∪ {P∞= (0,0,1)}

GF(q²) =GF(q)(i) with

i² =s with nonsquare s ∈GF(q) when q odd;

i²+i =δ with δ∈GF(q),Tr(δ) =1, when q even.

U := Hermitian unital associated to the unitaryU.

For p >2, U=





a11 a12+b12i a13+b13i a12−b12i a22 a23+b23i a₁₃−b₁₃i a₂₃−b₂₃i a₃₃





For p = 2,

Proof of Classification Ω ∩ U for p > 2

det((U)

For p >2, det(U) =

a11a22a33−a11a²₂₃+sa11b₂₃² −a22a²₁₃+sa22b₁₃² −a²₁₂a33+ 2a₁₂a₁₃a₂₃−2sa₁₂b₁₃b₂₃+sa₃₃b₁₂² + 2sa₁₃b₁₂b₂₃−2sa₂₃b₁₂b₁₃, For p = 2, det(U) =

a11a22a33+a11a²₂₃+a11a23b23+δa11b²₂₃+a22a₁₃² + a22a13b13+δa22b₁₃² +a²₁₂a33+a12a33b12+a12a13b23+ a₁₂a₂₃b₁₃+δa₃₃b₁₂² +a₁₃a₂₃b₁₂+a₁₃b₁₂b₂₃+δb₁₂b₁₃b₂₃

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Proof of Classification Ω ∩ U for p > 2

det((U)

For p >2, det(U) =

a11a22a33−a11a²₂₃+sa11b₂₃² −a22a²₁₃+sa22b₁₃² −a²₁₂a33+ 2a₁₂a₁₃a₂₃−2sa₁₂b₁₃b₂₃+sa₃₃b₁₂² + 2sa₁₃b₁₂b₂₃−2sa₂₃b₁₂b₁₃,

For p = 2, det(U) =

a11a22a33+a11a²₂₃+a11a23b23+δa11b²₂₃+a22a₁₃² + a22a13b13+δa22b₁₃² +a²₁₂a33+a12a33b12+a12a13b23+ a₁₂a₂₃b₁₃+δa₃₃b₁₂² +a₁₃a₂₃b₁₂+a₁₃b₁₂b₂₃+δb₁₂b₁₃b₂₃

Proof of Classification Ω ∩ U for p > 2

det((U)

For p >2, det(U) =

a11a22a33−a11a²₂₃+sa11b₂₃² −a22a²₁₃+sa22b₁₃² −a²₁₂a33+ 2a₁₂a₁₃a₂₃−2sa₁₂b₁₃b₂₃+sa₃₃b₁₂² + 2sa₁₃b₁₂b₂₃−2sa₂₃b₁₂b₁₃, For p = 2, det(U) =

a11a22a33+a11a²₂₃+a11a23b23+δa11b²₂₃+a22a₁₃² + a₂₂a₁₃b₁₃+δa₂₂b₁₃² +a²₁₂a₃₃+a₁₂a₃₃b₁₂+a₁₂a₁₃b₂₃+ a₁₂a₂₃b₁₃+δa₃₃b₁₂² +a₁₃a₂₃b₁₂+a₁₃b₁₂b₂₃+δb₁₂b₁₃b₂₃

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Proof of Classification Ω ∩ U

Let t=x+yi,t∈GF(q²),x,t∈GF(q) and P_t∈ C,t ∈GF(q²).

Then P_t ∈ U ⇐⇒ (1,t^q,t^2q)U



 1 t t²



= 0.

This condition is equivalent for p>2 to

e(x,y) = a33(x²−sy²)²+ 2(a23x+b23sy)(x²−sy²)+ (a₂₂+ 2a₁₃)x²+ 4b₁₃sxy +s(2a₁₃−a₂₂)y²+ 2a12x+ 2b12sy+a11= 0.

and forp = 2 to

e(x,y) = a₃₃(x²+xy+δy²)²+ (x²+xy +δy²) (b23x+ (a23+b23)y) + (a22+b13)x²+ a₂₂xy + (a₁₃+b₁₃+ (a₂₂+b₁₃)δ)y²+ b₁₂x+ (a₁₂+b₁₂)y+a₁₁= 0

E := curve of equation e(x,y) = 0.

