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