Proof of Classification Ω ∩ U

Let t=x+yi,t∈GF(q²),x,t∈GF(q) and P_t∈ C,t ∈GF(q²).Then P_t ∈ U ⇐⇒

(1,t^q,t^2q)U



 1 t t²



= 0.

This condition is equivalent for p>2 to

e(x,y) = a₃₃(x²−sy²)²+ 2(a₂₃x+b₂₃sy)(x²−sy²)+ (a₂₂+ 2a₁₃)x²+ 4b₁₃sxy +s(2a₁₃−a₂₂)y²+ 2a12x+ 2b12sy+a11= 0.

and forp = 2 to

e(x,y) = a₃₃(x²+xy+δy²)²+ (x²+xy +δy²) (b23x+ (a23+b23)y) + (a22+b13)x²+ a₂₂xy + (a₁₃+b₁₃+ (a₂₂+b₁₃)δ)y²+ b₁₂x+ (a₁₂+b₁₂)y+a₁₁= 0

E := curve of equation e(x,y) = 0.

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Proof of Classification Ω ∩ U

Let t=x+yi,t∈GF(q²),x,t∈GF(q) and P_t∈ C,t ∈GF(q²).Then P_t ∈ U ⇐⇒

(1,t^q,t^2q)U



 1 t t²



= 0.

This condition is equivalent for p>2 to

e(x,y) = a₃₃(x²−sy²)²+ 2(a₂₃x+b₂₃sy)(x²−sy²)+

(a₂₂+ 2a₁₃)x²+ 4b₁₃sxy +s(2a₁₃−a₂₂)y²+ 2a12x+ 2b12sy+a11= 0.

and forp = 2 to

e(x,y) = a₃₃(x²+xy+δy²)²+ (x²+xy +δy²) (b23x+ (a23+b23)y) + (a22+b13)x²+ a₂₂xy + (a₁₃+b₁₃+ (a₂₂+b₁₃)δ)y²+ b₁₂x+ (a₁₂+b₁₂)y+a₁₁= 0

E := curve of equation e(x,y) = 0.

Proof of Classification Ω ∩ U

Let t=x+yi,t∈GF(q²),x,t∈GF(q) and P_t∈ C,t ∈GF(q²).Then P_t ∈ U ⇐⇒

(1,t^q,t^2q)U



 1 t t²



= 0.

This condition is equivalent for p>2 to

e(x,y) = a₃₃(x²−sy²)²+ 2(a₂₃x+b₂₃sy)(x²−sy²)+

(a₂₂+ 2a₁₃)x²+ 4b₁₃sxy +s(2a₁₃−a₂₂)y²+ 2a12x+ 2b12sy+a11= 0.

and forp = 2 to

e(x,y) = a₃₃(x²+xy+δy²)²+ (x²+xy +δy²) (b23x+ (a23+b23)y) + (a22+b13)x²+ a₂₂xy + (a₁₃+b₁₃+ (a₂₂+b₁₃)δ)y²+ b₁₂x+ (a₁₂+b₁₂)y+a₁₁= 0

E := curve of equation e(x,y) = 0.

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Proof of Classification Ω ∩ U

Let t=x+yi,t∈GF(q²),x,t∈GF(q) and P_t∈ C,t ∈GF(q²).Then P_t ∈ U ⇐⇒

(1,t^q,t^2q)U



 1 t t²



= 0.

This condition is equivalent for p>2 to

e(x,y) = a₃₃(x²−sy²)²+ 2(a₂₃x+b₂₃sy)(x²−sy²)+

(a₂₂+ 2a₁₃)x²+ 4b₁₃sxy +s(2a₁₃−a₂₂)y²+ 2a12x+ 2b12sy+a11= 0.

and forp = 2 to

e(x,y) = a₃₃(x²+xy+δy²)²+ (x²+xy +δy²)

Proof of Classification Ω ∩ U

Number of points of the curveE

Let M be the number of points Q = (x,y)in AG(2,q) which lie on E.

Then M =

|Ω∩ U | P∞6∈ U;

|Ω∩ U | −1 P∞∈ U.

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Proof of Classification Ω ∩ U

Number of points of the curveE

Let M be the number of points Q = (x,y)in AG(2,q) which lie on E. Then

M =

|Ω∩ U | P∞6∈ U;

|Ω∩ U | −1 P∞∈ U.

Proof Classification Ω ∩ U

CaseM= 0

Proposition

M = 0 ⇐⇒ E splits into two distinct absolutely irreducible conics defined over a quadratic extension GF(q²) of GF(q) having no point in PG(2,q) and conjugate to each other over GF(q).

For p >2, this is the case⇐⇒

a₁₁=γ₁²−sγ₂², a₁₂=α₁γ₁−sα₂γ₂, a22= 2γ1+¹₂(α²₁−sα²₂−¹_s(β₁²−sβ₂²)), a23=α1,a33= 1, a13= ¹₄(α²₁−sα²₂+¹_s(β²₁−sβ₂²)), b12= ¹_s(β1γ1−sβ2γ2), b₁₃= _2s¹(α₁β₁−sα₂β₂), b₂₃= ¹_sβ₁.

Example: α1 =α2=β1 =β2 = 0, γ1=γ2 = 1,

U=





1−s 0 0

0 2 0

0 0 1



.

For p = 2, similar results hold.

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Proof Classification Ω ∩ U

CaseM= 0

Proposition

M = 0 ⇐⇒ E splits into two distinct absolutely irreducible conics defined over a quadratic extension GF(q²) of GF(q) having no point in PG(2,q) and conjugate to each other over GF(q).

For p >2, this is the case⇐⇒

a₁₁=γ₁²−sγ₂², a₁₂=α₁γ₁−sα₂γ₂, a22= 2γ1+¹₂(α²₁−sα²₂−¹_s(β₁²−sβ₂²)), a23=α1,a33= 1, a13= ¹₄(α²₁−sα²₂+¹_s(β²₁−sβ₂²)), b12= ¹_s(β1γ1−sβ2γ2), b₁₃= _2s¹(α₁β₁−sα₂β₂), b₂₃= ¹_sβ₁.

Example: α1 =α2=β1 =β2 = 0, γ1=γ2 = 1,

U=





1−s 0 0

0 2 0

0 0 1



.

For p = 2, similar results hold.

Proof Classification Ω ∩ U

CaseM= 0

Proposition

M = 0 ⇐⇒ E splits into two distinct absolutely irreducible conics defined over a quadratic extension GF(q²) of GF(q) having no point in PG(2,q) and conjugate to each other over GF(q).

For p >2, this is the case⇐⇒

a₁₁=γ₁²−sγ₂², a₁₂=α₁γ₁−sα₂γ₂, a22= 2γ1+¹₂(α²₁−sα²₂−¹_s(β₁²−sβ₂²)), a23=α1,a33= 1, a13= ¹₄(α²₁−sα²₂+¹_s(β²₁−sβ₂²)), b12= ¹_s(β1γ1−sβ2γ2), b₁₃= _2s¹(α₁β₁−sα₂β₂), b₂₃= ¹_sβ₁.

Example: α1 =α2=β1 =β2 = 0, γ1=γ2 = 1,

U=





1−s 0 0

0 2 0

0 0 1



.

For p = 2, similar results hold.

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Proof Classification Ω ∩ U

CaseM= 0

Proposition

M = 0 ⇐⇒ E splits into two distinct absolutely irreducible conics defined over a quadratic extension GF(q²) of GF(q) having no point in PG(2,q) and conjugate to each other over GF(q).

For p >2, this is the case⇐⇒

a₁₁=γ₁²−sγ₂², a₁₂=α₁γ₁−sα₂γ₂, a22= 2γ1+¹₂(α²₁−sα²₂−¹_s(β₁²−sβ₂²)), a23=α1,a33= 1, a13= ¹₄(α²₁−sα²₂+¹_s(β²₁−sβ₂²)), b12= ¹_s(β1γ1−sβ2γ2), b₁₃= _2s¹(α₁β₁−sα₂β₂), b₂₃= ¹_sβ₁.

Example: α1 =α2=β1 =β2 = 0, γ1=γ2 = 1,

Proof Classification Ω ∩ U

CaseM>0

P∞∈Ω∩ U is assumed =⇒ a33= 0 =⇒ degE ≤3. The goal is to computeM. For this purpose, the following cases are to be considered separately.

E is a line (this case cannot actually occur),

E is a conic, either absolutely irreducible or reducible; E is a cubic, either absolutely irreducible or reducible;

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Proof Classification Ω ∩ U

CaseM>0

P∞∈Ω∩ U is assumed =⇒ a33= 0 =⇒ degE ≤3.

The goal is to computeM. For this purpose, the following cases are to be considered separately.

E is a line (this case cannot actually occur),

E is a conic, either absolutely irreducible or reducible; E is a cubic, either absolutely irreducible or reducible;

Proof Classification Ω ∩ U

CaseM>0

P∞∈Ω∩ U is assumed =⇒ a33= 0 =⇒ degE ≤3.

The goal is to computeM. For this purpose, the following cases are to be considered separately.

E is a line (this case cannot actually occur),

E is a conic, either absolutely irreducible or reducible; E is a cubic, either absolutely irreducible or reducible;

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Proof Classification Ω ∩ U

CaseM>0

P∞∈Ω∩ U is assumed =⇒ a33= 0 =⇒ degE ≤3.

The goal is to computeM. For this purpose, the following cases are to be considered separately.

E is a line (this case cannot actually occur),

E is a conic, either absolutely irreducible or reducible; E is a cubic, either absolutely irreducible or reducible;

Proof Classification Ω ∩ U

CaseM>0

P∞∈Ω∩ U is assumed =⇒ a33= 0 =⇒ degE ≤3.

The goal is to computeM. For this purpose, the following cases are to be considered separately.

E is a line (this case cannot actually occur),

E is a conic, either absolutely irreducible or reducible;

E is a cubic, either absolutely irreducible or reducible;

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Proof Classification Ω ∩ U

CaseM>0

P∞∈Ω∩ U is assumed =⇒ a33= 0 =⇒ degE ≤3.

The goal is to computeM. For this purpose, the following cases are to be considered separately.

E is a line (this case cannot actually occur),

E is a conic, either absolutely irreducible or reducible;

E is a cubic, either absolutely irreducible or reducible;

Proof Classification Ω ∩ U

E is an absolutely reducible conic

Proposition

IfE is an absolutely reducible conic, then|U ∩Ω|can only assume a few values, namely1,2,q+ 1,2q,2q+ 1for p>2, and2,2q for p= 2. More precisely, one of the following cases occurs:

(i) E =`0∪`1 with `0, `1, defined over GF(q),|`<sub>0</sub>∩`1|= 1; Ω∩ U = Ω0∪Ω1,|Ω₀∩Ω1|= 2;

(ii) E =`0∪`1 with `0, `1 defined over GF(q),`0 k`1; Ω∩ U = Ω₀∪Ω₁,|Ω₀∩Ω₁|= 1 p>2;

(iii) E =`0∪`1 with `0, `1 defined over GF(q),`0 =`1; Ω∩ U is a classical Baer suboval ofΩ, p>2;

(iv) E =`<sub>0</sub>∪`₁ with `<sub>0</sub>, `₁ defined over GF(q²),|`<sub>0</sub>∩`₁|= 1; Ω∩ U ={P_∞,A};

(v) E =`<sub>0</sub>∪`₁ with `<sub>0</sub>, `₁ defined over GF(q²),`<sub>0</sub>k`₁; Ω∩ U ={P∞}; p >2.

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Proof Classification Ω ∩ U

E is an absolutely reducible conic

Proposition

IfE is an absolutely reducible conic, then|U ∩Ω|can only assume a few values, namely1,2,q+ 1,2q,2q+ 1for p>2, and 2,2q for p= 2. More precisely, one of the following cases occurs:

(i) E =`0∪`1 with `0, `1, defined over GF(q),|`<sub>0</sub>∩`1|= 1; Ω∩ U = Ω0∪Ω1,|Ω₀∩Ω1|= 2;

(ii) E =`0∪`1 with `0, `1 defined over GF(q),`0 k`1; Ω∩ U = Ω₀∪Ω₁,|Ω₀∩Ω₁|= 1 p>2;

(iii) E =`0∪`1 with `0, `1 defined over GF(q),`0 =`1; Ω∩ U is a classical Baer suboval ofΩ, p>2;

(iv) E =`<sub>0</sub>∪`₁ with `<sub>0</sub>, `₁ defined over GF(q²),|`<sub>0</sub>∩`₁|= 1; Ω∩ U ={P_∞,A};

(v) E =`<sub>0</sub>∪`₁ with `<sub>0</sub>, `₁ defined over GF(q²),`<sub>0</sub>k`₁; Ω∩ U ={P∞}; p >2.

Proof Classification Ω ∩ U

E is an absolutely reducible conic

Proposition

IfE is an absolutely reducible conic, then|U ∩Ω|can only assume a few values, namely1,2,q+ 1,2q,2q+ 1for p>2, and 2,2q for p= 2. More precisely, one of the following cases occurs:

(i) E =`0∪`1 with `0, `1, defined over GF(q),|`<sub>0</sub>∩`1|= 1;

Ω∩ U = Ω₀∪Ω₁,|Ω₀∩Ω₁|= 2;

(ii) E =`0∪`1 with `0, `1 defined over GF(q),`0 k`1; Ω∩ U = Ω₀∪Ω₁,|Ω₀∩Ω₁|= 1 p>2;

(iii) E =`0∪`1 with `0, `1 defined over GF(q),`0 =`1; Ω∩ U is a classical Baer suboval ofΩ, p>2;

(iv) E =`<sub>0</sub>∪`₁ with `<sub>0</sub>, `₁ defined over GF(q²),|`<sub>0</sub>∩`₁|= 1; Ω∩ U ={P_∞,A};

(v) E =`<sub>0</sub>∪`₁ with `<sub>0</sub>, `₁ defined over GF(q²),`<sub>0</sub>k`₁; Ω∩ U ={P∞}; p >2.

G´abor Korchm´aros Intersection of Ovals and Unitals in Desarguesian Planes

Proof Classification Ω ∩ U

E is an absolutely reducible conic

Proposition

IfE is an absolutely reducible conic, then|U ∩Ω|can only assume a few values, namely1,2,q+ 1,2q,2q+ 1for p>2, and 2,2q for p= 2. More precisely, one of the following cases occurs:

(i) E =`0∪`1 with `0, `1, defined over GF(q),|`<sub>0</sub>∩`1|= 1;

Ω∩ U = Ω₀∪Ω₁,|Ω₀∩Ω₁|= 2;

(ii) E =`0∪`1 with `0, `1 defined over GF(q),`0 k`1; Ω∩ U = Ω₀∪Ω₁,|Ω₀∩Ω₁|= 1 p>2;

(iii) E =`0∪`1 with `0, `1 defined over GF(q),`0 =`1; Ω∩ U is a classical Baer suboval ofΩ, p>2;

(iv) E =`<sub>0</sub>∪`₁ with `<sub>0</sub>, `₁ defined over GF(q²),|`<sub>0</sub>∩`₁|= 1; Ω∩ U ={P_∞,A};

(v) E =`<sub>0</sub>∪`₁ with `<sub>0</sub>, `₁ defined over GF(q²),`<sub>0</sub>k`₁; Ω∩ U ={P∞}; p >2.